Abstract group 2304433152.a: $\SL(2,8)\wr C_3.C_6$

• Label: 2304433152.a
• Order: \( 2^{10} \cdot 3^{8} \cdot 7^{3} \)
• Exponent: \( 2^{2} \cdot 3^{3} \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{8} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $27$
• Trans deg.: $27$
• Rank: $2$

Group information

Description:	$\SL(2,8)\wr C_3.C_6$
Order:	\(2304433152\)\(\medspace = 2^{10} \cdot 3^{8} \cdot 7^{3} \)
Exponent:	\(756\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7 \)
Automorphism group:	Group of order \(2304433152\)\(\medspace = 2^{10} \cdot 3^{8} \cdot 7^{3} \)
Composition factors:	$C_2$, $C_3$ x 2, $\SL(2,8)$ x 3
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	9	12	14	18	21	27	28	36	42	63	126
Elements	1	358911	2894696	6096384	267574104	10218312	127375416	261382464	103243896	546674184	339147648	256048128	20575296	144027072	41150592	54214272	123451776	2304433152
Conjugacy classes	1	5	8	2	20	7	20	5	10	27	7	3	1	3	3	18	9	149
Divisions	1	5	6	2	14	7	16	3	10	20	6	2	1	2	3	6	3	107
Autjugacy classes	1	5	8	2	20	7	20	5	10	27	7	3	1	3	3	18	9	149

Dimension	1	2	4	21	24	42	48	63	81	147	192	294	336	343	384	441	512	672	686	729	882	1008	1024	1029	1134	1176	1296	1344	1372	1458	2048	2058	2187	2352	2688	3087	3402	3528	3969	4032	4116	4374	5103	5184	5832	6174	6561	7056	7938	8748	9072	10206	11664	11907	15309	23814	27216	30618
Irr. complex chars.	6	3	0	6	6	0	0	2	2	6	6	0	3	6	0	2	6	0	3	2	2	1	3	2	4	6	1	6	0	1	0	4	2	0	0	8	0	2	8	2	0	4	8	2	2	1	4	2	6	0	4	4	1	0	0	0	0	0	149
Irr. rational chars.	2	3	1	2	2	2	2	2	2	2	2	2	1	2	2	2	2	1	3	2	2	1	3	2	1	2	1	2	1	1	1	2	2	2	2	8	1	2	2	2	1	2	2	2	2	1	4	2	0	1	1	1	1	2	2	2	1	1	107

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$27$
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

Permutation group:	Degree $27$ $\langle(1,24,15,5,20,10,4,21,16,8,26,18,2,23,13,3,22,14,7,27,17,9,25,12,6,19,11), (1,26,8,19,3,23,9,22,4,20,2,21,6,27,7,25,5,24)(10,16,14,17,13,11,12,18,15)\rangle$
Transitive group:	27T2338				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(\SL(2,8)\wr S_3)$ . $C_3$	$(\SL(2,8)\wr C_3)$ . $C_6$	$(\SL(2,8)^3.C_3^2)$ . $C_2$	$(\SL(2,8).\SL(2,8).\SL(2,8).C_3)$ . $S_3$	all 5

Elements of the group are displayed as permutations of degree 27.

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 7 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $\SL(2,8)\wr C_3$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9^3:C_3^2$

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 592704 (order at least 3888) are shown, as well as all normal subgroups of any index.

