Abstract group 3161088.a: $C_2^9.C_7^3:(C_3\times S_3)$

• Label: 3161088.a
• Order: \( 2^{10} \cdot 3^{2} \cdot 7^{3} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$C_2^9.C_7^3:(C_3\times S_3)$
Order:	\(3161088\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 7^{3} \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^9.C_7^3:(C_3\times S_3)$, of order \(3161088\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 7^{3} \)
Composition factors:	$C_2$ x 10, $C_3$ x 2, $C_7$ x 3
Derived length:	$4$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	7	12	14	21	28
Elements	1	1855	62720	9408	965888	117648	526848	517104	903168	56448	3161088
Conjugacy classes	1	5	5	2	13	29	4	29	6	2	96
Divisions	1	5	3	2	7	15	2	15	2	1	53
Autjugacy classes	1	5	5	2	13	29	4	29	6	2	96

Dimension	1	2	3	4	6	9	12	18	21	24	36	42	63	126	147	252	294	343	441	686	882	1372
Irr. complex chars.	6	3	4	0	8	28	0	11	6	0	0	0	4	7	6	0	0	6	4	3	0	0	96
Irr. rational chars.	2	3	0	1	2	0	2	15	2	1	5	2	0	3	2	3	2	2	0	3	2	1	53

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	21	21	21
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l \mid a^{6}=b^{21}=c^{7}=d^{14}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,10,21,5,15,23,2,9,17,8,14,18,3,13,22,6,12,19,7,16,20)(4,11,24), (1,7,8,2,4,3)(5,6)(9,18,14,17,15,21)(10,19,12,20,11,23)(13,24)(16,22)\rangle$
Transitive group:	24T22827	42T2254	42T2255		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^9.C_7\wr S_3)$ . $C_3$	$(C_2^9.C_7^3.C_3)$ . $S_3$	$(C_2^9.C_7\wr C_3)$ . $C_6$	$C_2^9$ . $(C_7^3:(C_3\times S_3))$	all 10
Aut. group:	$\Aut(C_2^9:(C_7^2:C_3))$	$\Aut(C_2^9.C_7^2:S_3)$	$\Aut(C_2^9.C_7^2:C_6)$	$\Aut(C_2^9.C_7\wr C_3)$	all 6

Elements of the group are displayed as permutations of degree 24.

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 12 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_2^9.C_7^3:(C_3\times S_3)$
Commutator:	$G' \simeq$ $C_2^9.C_7\wr C_3$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^9.C_7^3:(C_3\times S_3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^9$	$G/\operatorname{Fit} \simeq$ $C_7^3:(C_3\times S_3)$
Radical:	$R \simeq$ $C_2^9.C_7^3:(C_3\times S_3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^9$	$G/\operatorname{soc} \simeq$ $C_7^3:(C_3\times S_3)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\times C_2^6.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 18816 (order at least 168) are shown, as well as all normal subgroups of any index.

Order 3161088: $C_2^9.C_7^3:(C_3\times S_3)$

Order 1580544: $C_2^9.C_7^3:C_3^2$

Order 1053696: $C_2^9.C_7^3.C_6$, $C_2^9.C_7\wr S_3$

Order 526848: $C_2^9.C_7\wr C_3$ x 2, $C_2^9.C_7^3.C_3$

Order 451584: $C_2^9.C_7^2:(C_3\times S_3)$

Order 351232: $C_2^3:F_8.C_{14}\times F_8$

Order 225792: $C_2^9:(C_7^2:C_3^2)$ x 2

Order 175616: $C_2^6.C_7^2\times F_8$

Order 150528: $C_2^9.C_7.F_7$ x 4, $C_2^9.C_7^2.C_6$ x 2, $C_2\times C_2^6.C_{14}^2.C_6$, $C_2^9.C_7^2.C_6$, $C_2^9.C_7.F_7$, $C_2^9.C_7^2:S_3$

