Abstract group 108864.a: ${}^2G(2,3)\times \SOPlus(4,2)$

• Label: 108864.a
• Order: \( 2^{6} \cdot 3^{5} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{5} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $15$
• Trans deg.: $54$
• Rank: $2$

Group information

Description:	${}^2G(2,3)\times \SOPlus(4,2)$
Order:	\(108864\)\(\medspace = 2^{6} \cdot 3^{5} \cdot 7 \)
Exponent:	\(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \)
Automorphism group:	$\SL(2,8).C_3\times \PSU(3,2):C_2$, of order \(217728\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_3$ x 3, $\SL(2,8)$
Derived length:	$3$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	9	12	14	18	21	28	36	42
Elements	1	1407	2024	1152	39336	216	4536	13104	4536	22680	1728	3888	9072	5184	108864
Conjugacy classes	1	7	11	2	37	1	9	5	3	15	2	1	3	2	99
Divisions	1	7	8	2	24	1	6	3	3	10	2	1	2	2	72
Autjugacy classes	1	5	7	2	22	1	6	5	2	9	1	1	3	1	66

Dimension	1	2	4	7	8	14	16	21	27	28	32	42	54	56	64	84	108
Irr. complex chars.	12	3	12	12	12	3	3	4	4	12	12	1	1	0	0	4	4	99
Irr. rational chars.	4	5	5	4	8	5	5	4	4	5	5	1	1	4	4	4	4	72

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$54$
Rank:	$2$
Inequivalent generating pairs:	$13632$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	28	28	28
Arbitrary	11	11	11

Constructions

Permutation group:	Degree $15$ $\langle(1,2,6)(3,5,4), (2,6)(4,5), (3,4,5), (1,4,2,3,6,5)(8,14,12)(9,11,13), (1,4)(2,3,6,5), (4,5), (2,6)(3,4,5)(7,14,12)(8,10,13)(9,15,11)\rangle$
Direct product:	$\SOPlus(4,2)$ $\, \times\, $ ${}^2G(2,3)$
Semidirect product:	$(S_3^2\times \SL(2,8))$ $\,\rtimes\,$ $C_6$ (2)	$\SL(2,8)$ $\,\rtimes\,$ $(S_3^2:C_6)$	$S_3^2$ $\,\rtimes\,$ $(\SL(2,8):C_6)$ (2)	$(S_3^2\times {}^2G(2,3))$ $\,\rtimes\,$ $C_2$ (2)	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3:S_3)$ . $(C_2^2\times {}^2G(2,3))$	$(C_3:S_3\times \SL(2,8))$ . $(C_2\times C_6)$	$((C_3^2\times \SL(2,8)):C_6)$ . $C_2^2$		more information
Aut. group:	$\Aut(S_3^2\times \SL(2,8))$	$\Aut(S_3^2\times {}^2G(2,3))$

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 461196 subgroups in 1270 conjugacy classes, 21 normal (15 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ ${}^2G(2,3)\times \SOPlus(4,2)$
Commutator:	$G' \simeq$ $C_3:S_3\times \SL(2,8)$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ ${}^2G(2,3)\times \SOPlus(4,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2$	$G/\operatorname{Fit} \simeq$ $D_4\times {}^2G(2,3)$
Radical:	$R \simeq$ $\SOPlus(4,2)$	$G/R \simeq$ ${}^2G(2,3)$
Socle:	$\operatorname{soc} \simeq$ $C_3^2\times \SL(2,8)$	$G/\operatorname{soc} \simeq$ $C_3\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9:C_3^3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 108864: ${}^2G(2,3)\times \SOPlus(4,2)$

Order 54432: $S_3^2\times {}^2G(2,3)$ x 2, $(C_3^2\times \SL(2,8)):C_{12}$

Order 36288: $\SL(2,8)\times \SOPlus(4,2)$

Order 27216: $C_3\times S_3\times {}^2G(2,3)$ x 2, $(C_3^2\times \SL(2,8)):C_6$

Order 18144: $S_3^2\times \SL(2,8)$ x 2, $D_6\times {}^2G(2,3)$ x 2, $C_3^2:C_4\times \SL(2,8)$

Order 13608: $C_3^2\times {}^2G(2,3)$

Order 12096: $D_4\times {}^2G(2,3)$, $(S_3^2\times F_8):C_6$

Order 9072: $S_3\times {}^2G(2,3)$ x 6, $C_3\times S_3\times \SL(2,8)$ x 2, $C_6\times {}^2G(2,3)$ x 2, $C_3:S_3\times \SL(2,8)$