Order 2304433152: $\SL(2,8)\wr C_3.C_6$

Order 1152216576: $\SL(2,8)^3.C_3^2$

Order 768144384: $\SL(2,8).\SL(2,8)^2.C_6$, $\SL(2,8)\wr S_3$

Order 384072192: $\SL(2,8)\wr C_3$ x 2, $\SL(2,8).\SL(2,8).\SL(2,8).C_3$

Order 256048128: $\SL(2,8).\SL(2,8)^2.C_2$

Order 128024064: $\SL(2,8).\SL(2,8).\SL(2,8)$

Order 85349376: $F_8.\SL(2,8)^2.C_6$

Order 42674688: $F_8.\SL(2,8).\SL(2,8).C_3$

Order 28449792: $F_8.\SL(2,8)^2.C_2$

Order 27433728: $D_9.\SL(2,8)^2.C_6$

Order 21337344: $D_7.\SL(2,8)^2.C_6$

Order 14224896: $F_8.\SL(2,8).\SL(2,8)$

Order 13716864: $C_9.\SL(2,8)^2.C_6$ x 2, $D_9.\SL(2,8).\SL(2,8).C_3$

Order 12192768: $C_2^3.\SL(2,8)^2.C_6$

Order 10668672: $C_7.\SL(2,8)^2.C_6$ x 2, $D_7.\SL(2,8).\SL(2,8).C_3$

Order 9483264: $C_2^3:F_8.C_{14}.\SL(2,8).C_3$

Order 9144576: $S_3.\SL(2,8)^2.C_6$, $D_9.\SL(2,8)^2.C_2$

Order 7112448: $D_7.\SL(2,8)^2.C_2$

Order 6858432: $C_9.\SL(2,8).\SL(2,8).C_3$

Order 6096384: $C_2^2.\SL(2,8)^2.C_6$ x 2, $C_2^3.\SL(2,8).\SL(2,8).C_3$

Order 5334336: $C_7.\SL(2,8).\SL(2,8).C_3$

Order 4741632: $C_2^6.C_7^2.\SL(2,8).C_3$

Order 4572288: $C_3.\SL(2,8)^2.C_6$ x 3, $C_9.\SL(2,8)^2.C_2$ x 2, $\SL(2,8)^2\times D_9$, $S_3.\SL(2,8).\SL(2,8).C_3$

Order 4064256: $C_2^3.\SL(2,8)^2.C_2$

Order 3556224: $D_7.\SL(2,8).\SL(2,8)$, $C_7.\SL(2,8)^2.C_2$, $\GOrthPlus(4,8)$

Order 3161088: $C_2^3:F_8.C_{14}.\SL(2,8)$, $C_2^9.C_7^3:(C_3\times S_3)$

Order 3048192: $S_3.\SL(2,8)^2.C_2$, $C_2^2.\SL(2,8).\SL(2,8).C_3$, $\SOPlus(4,8):C_6$

Order 2286144: $C_3.\SL(2,8).\SL(2,8).C_3$ x 2, $C_9.\SL(2,8).\SL(2,8)$

Order 2032128: $C_2^2.\SL(2,8)^2.C_2$ x 2, $C_2^3\times \SL(2,8)^2$

Order 1778112: $C_7.\SL(2,8).\SL(2,8)$

Order 1580544: $C_2^6.C_7^2.\SL(2,8)$, $C_2^9.C_7^3:C_3^2$

Order 1524096: $C_3.\SL(2,8)^2.C_2$ x 2, $C_2.\SL(2,8).\SL(2,8).C_3$ x 2, $\SL(2,8)^2:C_6$ x 2, $S_3.\SL(2,8).\SL(2,8)$, $(C_9\times F_8).C_2.\SL(2,8).C_3$

Order 1354752: $C_2^6.D_7.\SL(2,8).C_3$, $C_2^6.C_{14}.\SL(2,8).C_3$

Order 1185408: $(C_7\times F_8).C_2.\SL(2,8).C_3$

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

Order 1016064: $\SL(2,8)^2:C_2^2$ x 2, $C_2^2.\SL(2,8).\SL(2,8)$

Order 979776: $D_9^2.C_2.\SL(2,8).C_3$

Order 762048: $\SL(2,8).\SL(2,8).C_3$ x 3, $C_3\times \SL(2,8)^2$, $(C_9\times F_8).\SL(2,8).C_3$

Order 677376: $C_2^6:C_7.\SL(2,8).C_3$ x 2, $C_2^3:F_8.\SL(2,8).C_3$ x 2, $(C_2^3\times F_8).\SL(2,8).C_3$