Order 131712: $C_2^6.C_7^3.C_6$

Order 75264: $C_2^9.C_7^2.C_3$ x 10, $C_2^9:(C_7:C_{21})$ x 3, $C_2^9.C_7^2.C_3$ x 2, $C_2^6.C_{14}^2.C_6$ x 2, $C_2^9.C_7^2.C_3$ x 2, $C_2^9:(C_7^2:C_3)$ x 2, $C_2^6.C_{14}^2.C_6$

Order 65856: $C_2^6.C_7^3.C_3$

Order 64512: $C_2^9.C_{21}:C_6$

Order 50176: $C_2^3:F_8.F_8.C_2$ x 6, $F_8\times C_2^6.D_7$, $F_8\times C_2^6.C_{14}$, $C_2^3\times C_2^3:F_8.C_{14}$

Order 43904: $C_7\times C_2^3:F_8.C_{14}$

Order 37632: $C_2^6.C_7^2.A_4$, $F_8\wr C_2:C_6$

Order 32256: $C_2^9:(C_{21}:C_3)$ x 2

Order 25088: $C_2^9.C_7^2$ x 10, $C_2^9.C_7^2$ x 2, $C_2^2\times C_2^3:F_8.C_{14}$ x 2, $C_2^6.C_{14}^2.C_2$, $C_2^9.C_7^2$, $C_2^3:F_8\times F_8$

Order 21952: $C_7\times F_8\times F_8$

Order 21504: $C_2^9.C_7.C_6$ x 3, $C_2^6.F_8:C_6$ x 3, $C_2^6:C_7.A_4.C_2^2$, $F_8.C_2^5.C_6.C_2$, $C_2^9.F_7$, $C_2^3\wr C_3.C_{14}$

Order 18816: $F_8\wr C_2:C_3$ x 8, $C_2^6.C_7^2.C_6$, $C_2\times C_2^6.C_7^2.C_3$, $C_2^6.C_7^2.C_6$, $C_2^6.C_7^2.C_6$

Order 16464: $F_8.C_7^2.C_6$

Order 12544: $C_2\times F_8\wr C_2$ x 2, $C_2^2\times F_8^2$

Order 10752: $C_2^6.F_8:C_3$ x 7, $C_2^9:C_7:C_3$ x 6, $C_2^6:C_7.A_4.C_2$ x 2, $C_2^9:C_{21}$ x 2, $C_2\wr C_7.A_4$ x 2, $C_2^4.F_8:A_4$ x 2, $C_2^8.F_7$ x 2, $F_8.C_2^4.C_6.C_2$, $F_8.C_2^4.C_6.C_2$, $F_8.C_2^4.C_{12}$

Order 9408: $C_2^6.C_7^2.C_3$ x 30, $C_2^6.C_7^2.C_3$ x 2, $C_2^6.C_7^2.C_3$ x 2, $F_8.C_7:A_4.C_2$

Order 9216: $C_2^5.(A_4\times S_4)$

Order 8232: $C_2^3:(C_7^3:C_3)$

Order 8064: $C_2^6:C_{21}:C_6$

Order 7168: $C_2^9.C_{14}$ x 3, $C_2^9.C_{14}$ x 3, $C_2^3\times C_2^6.C_{14}$, $C_2^6.C_2\times F_8$, $C_2^3\times C_2^6.D_7$

Order 6272: $F_8\wr C_2$ x 8, $C_2\times F_8\times F_8$ x 2, $C_7\times C_2^6.C_{14}$, $C_7\times C_2^6.D_7$

Order 6174: $C_7^3:(C_3\times S_3)$

Order 5488: $F_8\times C_7\times D_7$

Order 5376: $F_8.C_2^4.C_6$ x 2, $C_2^3.F_8:A_4$ x 2, $C_2^3.F_8:A_4$ x 2, $F_8.C_2^4.C_6$, $F_8.(D_4\times A_4)$, $C_2^4:F_8:C_6$, $C_2\wr F_7$