Order 6048: $C_2^2\times {}^2G(2,3)$ x 2, $F_8:C_3\times S_3^2$ x 2, $D_6\times \SL(2,8)$ x 2, $\SL(2,8):C_{12}$, $(C_3^2\times F_8):C_{12}$

Order 4536: $C_3\times {}^2G(2,3)$ x 4, $C_3^2\times \SL(2,8)$

Order 4032: $F_8\times \SOPlus(4,2)$, $D_4\times \SL(2,8)$

Order 3888: $(D_9\times S_3^2):C_6$

Order 3024: $\SL(2,8):C_6$ x 5, $S_3\times \SL(2,8)$ x 4, $S_3\times F_8:C_3^2$ x 2, $C_6\times \SL(2,8)$ x 2, $F_7\times \SOPlus(4,2)$, $(C_3^2\times F_8):C_6$

Order 2016: $S_3^2\times F_8$ x 2, $F_8:C_6\times S_3$ x 2, $C_2^2\times \SL(2,8)$ x 2, $C_3^2:C_4\times F_8$, $C_4\times \SL(2,8)$

Order 1944: $C_3^2.S_3^3$ x 2, $(C_9\times S_3^2):C_6$ x 2, $C_3^4.D_{12}$, $(C_9\times S_3^2):C_6$, $(C_3^2\times D_9):C_{12}$

Order 1728: $C_2\times D_6^2:C_6$

Order 1512: ${}^2G(2,3)$ x 3, $S_3^2\times F_7$ x 2, $S_3^2:F_7$ x 2, $C_3\times \SL(2,8)$ x 2, $F_8:C_3^3$, $C_3^2:C_{28}:C_6$, $(C_7\times S_3^2):C_6$, $C_3^2:C_4\times F_7$

Order 1344: $D_4\times F_8:C_3$

Order 1296: $S_3^3:C_6$, $S_3^2:D_{18}$

Order 1008: $F_8:C_3\times S_3$ x 6, $C_2\times \SL(2,8)$ x 3, $C_3\times S_3\times F_8$ x 2, $C_3\times F_8:C_6$ x 2, $C_3:S_3\times F_8$, $S_3^2:D_{14}$

Order 972: $C_3^3.S_3^2$ x 2, $C_3^3.S_3^2$ x 2, $C_3^3.C_6^2$ x 2, $C_3^3.S_3^2$ x 2, $C_3^3.S_3^2$, $(C_3^2\times C_9):C_{12}$, $C_3^4.C_{12}$

Order 864: $D_6^2:C_6$ x 4, $D_6^2:C_6$ x 2, $C_2\times C_6^2:C_{12}$

Order 756: $C_{21}:(C_6\times S_3)$ x 2, $C_{21}:C_6\times S_3$ x 2, $C_{21}:C_6\times S_3$ x 2, $C_{21}:C_6^2$ x 2, $C_3:S_3\times F_7$, $(C_3\times C_{21}):C_{12}$, $(C_3\times C_{21}):C_{12}$

Order 672: $D_6\times F_8$ x 2, $C_2\times F_8:C_6$ x 2, $F_8:C_{12}$

Order 648: $C_3\times S_3^3$ x 2, $C_3\wr D_4$ x 2, $C_{18}:C_6\times S_3$ x 2, $D_9\times S_3^2$ x 2, $S_3^2:D_9$ x 2, $S_3^2:C_{18}$ x 2, $C_3^3:D_{12}$, $C_3^4:D_4$, $S_3\times C_3^2:C_{12}$, $C_3^2:D_{36}$, $C_3^2:C_4\times D_9$

Order 576: $D_6^2:C_2^2$

Order 504: $F_8:C_3^2$ x 4, $D_7\times S_3^2$ x 2, $S_3^2:D_7$ x 2, $D_6\times F_7$ x 2, $C_3^2\times F_8$, $C_3^2:D_{28}$, $S_3^2:C_{14}$, $C_3^2:C_4\times D_7$, $\SL(2,8)$

Order 486: $C_3^4.S_3$ x 2, $C_3^4.C_6$ x 2, $C_3^4.S_3$, $C_3^4.C_6$, $D_9:C_3^3$