Order 592704: $D_7\wr C_2.\SL(2,8).C_3$, $(C_7\times F_8).\SL(2,8).C_3$

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

Order 508032: $C_2.\SL(2,8).\SL(2,8)$ x 2, $\SOPlus(4,8)$ x 2, $(S_3\times F_8).\SL(2,8).C_3$, $(C_9\times F_8).C_2.\SL(2,8)$

Order 489888: $D_9^2.\SL(2,8).C_3$ x 2, $C_9^2:C_4.\SL(2,8).C_3$

Order 451584: $C_2^6.D_7.\SL(2,8)$, $C_2^6.C_{14}.\SL(2,8)$, $C_2^9.C_7^2:(C_3\times S_3)$

Order 395136: $(C_7\times F_8).C_2.\SL(2,8)$

Order 381024: $(D_7\times D_9).\SL(2,8).C_3$

Order 351232: $C_2^3:F_8.(C_7\times F_8).C_2$

Order 338688: $C_2^3:F_8.C_{63}.C_6.C_2$, $(C_2^2\times F_8).\SL(2,8).C_3$

Order 326592: $D_9^2.C_2.\SL(2,8)$

Order 296352: $D_7^2.\SL(2,8).C_3$ x 2, $C_7:D_7.C_2.\SL(2,8).C_3$

Order 263424: $C_2^6.C_7^3.C_6.C_2$

Order 254016: $\SL(2,8)^2$ x 3, $(C_3\times F_8).\SL(2,8).C_3$ x 2, $(C_9\times F_8).\SL(2,8)$

Order 244944: $(C_9\times D_9).\SL(2,8).C_3$ x 2, $C_9:D_9.\SL(2,8).C_3$

Order 225792: $C_2^6:C_7.\SL(2,8)$ x 2, $C_2^3:F_8.\SL(2,8)$ x 2, $C_2^9:(C_7^2:C_3^2)$ x 2, $(C_2^3\times F_8).\SL(2,8)$

Order 217728: $(C_2^3\times D_9).\SL(2,8).C_3$

Order 197568: $D_7\wr C_2.\SL(2,8)$, $(C_7\times F_8).\SL(2,8)$

Order 193536: $C_2^4.C_2^3.\SL(2,8).C_3$

Order 190512: $D_{63}.\SL(2,8).C_3$, $(C_9\times D_7).\SL(2,8).C_3$, $(C_7\times D_9).\SL(2,8).C_3$

Order 175616: $C_2^9.C_7^3$

Order 169344: $(C_2\times F_8).\SL(2,8).C_3$ x 3, $C_2^6:C_7.C_{63}.C_6$, $C_2^6:C_7.C_9:F_7$, $C_2^6.C_7^2.C_9.C_6$, $(S_3\times F_8).\SL(2,8)$, $(C_2^3\times D_7).\SL(2,8).C_3$

Order 163296: $D_9^2.\SL(2,8)$ x 2, $(S_3\times D_9).\SL(2,8).C_3$ x 2, $C_9^2:C_4.\SL(2,8)$

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

Order 148176: $(C_7\times D_7).\SL(2,8).C_3$ x 2, $C_7:D_7.\SL(2,8).C_3$

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

Order 127008: $(S_3\times D_7).\SL(2,8).C_3$, $(D_7\times D_9).\SL(2,8)$

Order 122472: $C_9^2.\SL(2,8).C_3$

Order 112896: $C_2^6.C_{63}.C_{14}.C_2$, $C_2^3:F_8.C_{21}.C_6.C_2$, $(C_2^2\times F_8).\SL(2,8)$

Order 108864: $(C_2^2\times D_9).\SL(2,8).C_3$ x 2, $C_9^2.F_8.C_6.C_2^2$, $(C_2^2\times C_{18}).\SL(2,8).C_3$, ${}^2G(2,3)\times \SOPlus(4,2)$