Order 4704: $(C_7\times F_8).A_4$

Order 4608: $C_2^4.A_4^2.C_2$, $C_2^4:(A_4\times S_4)$, $C_2^9:C_3^2$

Order 4032: $C_2^6:C_7.C_3^2$ x 2

Order 3584: $C_2^9:C_7$ x 6, $C_2^9.C_7$ x 4, $C_2^2\times C_2^6.C_{14}$ x 2, $C_2^7:D_{14}$ x 2, $C_2^3\times C_2^3:F_8$, $C_2^3:F_8.C_2^3$, $(C_2^3\times F_8).C_2^3$, $C_2^6:C_7.C_2^3$, $C_2^3\times C_2^6:C_7$, $C_2^3\times C_2^3\times F_8$, $(C_2^3\times F_8).C_2^3$

Order 3136: $C_2^6.C_7^2$ x 24, $F_8^2$ x 6, $C_7\times C_2^3:F_8$ x 2, $C_2^6:C_7^2$ x 2, $C_2^2\times C_{14}\times F_8$

Order 3087: $C_7^3:C_3^2$

Order 3072: $C_2^7.C_6.C_2^2$, $C_2^7.S_4$

Order 2744: $C_7\times C_7\times F_8$

Order 2688: $C_2^3:F_8:C_6$ x 11, $C_2^3:F_8:C_6$ x 5, $C_2\wr C_7:C_3$ x 5, $F_8.A_4.C_2^2$ x 3, $C_2\times F_8.A_4.C_2$ x 2, $C_2^6:F_7$ x 2, $F_8.C_2^4.C_3$, $F_8.A_4.C_4$, $(C_2^4\times C_{14}).C_6.C_2$, $C_2^6:(S_3\times C_7)$

Order 2352: $F_8:F_7$ x 7, $C_2^4:(C_7^2:C_3)$ x 3, $C_7^2:(C_2^2\times A_4)$

Order 2304: $C_2^8.C_3^2$, $C_2^4:(C_6\times S_4)$, $C_2^4:(A_4\times D_6)$

Order 2058: $C_7^3:C_6$, $C_7\wr S_3$

Order 1792: $C_2\wr D_7$ x 2, $C_2^5:F_8$ x 2, $C_2^5:F_8$ x 2, $C_2^8.C_7$ x 2, $C_2^7.C_{14}$ x 2, $C_2^5:F_8$ x 2, $C_2\times (D_4\times F_8).C_2$, $C_2^7.C_{14}$

Order 1568: $C_2^3:C_{14}^2$

Order 1536: $C_2^8:C_6$ x 4, $C_2^3\wr C_3$ x 3, $C_2^6:S_4$ x 2, $C_2^7:A_4$, $C_2^7:C_{12}$

Order 1344: $C_2^3:F_8:C_3$ x 27, $C_2\times F_8:A_4$ x 19, $C_2^3:F_8:C_3$ x 17, $C_2^6:C_{21}$ x 2, $(C_2^2\times C_{28}):A_4$, $(C_2^3\times C_{14}):A_4$, $D_4\times F_8:C_3$, $(C_2^3\times C_{14}):A_4$

Order 1176: $(C_7\times F_8):C_3$ x 34, $C_{14}^2:C_6$ x 2, $C_{14}^2:C_6$

Order 1152: $C_2^3:A_4^2$ x 3, $C_2^4:(S_3\times A_4)$ x 2, $C_2^4:(C_3\times S_4)$ x 2

Order 1029: $C_7\wr C_3$ x 2, $C_7^3:C_3$

Order 1024: $C_2^3\times C_2^6.C_2$

Order 1008: $F_8:C_3\times S_3$

Order 896: $C_2^4:F_8$ x 11, $C_2^4\times F_8$ x 8, $C_2\wr C_7$ x 5, $C_2^4:F_8$ x 5, $C_2^6:D_7$ x 2, $C_2^6:C_{14}$ x 2, $C_2^2:C_4\times F_8$ x 2, $C_2^2\times C_4\times F_8$, $C_7\times C_2^3\wr C_2$