Order 448: $D_4\times F_8$

Order 432: $A_4\times S_3^2$ x 6, $C_6\times S_3\times A_4$ x 2, $C_6^2:C_{12}$ x 2, $C_6:S_3\times A_4$, $C_6\times \SOPlus(4,2)$, $S_3^3:C_2$, $D_{36}:C_6$

Order 378: $C_3^2:F_7$ x 2, $C_3\times C_{21}:C_6$ x 2, $C_3^2:F_7$, $(C_3\times C_{21}):C_6$, $C_3^2\times F_7$

Order 336: $F_8:C_6$ x 5, $S_3\times F_8$ x 4, $C_6\times F_8$ x 2, $D_4\times F_7$

Order 324: $C_3^2.S_3^2$ x 12, $C_3^2\times S_3^2$ x 4, $C_9\times S_3^2$ x 4, $C_3^2:S_3^2$ x 3, $C_3\wr C_2^2$ x 2, $C_3^3.D_6$ x 2, $C_3^2.C_6^2$ x 2, $C_9:C_6^2$ x 2, $C_9:S_3^2$ x 2, $C_9:S_3^2$ x 2, $C_3^2:D_{18}$ x 2, $C_3^2:C_{36}$ x 2, $C_3\wr C_4$, $C_3^3:C_{12}$, $C_3^2:D_{18}$, $(C_3^2\times C_9):C_4$

Order 288: $C_6^2:D_4$ x 4, $C_2\times D_6^2$ x 2, $C_2\times A_4\times D_6$ x 2, $C_2\times C_6^2:C_4$

Order 252: $S_3\times F_7$ x 10, $C_{21}:D_6$ x 2, $S_3\times D_{21}$ x 2, $C_7\times S_3^2$ x 2, $C_{21}:D_6$ x 2, $C_{42}:C_6$ x 2, $C_{42}:C_6$ x 2, $C_6\times F_7$ x 2, $C_{21}:D_6$, $(C_3\times C_{21}):C_4$, $C_3^2:C_{28}$

Order 243: $C_9:C_3^3$

Order 224: $C_2^2\times F_8$ x 2, $C_4\times F_8$

Order 216: $S_3^2:C_6$ x 5, $C_6\times S_3^2$ x 4, $C_6^2:C_6$ x 4, $D_{18}:C_6$ x 2, $C_6^2:C_6$ x 2, $S_3^3$ x 2, $S_3^2:S_3$ x 2, $D_{18}:C_6$ x 2, $S_3\times D_{18}$ x 2, $C_{36}:C_6$, $C_{36}:C_6$, $C_{36}:C_6$, $C_6^2:C_6$, $C_2\times C_3^2:C_{12}$, $C_3^2:D_{12}$, $C_3^2:C_4\times S_3$

Order 192: $C_2^5:C_6$

Order 189: $C_7:C_3^3$

Order 168: $F_8:C_3$ x 3, $S_3\times D_{14}$ x 2, $C_2^2\times F_7$ x 2, $C_3\times F_8$ x 2, $D_{14}:C_6$ x 2, $C_{28}:C_6$, $C_4\times F_7$, $C_{28}:C_6$

Order 162: $C_3^3.C_6$ x 10, $C_3^3.S_3$ x 8, $D_9:C_3^2$ x 8, $C_3^2:C_{18}$ x 4, $C_3^2\wr C_2$ x 3, $S_3\times C_3^3$ x 3, $C_{18}:C_3^2$ x 2, $C_3^2:C_{18}$ x 2, $C_3^2:D_9$ x 2, $C_3^3:C_6$, $C_3^2:D_9$, $C_3^2\times D_9$

Order 144: $A_4\times D_6$ x 14, $D_6^2$ x 6, $S_3^2:C_2^2$ x 6, $C_6^2:C_2^2$ x 2, $C_2^2:C_6^2$ x 2, $C_6^2:C_4$ x 2, $D_4\times D_9$, $C_6^2:C_2^2$, $C_{12}:D_6$

Order 126: $C_{21}:C_6$ x 6, $C_{21}:C_6$ x 6, $C_3\times F_7$ x 6, $C_3\times D_{21}$ x 2, $S_3\times C_{21}$ x 2, $C_{21}:C_6$ x 2, $C_3:D_{21}$, $C_{21}:S_3$, $C_3^2\times D_7$