Order 104976: $C_9^3:(C_6\times S_4)$

Order 98784: $D_7^2.\SL(2,8)$ x 2, $C_7:D_7.C_2.\SL(2,8)$

Order 96768: $C_2^6.\SL(2,8).C_3$, $C_2^5.C_2.\SL(2,8).C_3$, $(C_2\times C_2^2\wr C_2).\SL(2,8).C_3$

Order 95256: $C_{63}.\SL(2,8).C_3$

Order 87808: $C_2^6.C_7^3.C_2^2$

Order 84672: $F_8.\SL(2,8).C_3$ x 8, $(C_2^2\times D_7).\SL(2,8).C_3$ x 2, $C_2^6.C_7^2.C_9:C_3$, $(C_3\times F_8).\SL(2,8)$, $(C_2^2\times C_{14}).\SL(2,8).C_3$

Order 81648: $(C_3\times D_9).\SL(2,8).C_3$ x 3, $C_3:D_9.\SL(2,8).C_3$ x 2, $(C_9\times S_3).\SL(2,8).C_3$ x 2, $(C_9\times D_9).\SL(2,8)$ x 2, $C_9:D_9.\SL(2,8)$, $C_3\times D_9.\SL(2,8).C_3$

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

Order 74088: $C_7^2.\SL(2,8).C_3$

Order 72576: $(C_2^3\times S_3).\SL(2,8).C_3$, $(C_2^3\times D_9).\SL(2,8)$

Order 65856: $C_7^2.F_8.C_6.C_2^2$, $C_2^6.C_7^3.C_3$

Order 64512: $C_2^6.C_2\times \SL(2,8)$, $C_2^9.C_{21}:C_6$

Order 63504: $D_{63}.\SL(2,8)$, $D_{21}.\SL(2,8).C_3$, $C_3\times D_7.\SL(2,8).C_3$, $(C_9\times D_7).\SL(2,8)$, $(C_7\times S_3).\SL(2,8).C_3$, $(C_7\times D_9).\SL(2,8)$, $(C_3\times D_7).\SL(2,8).C_3$

Order 56448: $(C_2\times F_8).\SL(2,8)$ x 3, $C_2^6:C_7.C_{21}.C_6$ x 2, $C_9.(C_2^3\times F_8).C_{14}$, $C_2^6.C_{63}.C_{14}$, $C_2^6.C_7^2.C_{18}$, $C_2^6.C_7^2.C_3.C_6$, $C_2^3:F_8.C_{126}$, $(C_2^3\times D_7).\SL(2,8)$

Order 54432: $D_{18}.\SL(2,8).C_3$ x 3, $F_8.C_9^2.C_6.C_2$ x 2, $(S_3\times D_9).\SL(2,8)$ x 2, $S_3^2\times {}^2G(2,3)$ x 2, $F_8.C_9^2.C_{12}$, $(C_2\times C_{18}).\SL(2,8).C_3$, $(C_3^2\times \SL(2,8)):C_{12}$

Order 52488: $D_9^3:C_3^2$, $C_9^3:(C_3\times S_4)$, $C_9^3:(C_3\times S_4)$

Order 50176: $C_2^3:F_8.F_8.C_2$ x 7, $C_2^9.C_7.C_{14}$, $C_2^6.C_7^2.C_2^4$

Order 49392: $(C_7\times D_7).\SL(2,8)$ x 2, $C_7:D_7.\SL(2,8)$, $C_7^3:(C_6\times S_4)$

Order 48384: $C_2^6.C_{63}.C_6.C_2$ x 2, $C_2^5.\SL(2,8).C_3$ x 2, $C_2^2\wr C_2.\SL(2,8).C_3$, $(C_2^2\times D_4).\SL(2,8).C_3$

Order 43904: $C_7\times C_2^3:F_8.C_{14}$, $C_7.(C_2^3\times F_8).C_{14}$, $C_2^6.C_7^3.C_2$

Order 42336: $D_{14}.\SL(2,8).C_3$ x 3, $(S_3\times D_7).\SL(2,8)$, $(C_7\times F_8).C_9.C_6.C_2$, $(C_2\times C_{14}).\SL(2,8).C_3$