Order 882: $C_7^2:(C_3\times S_3)$

Order 784: $D_7\times F_8$ x 7, $C_{14}\times F_8$ x 3, $C_{14}^2:C_2^2$

Order 768: $C_2^6:A_4$ x 5, $C_2^6:A_4$ x 4, $C_2^7:C_6$ x 4, $C_2^5:S_4$ x 2, $C_2^6:C_{12}$ x 2, $C_2^5:S_4$, $C_2^7:C_6$

Order 686: $D_7\times C_7^2$

Order 672: $F_8:A_4$ x 11, $C_2\times F_8:C_6$ x 3, $(C_2^2\times C_{14}):A_4$ x 2, $(C_2^2\times C_{14}):A_4$, $D_4\times C_7:A_4$, $F_8:C_{12}$

Order 588: $C_{14}^2:C_3$, $C_{14}:F_7$

Order 576: $C_2^2:A_4^2$ x 3, $A_4^2:C_2^2$, $A_4^2:C_2^2$

Order 512: $C_2^8.C_2$ x 4, $C_2^8.C_2$, $C_2^9$, $C_2^6:D_4$

Order 504: $F_8:C_3^2$ x 2

Order 448: $C_2^3:F_8$ x 24, $C_2^3\times F_8$ x 23, $C_2^3:F_8$ x 17, $D_4\times F_8$ x 2, $C_2^4:C_{28}$, $C_2^5\times C_{14}$, $C_2^4:C_{28}$, $C_2^5:C_{14}$

Order 441: $C_7^2:C_3^2$ x 2

Order 392: $C_7\times F_8$ x 34, $C_{14}\times D_{14}$ x 2, $C_2\times C_{14}^2$

Order 384: $C_2\wr C_6$ x 16, $C_2^5:A_4$ x 13, $C_2^2\wr S_3$ x 6, $C_2^5:A_4$ x 5, $C_2^5:C_{12}$ x 4, $C_2^6:C_6$ x 4, $C_2^5:A_4$ x 4, $C_2^4:S_4$ x 2, $C_2^5:A_4$ x 2, $C_2^5:C_{12}$

Order 343: $C_7^3$

Order 336: $F_8:C_6$ x 28, $C_2\times C_{14}:A_4$ x 5, $(C_2\times C_{14}):A_4$, $D_{14}:A_4$, $S_3\times F_8$, $C_{28}:A_4$

Order 294: $C_7:F_7$ x 9, $C_7^2:C_6$ x 2, $C_7^2:S_3$

Order 288: $C_2\times A_4^2$ x 4, $A_4\wr C_2$ x 2, $A_4\times S_4$ x 2

Order 256: $C_2^5:D_4$ x 16, $C_2^5:D_4$ x 8, $C_2^6:C_4$ x 4, $C_2^8$ x 3, $D_4\times C_2^5$, $C_2^6:C_4$

Order 224: $C_2^2\times F_8$ x 17, $C_2^4\times C_{14}$ x 2, $C_2^4:C_{14}$ x 2, $C_{28}:C_2^3$, $C_4\times F_8$, $C_2^3:C_{28}$

Order 196: $C_7\times D_{14}$ x 2, $C_{14}^2$

Order 192: $C_2^2\wr C_3$ x 52, $C_2^4:A_4$ x 16, $C_2^3:S_4$ x 10, $C_2^4:C_{12}$ x 8, $C_2^5:C_6$ x 5, $C_2^4:A_4$ x 3, $C_2^4:C_{12}$ x 2, $C_2^3\times S_4$

Order 168: $F_8:C_3$ x 49, $C_{14}:A_4$ x 15, $D_7:A_4$ x 2, $C_3\times F_8$ x 2, $C_{28}:C_6$

Order 9: $C_3^2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 18816 (order at least 168) are shown, as well as all normal subgroups of any index.