Order 112: $C_2\times F_8$ x 3, $D_4\times D_7$

Order 108: $C_3\times S_3^2$ x 22, $C_{18}:C_6$ x 11, $S_3\times D_9$ x 10, $C_3^2\times D_6$ x 6, $S_3\times C_{18}$ x 4, $C_3^2:D_6$ x 3, $C_3:S_3^2$ x 3, $C_3^2:C_{12}$ x 3, $C_3:S_3^2$ x 2, $C_{18}:C_6$ x 2, $C_3:D_{18}$ x 2, $C_3\times D_{18}$ x 2, $C_9:C_{12}$, $C_3^2\times A_4$, $C_3^3:C_4$, $C_9:C_{12}$

Order 96: $D_4\times A_4$ x 4, $C_2^3\times D_6$ x 2, $C_2^3\times A_4$ x 2, $C_2^3:C_{12}$

Order 84: $C_2\times F_7$ x 9, $S_3\times D_7$ x 8, $C_{14}:C_6$ x 2, $D_{42}$ x 2, $S_3\times C_{14}$ x 2, $C_3\times D_{14}$ x 2, $C_7:C_{12}$, $C_7:C_{12}$

Order 81: $C_9:C_3^2$ x 8, $C_3^2\times C_9$ x 2, $C_3^4$

Order 72: $S_3\times A_4$ x 12, $S_3\times D_6$ x 10, $C_6\times D_6$ x 8, $C_6\times A_4$ x 8, $\SOPlus(4,2)$ x 6, $C_9:D_4$ x 2, $C_6:D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $C_6\wr C_2$ x 2, $C_2\times D_{18}$ x 2, $D_4\times C_9$ x 2, $D_{36}$, $C_2\times C_6^2$, $C_4\times D_9$, $D_4\times C_3^2$, $C_3\times D_{12}$, $S_3\times C_{12}$

Order 64: $D_4\times C_2^3$

Order 63: $C_{21}:C_3$ x 4, $C_3\times C_{21}$

Order 56: $C_7:D_4$ x 2, $C_2\times D_{14}$ x 2, $C_7\times D_4$, $D_{28}$, $C_4\times D_7$, $F_8$

Order 54: $S_3\times C_3^2$ x 23, $C_9:C_6$ x 14, $C_3^2:C_6$ x 13, $S_3\times C_9$ x 12, $C_9:C_6$ x 9, $C_3:D_9$ x 6, $C_3\times D_9$ x 6, $C_3\times C_{18}$ x 4, $C_3^2\times C_6$ x 3, $C_3^2:S_3$

Order 48: $C_2^2\times D_6$ x 12, $C_2^2\times A_4$ x 11, $C_2^3\times C_6$ x 2, $C_4\times A_4$ x 2, $C_6\times D_4$, $S_3\times D_4$

Order 42: $F_7$ x 8, $C_7:C_6$ x 5, $D_{21}$ x 4, $C_3\times D_7$ x 4, $S_3\times C_7$ x 4, $C_{42}$ x 2

Order 36: $C_6\times S_3$ x 33, $S_3^2$ x 18, $D_{18}$ x 9, $C_6^2$ x 5, $C_2\times C_{18}$ x 4, $C_6:S_3$ x 4, $C_3\times A_4$ x 4, $C_3^2:C_4$ x 2, $C_{36}$ x 2, $C_3\times C_{12}$, $C_3:C_{12}$, $C_9:C_4$

Order 32: $C_2^2\times D_4$ x 4, $C_2^5$ x 2, $C_2^3\times C_4$

Order 28: $D_{14}$ x 7, $C_2\times C_{14}$ x 2, $C_{28}$, $C_7:C_4$

Order 27: $C_9:C_3$ x 11, $C_3^3$ x 8, $C_3\times C_9$ x 8

Order 24: $C_2\times D_6$ x 22, $C_2\times A_4$ x 13, $C_2^2\times C_6$ x 8, $C_3\times D_4$ x 5, $C_3:D_4$ x 2, $C_2\times C_{12}$, $D_{12}$, $C_4\times S_3$

Order 21: $C_7:C_3$ x 3, $C_{21}$ x 2

Order 18: $C_3\times S_3$ x 42, $C_3\times C_6$ x 18, $C_{18}$ x 10, $C_3:S_3$ x 8, $D_9$ x 8