Order 40824: $(C_3\times C_9).\SL(2,8).C_3$ x 4, $C_9^2.\SL(2,8)$

Order 37632: $C_2^6:C_7.F_7.C_2$, $C_2^6.C_{21}.C_{14}.C_2$, $C_2^6.C_7^2.C_6.C_2$, $C_2^6.C_7^2.A_4$, $F_8\wr C_2:C_6$

Order 36288: $(C_2^2\times S_3).\SL(2,8).C_3$ x 2, $(C_2^2\times D_9).\SL(2,8)$ x 2, $(C_2^2\times C_6).\SL(2,8).C_3$ x 2, $C_9^2.(D_4\times F_8)$, $(C_2^2\times C_{18}).\SL(2,8)$, $\SL(2,8)\times \SOPlus(4,2)$

Order 34992: $D_9^3:C_6$, $D_9\wr S_3$

Order 32928: $F_8.C_7^2.C_6.C_2$ x 2, $F_8.C_7^2.C_{12}$

Order 32256: $C_2^9:(C_{21}:C_3)$ x 2, $C_2^6.\SL(2,8)$, $C_2^5.C_2.\SL(2,8)$, $(C_2\times C_2^2\wr C_2).\SL(2,8)$

Order 31752: $C_{21}.\SL(2,8).C_3$, $C_3\times C_7.\SL(2,8).C_3$, $C_9\times \GL(2,8)$

Order 28224: $F_8.\SL(2,8)$ x 8, $C_2^6.C_7^2.C_9$ x 2, $C_2^2\times D_7.\SL(2,8)$ x 2, $C_2^6.C_7^2.C_3^2$, $(C_2^2\times C_{14}).\SL(2,8)$

Order 27216: $D_9.\SL(2,8).C_3$ x 9, $C_{18}.\SL(2,8).C_3$ x 3, $F_8.C_9^2.C_6$ x 2, $C_3:D_9.\SL(2,8)$ x 2, $(C_9\times S_3).\SL(2,8)$ x 2, $(C_3\times S_3).\SL(2,8).C_3$ x 2, $C_3\times S_3\times {}^2G(2,3)$ x 2, $F_8.(C_9\times D_9):C_3$, $C_9^2.(D_4\times F_7)$, $C_3\times D_9.\SL(2,8)$, $(C_3\times D_9).\SL(2,8)$, $(C_3^2\times \SL(2,8)):C_6$

Order 26244: $C_9^3:(C_3\times A_4)$, $C_9^3:(C_6\times S_3)$

Order 25088: $C_2^9.C_7^2$ x 13, $C_2^2\times C_2^3:F_8.C_{14}$ x 2, $F_8\times C_2^3:F_8$, $C_2^6.C_{14}^2.C_2$

Order 24696: $C_7^2.\SL(2,8)$, $C_7^3:(C_3\times S_4)$, $D_7^3:C_3^2$, $C_7^3:(C_3\times S_4)$

Order 24192: $C_2^6.C_{63}.C_6$ x 7, $C_2^4.\SL(2,8).C_3$ x 6, $F_8.C_9:A_4.C_2^2$, $C_2^6.C_9:F_7$, $(C_2^3\times S_3).\SL(2,8)$, $(C_2^2\times C_4).\SL(2,8).C_3$

Order 21952: $F_8\times C_7\times F_8$, $C_7^2.(D_4\times F_8)$

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

Order 21168: $D_7.\SL(2,8).C_3$ x 8, $C_{14}.\SL(2,8).C_3$ x 3, $F_8.C_{63}.C_6$, $F_8.C_9:F_7$, $D_{21}.\SL(2,8)$, $C_7^2.(D_4\times D_9:C_3)$, $(C_7\times S_3).\SL(2,8)$, $(C_7\times F_8).C_9.C_6$, $(C_3\times D_7).\SL(2,8)$