Order 3161088: $C_2^9.C_7^3:(C_3\times S_3)$

Order 1580544: $C_2^9.C_7^3:C_3^2$

Order 1053696: $C_2^9.C_7^3.C_6$, $C_2^9.C_7\wr S_3$

Order 526848: $C_2^9.C_7\wr C_3$ x 2, $C_2^9.C_7^3.C_3$

Order 451584: $C_2^9.C_7^2:(C_3\times S_3)$

Order 351232: $C_2^3:F_8.C_{14}\times F_8$

Order 225792: $C_2^9:(C_7^2:C_3^2)$ x 2

Order 175616: $C_2^6.C_7^2\times F_8$

Order 150528: $C_2^9.C_7.F_7$ x 4, $C_2^9.C_7^2.C_6$ x 2, $C_2\times C_2^6.C_{14}^2.C_6$, $C_2^9.C_7^2.C_6$, $C_2^9.C_7.F_7$, $C_2^9.C_7^2:S_3$

Order 131712: $C_2^6.C_7^3.C_6$

Order 75264: $C_2^9.C_7^2.C_3$ x 10, $C_2^9:(C_7:C_{21})$ x 3, $C_2^9.C_7^2.C_3$ x 2, $C_2^6.C_{14}^2.C_6$ x 2, $C_2^9.C_7^2.C_3$ x 2, $C_2^9:(C_7^2:C_3)$ x 2, $C_2^6.C_{14}^2.C_6$

Order 65856: $C_2^6.C_7^3.C_3$

Order 64512: $C_2^9.C_{21}:C_6$

Order 50176: $C_2^3:F_8.F_8.C_2$ x 6, $F_8\times C_2^6.D_7$, $F_8\times C_2^6.C_{14}$, $C_2^3\times C_2^3:F_8.C_{14}$

Order 43904: $C_7\times C_2^3:F_8.C_{14}$

Order 37632: $C_2^6.C_7^2.A_4$, $F_8\wr C_2:C_6$

Order 32256: $C_2^9:(C_{21}:C_3)$ x 2

Order 25088: $C_2^9.C_7^2$ x 10, $C_2^9.C_7^2$ x 2, $C_2^2\times C_2^3:F_8.C_{14}$ x 2, $C_2^6.C_{14}^2.C_2$, $C_2^9.C_7^2$, $C_2^3:F_8\times F_8$

Order 21952: $C_7\times F_8\times F_8$

Order 21504: $C_2^9.C_7.C_6$ x 3, $C_2^6.F_8:C_6$ x 3, $C_2^6:C_7.A_4.C_2^2$, $F_8.C_2^5.C_6.C_2$, $C_2^9.F_7$, $C_2^3\wr C_3.C_{14}$

Order 18816: $F_8\wr C_2:C_3$ x 8, $C_2^6.C_7^2.C_6$, $C_2\times C_2^6.C_7^2.C_3$, $C_2^6.C_7^2.C_6$, $C_2^6.C_7^2.C_6$

Order 16464: $F_8.C_7^2.C_6$

Order 12544: $C_2\times F_8\wr C_2$ x 2, $C_2^2\times F_8^2$

Order 10752: $C_2^6.F_8:C_3$ x 7, $C_2^9:C_7:C_3$ x 6, $C_2^6:C_7.A_4.C_2$ x 2, $C_2^9:C_{21}$ x 2, $C_2\wr C_7.A_4$ x 2, $C_2^4.F_8:A_4$ x 2, $C_2^8.F_7$ x 2, $F_8.C_2^4.C_6.C_2$, $F_8.C_2^4.C_6.C_2$, $F_8.C_2^4.C_{12}$

Order 9408: $C_2^6.C_7^2.C_3$ x 30, $C_2^6.C_7^2.C_3$ x 2, $C_2^6.C_7^2.C_3$ x 2, $F_8.C_7:A_4.C_2$