Order 16: $C_2^4$ x 9, $C_2\times D_4$ x 6, $C_2^2\times C_4$ x 2

Order 14: $D_7$ x 4, $C_{14}$ x 3

Order 12: $D_6$ x 29, $C_2\times C_6$ x 19, $A_4$ x 3, $C_{12}$ x 3, $C_3:C_4$

Order 9: $C_3^2$ x 16, $C_9$ x 6

Order 8: $C_2^3$ x 15, $D_4$ x 6, $C_2\times C_4$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 24, $S_3$ x 16

Order 4: $C_2^2$ x 15, $C_4$ x 2

Order 3: $C_3$ x 8

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 108864: ${}^2G(2,3)\times \SOPlus(4,2)$

Order 54432: $(C_3^2\times \SL(2,8)):C_{12}$, $S_3^2\times {}^2G(2,3)$

Order 36288: $\SL(2,8)\times \SOPlus(4,2)$

Order 27216: $(C_3^2\times \SL(2,8)):C_6$, $C_3\times S_3\times {}^2G(2,3)$

Order 18144: $C_3^2:C_4\times \SL(2,8)$, $S_3^2\times \SL(2,8)$, $D_6\times {}^2G(2,3)$

Order 13608: $C_3^2\times {}^2G(2,3)$

Order 12096: $D_4\times {}^2G(2,3)$, $(S_3^2\times F_8):C_6$

Order 9072: $S_3\times {}^2G(2,3)$ x 3, $C_3\times S_3\times \SL(2,8)$, $C_3:S_3\times \SL(2,8)$, $C_6\times {}^2G(2,3)$

Order 6048: $C_2^2\times {}^2G(2,3)$, $\SL(2,8):C_{12}$, $(C_3^2\times F_8):C_{12}$, $F_8:C_3\times S_3^2$, $D_6\times \SL(2,8)$

Order 4536: $C_3\times {}^2G(2,3)$ x 2, $C_3^2\times \SL(2,8)$

Order 4032: $F_8\times \SOPlus(4,2)$, $D_4\times \SL(2,8)$

Order 3888: $(D_9\times S_3^2):C_6$

Order 3024: $\SL(2,8):C_6$ x 3, $S_3\times \SL(2,8)$ x 2, $F_7\times \SOPlus(4,2)$, $S_3\times F_8:C_3^2$, $(C_3^2\times F_8):C_6$, $C_6\times \SL(2,8)$

Order 2016: $S_3^2\times F_8$, $C_3^2:C_4\times F_8$, $F_8:C_6\times S_3$, $C_2^2\times \SL(2,8)$, $C_4\times \SL(2,8)$

Order 1944: $C_3^2.S_3^3$, $C_3^4.D_{12}$, $(C_9\times S_3^2):C_6$, $(C_9\times S_3^2):C_6$, $(C_3^2\times D_9):C_{12}$

Order 1728: $C_2\times D_6^2:C_6$

Order 1512: ${}^2G(2,3)$ x 2, $F_8:C_3^3$, $S_3^2\times F_7$, $C_3^2:C_{28}:C_6$, $S_3^2:F_7$, $(C_7\times S_3^2):C_6$, $C_3^2:C_4\times F_7$, $C_3\times \SL(2,8)$

Order 1344: $D_4\times F_8:C_3$

Order 1296: $S_3^3:C_6$, $S_3^2:D_{18}$

Order 1008: $F_8:C_3\times S_3$ x 3, $C_2\times \SL(2,8)$ x 2, $C_3:S_3\times F_8$, $C_3\times S_3\times F_8$, $S_3^2:D_{14}$, $C_3\times F_8:C_6$

Order 972: $C_3^3.S_3^2$, $C_3^3.S_3^2$, $C_3^3.S_3^2$, $C_3^3.C_6^2$, $C_3^3.S_3^2$, $(C_3^2\times C_9):C_{12}$, $C_3^4.C_{12}$

Order 864: $D_6^2:C_6$ x 3, $D_6^2:C_6$, $C_2\times C_6^2:C_{12}$

Order 756: $C_{21}:(C_6\times S_3)$, $C_{21}:C_6\times S_3$, $C_{21}:C_6\times S_3$, $C_3:S_3\times F_7$, $C_{21}:C_6^2$, $(C_3\times C_{21}):C_{12}$, $(C_3\times C_{21}):C_{12}$