Order 9216: $C_2^5.(A_4\times S_4)$

Order 8232: $C_2^3:(C_7^3:C_3)$

Order 8064: $C_2^6:C_{21}:C_6$

Order 7168: $C_2^9.C_{14}$ x 3, $C_2^9.C_{14}$ x 3, $C_2^3\times C_2^6.C_{14}$, $C_2^6.C_2\times F_8$, $C_2^3\times C_2^6.D_7$

Order 6272: $F_8\wr C_2$ x 8, $C_2\times F_8\times F_8$ x 2, $C_7\times C_2^6.C_{14}$, $C_7\times C_2^6.D_7$

Order 6174: $C_7^3:(C_3\times S_3)$

Order 5488: $F_8\times C_7\times D_7$

Order 5376: $F_8.C_2^4.C_6$ x 2, $C_2^3.F_8:A_4$ x 2, $C_2^3.F_8:A_4$ x 2, $F_8.C_2^4.C_6$, $F_8.(D_4\times A_4)$, $C_2^4:F_8:C_6$, $C_2\wr F_7$

Order 4704: $(C_7\times F_8).A_4$

Order 4608: $C_2^4.A_4^2.C_2$, $C_2^4:(A_4\times S_4)$, $C_2^9:C_3^2$

Order 4032: $C_2^6:C_7.C_3^2$ x 2

Order 3584: $C_2^9:C_7$ x 6, $C_2^9.C_7$ x 4, $C_2^2\times C_2^6.C_{14}$ x 2, $C_2^7:D_{14}$ x 2, $C_2^3\times C_2^3:F_8$, $C_2^3:F_8.C_2^3$, $(C_2^3\times F_8).C_2^3$, $C_2^6:C_7.C_2^3$, $C_2^3\times C_2^6:C_7$, $C_2^3\times C_2^3\times F_8$, $(C_2^3\times F_8).C_2^3$

Order 3136: $C_2^6.C_7^2$ x 24, $F_8^2$ x 6, $C_7\times C_2^3:F_8$ x 2, $C_2^6:C_7^2$ x 2, $C_2^2\times C_{14}\times F_8$

Order 3087: $C_7^3:C_3^2$

Order 3072: $C_2^7.C_6.C_2^2$, $C_2^7.S_4$

Order 2744: $C_7\times C_7\times F_8$

Order 2688: $C_2^3:F_8:C_6$ x 11, $C_2^3:F_8:C_6$ x 5, $C_2\wr C_7:C_3$ x 5, $F_8.A_4.C_2^2$ x 3, $C_2\times F_8.A_4.C_2$ x 2, $C_2^6:F_7$ x 2, $F_8.C_2^4.C_3$, $F_8.A_4.C_4$, $(C_2^4\times C_{14}).C_6.C_2$, $C_2^6:(S_3\times C_7)$

Order 2352: $F_8:F_7$ x 7, $C_2^4:(C_7^2:C_3)$ x 3, $C_7^2:(C_2^2\times A_4)$

Order 2304: $C_2^8.C_3^2$, $C_2^4:(C_6\times S_4)$, $C_2^4:(A_4\times D_6)$

Order 2058: $C_7^3:C_6$, $C_7\wr S_3$

Order 1792: $C_2\wr D_7$ x 2, $C_2^5:F_8$ x 2, $C_2^5:F_8$ x 2, $C_2^8.C_7$ x 2, $C_2^7.C_{14}$ x 2, $C_2^5:F_8$ x 2, $C_2\times (D_4\times F_8).C_2$, $C_2^7.C_{14}$

Order 1568: $C_2^3:C_{14}^2$

Order 1536: $C_2^8:C_6$ x 4, $C_2^3\wr C_3$ x 3, $C_2^6:S_4$ x 2, $C_2^7:A_4$, $C_2^7:C_{12}$

Order 1344: $C_2^3:F_8:C_3$ x 27, $C_2\times F_8:A_4$ x 19, $C_2^3:F_8:C_3$ x 17, $C_2^6:C_{21}$ x 2, $(C_2^2\times C_{28}):A_4$, $(C_2^3\times C_{14}):A_4$, $D_4\times F_8:C_3$, $(C_2^3\times C_{14}):A_4$

Order 1176: $(C_7\times F_8):C_3$ x 34, $C_{14}^2:C_6$ x 2, $C_{14}^2:C_6$

Order 1152: $C_2^3:A_4^2$ x 3, $C_2^4:(S_3\times A_4)$ x 2, $C_2^4:(C_3\times S_4)$ x 2

Order 1029: $C_7\wr C_3$ x 2, $C_7^3:C_3$

Order 1024: $C_2^3\times C_2^6.C_2$

Order 1008: $F_8:C_3\times S_3$

Order 896: $C_2^4:F_8$ x 11, $C_2^4\times F_8$ x 8, $C_2\wr C_7$ x 5, $C_2^4:F_8$ x 5, $C_2^6:D_7$ x 2, $C_2^6:C_{14}$ x 2, $C_2^2:C_4\times F_8$ x 2, $C_2^2\times C_4\times F_8$, $C_7\times C_2^3\wr C_2$

Order 882: $C_7^2:(C_3\times S_3)$

Order 784: $D_7\times F_8$ x 7, $C_{14}\times F_8$ x 3, $C_{14}^2:C_2^2$

Order 768: $C_2^6:A_4$ x 5, $C_2^6:A_4$ x 4, $C_2^7:C_6$ x 4, $C_2^5:S_4$ x 2, $C_2^6:C_{12}$ x 2, $C_2^5:S_4$, $C_2^7:C_6$

Order 686: $D_7\times C_7^2$

Order 672: $F_8:A_4$ x 11, $C_2\times F_8:C_6$ x 3, $(C_2^2\times C_{14}):A_4$ x 2, $(C_2^2\times C_{14}):A_4$, $D_4\times C_7:A_4$, $F_8:C_{12}$

Order 588: $C_{14}^2:C_3$, $C_{14}:F_7$

Order 576: $C_2^2:A_4^2$ x 3, $A_4^2:C_2^2$, $A_4^2:C_2^2$

Order 512: $C_2^8.C_2$ x 4, $C_2^8.C_2$, $C_2^9$, $C_2^6:D_4$

Order 504: $F_8:C_3^2$ x 2

Order 448: $C_2^3:F_8$ x 24, $C_2^3\times F_8$ x 23, $C_2^3:F_8$ x 17, $D_4\times F_8$ x 2, $C_2^4:C_{28}$, $C_2^5\times C_{14}$, $C_2^4:C_{28}$, $C_2^5:C_{14}$

Order 441: $C_7^2:C_3^2$ x 2

Order 392: $C_7\times F_8$ x 34, $C_{14}\times D_{14}$ x 2, $C_2\times C_{14}^2$

Order 384: $C_2\wr C_6$ x 16, $C_2^5:A_4$ x 13, $C_2^2\wr S_3$ x 6, $C_2^5:A_4$ x 5, $C_2^5:C_{12}$ x 4, $C_2^6:C_6$ x 4, $C_2^5:A_4$ x 4, $C_2^4:S_4$ x 2, $C_2^5:A_4$ x 2, $C_2^5:C_{12}$

Order 343: $C_7^3$

Order 336: $F_8:C_6$ x 28, $C_2\times C_{14}:A_4$ x 5, $(C_2\times C_{14}):A_4$, $D_{14}:A_4$, $S_3\times F_8$, $C_{28}:A_4$

Order 294: $C_7:F_7$ x 9, $C_7^2:C_6$ x 2, $C_7^2:S_3$

Order 288: $C_2\times A_4^2$ x 4, $A_4\wr C_2$ x 2, $A_4\times S_4$ x 2

Order 256: $C_2^5:D_4$ x 16, $C_2^5:D_4$ x 8, $C_2^6:C_4$ x 4, $C_2^8$ x 3, $D_4\times C_2^5$, $C_2^6:C_4$