Order 672: $D_6\times F_8$, $C_2\times F_8:C_6$, $F_8:C_{12}$

Order 648: $S_3^2:C_{18}$ x 2, $C_3\times S_3^3$, $C_3^3:D_{12}$, $C_3\wr D_4$, $C_3^4:D_4$, $S_3\times C_3^2:C_{12}$, $C_{18}:C_6\times S_3$, $D_9\times S_3^2$, $C_3^2:D_{36}$, $S_3^2:D_9$, $C_3^2:C_4\times D_9$

Order 576: $D_6^2:C_2^2$

Order 504: $F_8:C_3^2$ x 2, $C_3^2\times F_8$, $D_7\times S_3^2$, $C_3^2:D_{28}$, $S_3^2:D_7$, $S_3^2:C_{14}$, $C_3^2:C_4\times D_7$, $D_6\times F_7$, $\SL(2,8)$

Order 486: $C_3^4.S_3$, $C_3^4.C_6$, $C_3^4.S_3$, $C_3^4.C_6$, $D_9:C_3^3$

Order 448: $D_4\times F_8$

Order 432: $A_4\times S_3^2$ x 3, $C_6^2:C_{12}$ x 2, $C_6:S_3\times A_4$, $C_6\times S_3\times A_4$, $C_6\times \SOPlus(4,2)$, $S_3^3:C_2$, $D_{36}:C_6$

Order 378: $C_3^2:F_7$, $(C_3\times C_{21}):C_6$, $C_3^2:F_7$, $C_3\times C_{21}:C_6$, $C_3^2\times F_7$

Order 336: $F_8:C_6$ x 3, $S_3\times F_8$ x 2, $C_6\times F_8$, $D_4\times F_7$

Order 324: $C_3^2.S_3^2$ x 6, $C_3^2:S_3^2$ x 2, $C_3^2\times S_3^2$ x 2, $C_9\times S_3^2$ x 2, $C_3^2:C_{36}$ x 2, $C_3\wr C_2^2$, $C_3\wr C_4$, $C_3^3:C_{12}$, $C_3^3.D_6$, $C_3^2.C_6^2$, $C_9:C_6^2$, $C_9:S_3^2$, $C_9:S_3^2$, $C_3^2:D_{18}$, $C_3^2:D_{18}$, $(C_3^2\times C_9):C_4$

Order 288: $C_6^2:D_4$ x 3, $C_2\times D_6^2$, $C_2\times C_6^2:C_4$, $C_2\times A_4\times D_6$

Order 252: $S_3\times F_7$ x 5, $C_{21}:D_6$, $S_3\times D_{21}$, $C_7\times S_3^2$, $C_{21}:D_6$, $C_{21}:D_6$, $(C_3\times C_{21}):C_4$, $C_3^2:C_{28}$, $C_{42}:C_6$, $C_{42}:C_6$, $C_6\times F_7$

Order 243: $C_9:C_3^3$

Order 224: $C_2^2\times F_8$, $C_4\times F_8$

Order 216: $S_3^2:C_6$ x 4, $C_6\times S_3^2$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$ x 2, $C_{36}:C_6$, $D_{18}:C_6$, $C_{36}:C_6$, $C_{36}:C_6$, $C_6^2:C_6$, $C_2\times C_3^2:C_{12}$, $S_3^3$, $C_3^2:D_{12}$, $S_3^2:S_3$, $C_3^2:C_4\times S_3$, $D_{18}:C_6$, $S_3\times D_{18}$

Order 192: $C_2^5:C_6$

Order 189: $C_7:C_3^3$

Order 168: $F_8:C_3$ x 2, $C_{28}:C_6$, $C_4\times F_7$, $S_3\times D_{14}$, $C_2^2\times F_7$, $C_3\times F_8$, $C_{28}:C_6$, $D_{14}:C_6$

Order 162: $C_3^3.C_6$ x 5, $C_3^3.S_3$ x 4, $D_9:C_3^2$ x 4, $C_3^2\wr C_2$ x 2, $S_3\times C_3^3$ x 2, $C_3^2:C_{18}$ x 2, $C_3^2:C_{18}$ x 2, $C_3^3:C_6$, $C_{18}:C_3^2$, $C_3^2:D_9$, $C_3^2:D_9$, $C_3^2\times D_9$

Order 144: $A_4\times D_6$ x 7, $S_3^2:C_2^2$ x 4, $D_6^2$ x 3, $C_6^2:C_4$ x 2, $D_4\times D_9$, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_2^2:C_6^2$, $C_{12}:D_6$

Order 126: $C_{21}:C_6$ x 3, $C_{21}:C_6$ x 3, $C_3\times F_7$ x 3, $C_3:D_{21}$, $C_{21}:S_3$, $C_3\times D_{21}$, $S_3\times C_{21}$, $C_3^2\times D_7$, $C_{21}:C_6$

Order 112: $C_2\times F_8$ x 2, $D_4\times D_7$

Order 108: $C_3\times S_3^2$ x 11, $C_{18}:C_6$ x 6, $S_3\times D_9$ x 5, $C_3^2\times D_6$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$ x 2, $C_3:S_3^2$ x 2, $S_3\times C_{18}$ x 2, $C_9:C_{12}$, $C_3^2\times A_4$, $C_3:S_3^2$, $C_3^3:C_4$, $C_{18}:C_6$, $C_3:D_{18}$, $C_3\times D_{18}$, $C_9:C_{12}$

Order 96: $D_4\times A_4$ x 3, $C_2^3\times D_6$, $C_2^3\times A_4$, $C_2^3:C_{12}$

Order 84: $C_2\times F_7$ x 5, $S_3\times D_7$ x 4, $C_{14}:C_6$, $C_7:C_{12}$, $D_{42}$, $S_3\times C_{14}$, $C_3\times D_{14}$, $C_7:C_{12}$

Order 81: $C_9:C_3^2$ x 4, $C_3^2\times C_9$ x 2, $C_3^4$

Order 72: $S_3\times A_4$ x 6, $S_3\times D_6$ x 5, $C_6\times D_6$ x 4, $C_6\times A_4$ x 4, $\SOPlus(4,2)$ x 4, $C_6:D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $D_4\times C_9$ x 2, $C_9:D_4$, $D_{36}$, $C_2\times C_6^2$, $C_4\times D_9$, $D_4\times C_3^2$, $C_6\wr C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_2\times D_{18}$

Order 64: $D_4\times C_2^3$

Order 63: $C_{21}:C_3$ x 2, $C_3\times C_{21}$

Order 56: $C_7\times D_4$, $C_7:D_4$, $D_{28}$, $C_4\times D_7$, $C_2\times D_{14}$, $F_8$

Order 54: $S_3\times C_3^2$ x 12, $C_9:C_6$ x 8, $C_3^2:C_6$ x 8, $S_3\times C_9$ x 6, $C_9:C_6$ x 5, $C_3:D_9$ x 3, $C_3\times D_9$ x 3, $C_3\times C_{18}$ x 2, $C_3^2\times C_6$ x 2, $C_3^2:S_3$

Order 48: $C_2^2\times D_6$ x 6, $C_2^2\times A_4$ x 6, $C_4\times A_4$ x 2, $C_2^3\times C_6$, $C_6\times D_4$, $S_3\times D_4$

Order 42: $F_7$ x 5, $C_7:C_6$ x 3, $D_{21}$ x 2, $C_3\times D_7$ x 2, $S_3\times C_7$ x 2, $C_{42}$

Order 36: $C_6\times S_3$ x 17, $S_3^2$ x 9, $D_{18}$ x 5, $C_6^2$ x 3, $C_6:S_3$ x 3, $C_3^2:C_4$ x 2, $C_2\times C_{18}$ x 2, $C_{36}$ x 2, $C_3\times A_4$ x 2, $C_3\times C_{12}$, $C_3:C_{12}$, $C_9:C_4$

Order 32: $C_2^2\times D_4$ x 3, $C_2^5$, $C_2^3\times C_4$

Order 28: $D_{14}$ x 4, $C_2\times C_{14}$, $C_{28}$, $C_7:C_4$

Order 27: $C_9:C_3$ x 6, $C_3^3$ x 5, $C_3\times C_9$ x 4

Order 24: $C_2\times D_6$ x 11, $C_2\times A_4$ x 8, $C_2^2\times C_6$ x 4, $C_3\times D_4$ x 4, $C_2\times C_{12}$, $C_3:D_4$, $D_{12}$, $C_4\times S_3$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 18: $C_3\times S_3$ x 22, $C_3\times C_6$ x 10, $C_{18}$ x 6, $C_3:S_3$ x 5, $D_9$ x 5