Order 224: $C_2^2\times F_8$ x 17, $C_2^4\times C_{14}$ x 2, $C_2^4:C_{14}$ x 2, $C_{28}:C_2^3$, $C_4\times F_8$, $C_2^3:C_{28}$

Order 196: $C_7\times D_{14}$ x 2, $C_{14}^2$

Order 192: $C_2^2\wr C_3$ x 52, $C_2^4:A_4$ x 16, $C_2^3:S_4$ x 10, $C_2^4:C_{12}$ x 8, $C_2^5:C_6$ x 5, $C_2^4:A_4$ x 3, $C_2^4:C_{12}$ x 2, $C_2^3\times S_4$

Order 168: $F_8:C_3$ x 49, $C_{14}:A_4$ x 15, $D_7:A_4$ x 2, $C_3\times F_8$ x 2, $C_{28}:C_6$

Order 9: $C_3^2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3161088: $C_2^9.C_7^3:(C_3\times S_3)$ ($C_1$)

Order 1580544: $C_2^9.C_7^3:C_3^2$ ($C_2$)

Order 1053696: $C_2^9.C_7\wr S_3$ ($C_3$)

Order 526848: $C_2^9.C_7^3.C_3$ ($S_3$), $C_2^9.C_7\wr C_3$ ($C_6$)

Order 175616: $C_2^6.C_7^2\times F_8$ ($C_3\times S_3$)

Order 150528: $C_2^9.C_7^2:S_3$ ($C_7:C_3$)

Order 75264: $C_2^9:(C_7^2:C_3)$ ($C_7:C_6$)

Order 25088: $C_2^9.C_7^2$ ($C_{21}:C_6$)

Order 3584: $C_2^9.C_7$ ($C_7^2:(C_3\times S_3)$)

Order 512: $C_2^9$ ($C_7^3:(C_3\times S_3)$)

Order 1: $C_1$ ($C_2^9.C_7^3:(C_3\times S_3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3161088: $C_2^9.C_7^3:(C_3\times S_3)$ ($C_1$)

Order 1580544: $C_2^9.C_7^3:C_3^2$ ($C_2$)

Order 1053696: $C_2^9.C_7\wr S_3$ ($C_3$)

Order 526848: $C_2^9.C_7^3.C_3$ ($S_3$), $C_2^9.C_7\wr C_3$ ($C_6$)

Order 175616: $C_2^6.C_7^2\times F_8$ ($C_3\times S_3$)

Order 150528: $C_2^9.C_7^2:S_3$ ($C_7:C_3$)

Order 75264: $C_2^9:(C_7^2:C_3)$ ($C_7:C_6$)

Order 25088: $C_2^9.C_7^2$ ($C_{21}:C_6$)

Order 3584: $C_2^9.C_7$ ($C_7^2:(C_3\times S_3)$)

Order 512: $C_2^9$ ($C_7^3:(C_3\times S_3)$)

Order 1: $C_1$ ($C_2^9.C_7^3:(C_3\times S_3)$)

Series

Derived series

$C_2^9.C_7^3:(C_3\times S_3)$

$\rhd$

$C_2^9.C_7\wr C_3$

$\rhd$

$C_2^9.C_7^2$

$\rhd$

$C_2^9$

$\rhd$

$C_1$

Chief series

$C_2^9.C_7^3:(C_3\times S_3)$

$\rhd$

$C_2^9.C_7^3:C_3^2$

$\rhd$

$C_2^9.C_7\wr C_3$

$\rhd$

$C_2^6.C_7^2\times F_8$

$\rhd$

$C_2^9.C_7^2$

$\rhd$

$C_2^9$

$\rhd$

$C_1$

Lower central series

$C_2^9.C_7^3:(C_3\times S_3)$

$\rhd$

$C_2^9.C_7\wr C_3$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $96 \times 96$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $53 \times 53$ rational character table.