Order 16: $C_2^4$ x 5, $C_2\times D_4$ x 4, $C_2^2\times C_4$ x 2

Order 14: $D_7$ x 3, $C_{14}$ x 2

Order 12: $D_6$ x 15, $C_2\times C_6$ x 10, $C_{12}$ x 3, $A_4$ x 2, $C_3:C_4$

Order 9: $C_3^2$ x 9, $C_9$ x 4

Order 8: $C_2^3$ x 9, $D_4$ x 4, $C_2\times C_4$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 14, $S_3$ x 9

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 108864: ${}^2G(2,3)\times \SOPlus(4,2)$ ($C_1$)

Order 54432: $S_3^2\times {}^2G(2,3)$ ($C_2$) x 2, $(C_3^2\times \SL(2,8)):C_{12}$ ($C_2$)

Order 36288: $\SL(2,8)\times \SOPlus(4,2)$ ($C_3$)

Order 27216: $(C_3^2\times \SL(2,8)):C_6$ ($C_2^2$)

Order 18144: $S_3^2\times \SL(2,8)$ ($C_6$) x 2, $C_3^2:C_4\times \SL(2,8)$ ($C_6$)

Order 13608: $C_3^2\times {}^2G(2,3)$ ($D_4$)

Order 9072: $C_3:S_3\times \SL(2,8)$ ($C_2\times C_6$)

Order 4536: $C_3^2\times \SL(2,8)$ ($C_3\times D_4$)

Order 1512: ${}^2G(2,3)$ ($\SOPlus(4,2)$)

Order 504: $\SL(2,8)$ ($S_3^2:C_6$)

Order 72: $\SOPlus(4,2)$ (${}^2G(2,3)$)

Order 36: $S_3^2$ ($\SL(2,8):C_6$) x 2, $C_3^2:C_4$ ($\SL(2,8):C_6$)

Order 18: $C_3:S_3$ ($C_2^2\times {}^2G(2,3)$)

Order 9: $C_3^2$ ($D_4\times {}^2G(2,3)$)

Order 1: $C_1$ (${}^2G(2,3)\times \SOPlus(4,2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 108864: ${}^2G(2,3)\times \SOPlus(4,2)$ ($C_1$)

Order 54432: $(C_3^2\times \SL(2,8)):C_{12}$ ($C_2$), $S_3^2\times {}^2G(2,3)$ ($C_2$)

Order 36288: $\SL(2,8)\times \SOPlus(4,2)$ ($C_3$)

Order 27216: $(C_3^2\times \SL(2,8)):C_6$ ($C_2^2$)

Order 18144: $C_3^2:C_4\times \SL(2,8)$ ($C_6$), $S_3^2\times \SL(2,8)$ ($C_6$)

Order 13608: $C_3^2\times {}^2G(2,3)$ ($D_4$)

Order 9072: $C_3:S_3\times \SL(2,8)$ ($C_2\times C_6$)

Order 4536: $C_3^2\times \SL(2,8)$ ($C_3\times D_4$)

Order 1512: ${}^2G(2,3)$ ($\SOPlus(4,2)$)

Order 504: $\SL(2,8)$ ($S_3^2:C_6$)

Order 72: $\SOPlus(4,2)$ (${}^2G(2,3)$)

Order 36: $C_3^2:C_4$ ($\SL(2,8):C_6$), $S_3^2$ ($\SL(2,8):C_6$)

Order 18: $C_3:S_3$ ($C_2^2\times {}^2G(2,3)$)

Order 9: $C_3^2$ ($D_4\times {}^2G(2,3)$)

Order 1: $C_1$ (${}^2G(2,3)\times \SOPlus(4,2)$)

Series

Derived series

${}^2G(2,3)\times \SOPlus(4,2)$

$\rhd$

$C_3:S_3\times \SL(2,8)$

$\rhd$

$C_3^2\times \SL(2,8)$

$\rhd$

$\SL(2,8)$

Chief series

${}^2G(2,3)\times \SOPlus(4,2)$

$\rhd$

$\SL(2,8)\times \SOPlus(4,2)$

$\rhd$

$\SOPlus(4,2)$

$\rhd$

$S_3^2$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

${}^2G(2,3)\times \SOPlus(4,2)$

$\rhd$

$C_3:S_3\times \SL(2,8)$

$\rhd$

$C_3^2\times \SL(2,8)$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $99 \times 99$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $72 \times 72$ rational character table.