Order 18816: $C_2^6.C_7^2.C_6$ x 9, $F_8\wr C_2:C_3$ x 8, $F_8.C_7:A_4.C_2^2$, $C_2^6:C_7.F_7$, $C_2^6.C_{21}.C_{14}$, $C_2^3:F_8.C_{42}$, $(C_2^5\times C_6).C_7.C_{14}$

Order 18144: $D_{18}.\SL(2,8)$ x 4, $D_6\times {}^2G(2,3)$ x 3, $F_8.(S_3\times D_9:C_3)$ x 2, $C_9^2.F_8.C_2^2$ x 2, $(C_2\times C_6).\SL(2,8).C_3$ x 2, $S_3^2\times \SL(2,8)$ x 2, $C_9^2.F_8.C_4$, $(C_2\times C_{18}).\SL(2,8)$, $C_3^2:C_4\times \SL(2,8)$

Order 17496: $D_9^3:C_3$ x 2, $(C_9\times D_9^2):C_6$ x 2, $D_9\wr C_3$ x 2, $(C_9\times D_9^2):C_6$, $C_9^3:(C_2\times C_{12})$, $C_9^3:(C_3\times D_4)$, $C_9^3:S_4$, $C_9^3:S_4$

Order 16464: $C_2^3.C_7^3.C_6$ x 3, $D_7^3:C_6$, $D_7\wr S_3$

Order 16128: $C_2^5.\SL(2,8)$ x 2, $C_2^2\wr C_2.\SL(2,8)$ x 2, $C_9.C_2^6:C_7.C_2^2$, $C_2^6.C_{21}.D_6$, $C_2^6.C_{21}.C_6.C_2$, $(C_2^2\times D_4).\SL(2,8)$, $(C_2\times C_2^2:C_4).\SL(2,8)$, $S_3\times C_2^6:F_7$

Order 15552: $C_9:D_9.A_4.C_2^3$

Order 14112: $D_{14}.\SL(2,8)$ x 4, $F_8.(S_3\times F_7)$, $C_{63}.F_8.C_2^2$, $(C_2\times C_{14}).\SL(2,8)$

Order 13608: $C_9.\SL(2,8).C_3$ x 6, $(C_9\times C_{63}).C_6.C_2^2$ x 3, $C_9^2.C_2^2:F_7$ x 2, $C_3^2.\SL(2,8).C_3$ x 2, $(C_3\times C_9).\SL(2,8)$ x 2, $F_8.C_9^2.C_3$, $C_9^2.D_{28}:C_3$, $C_9^2.C_{14}.C_6.C_2$, $C_3^2\times {}^2G(2,3)$

Order 13122: $C_9^3:(C_3\times C_6)$, $C_9^3:(C_3\times S_3)$, $C_9^3:(C_3\times S_3)$

Order 12544: $C_2\times F_8\wr C_2$ x 2, $C_7.C_2^6:C_7.C_2^2$, $C_2^6.D_7^2$, $C_2^2\times F_8^2$

Order 12348: $C_7^3:(C_6\times S_3)$, $C_7^3:(C_3\times A_4)$

Order 12096: $C_2^3:{}^2G(2,3)$ x 13, $C_2^6.C_{21}.C_3^2$ x 4, $F_8.D_9:A_4$ x 2, $(C_2^2\times S_3).\SL(2,8)$ x 2, $F_8.C_9:A_4.C_2$, $(C_2^2\times C_6).\SL(2,8)$, $D_4\times {}^2G(2,3)$, $(S_3^2\times F_8):C_6$

Order 11664: $D_9^2:D_{18}$, $D_9^2:C_6\times S_3$

Order 10976: $C_7^2.F_8.C_2^2$ x 2, $C_7^2.F_8.C_4$

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

Order 10584: $C_7.\SL(2,8).C_3$ x 5, $C_7^2.C_{18}.C_6.C_2$ x 3, $F_8.C_{21}.C_3^2$, $C_7^2.D_{36}:C_3$, $C_7^2.D_9:C_3.C_2^2$, $C_7^2.C_9:C_3.C_2^3$, $C_7^2.(C_4\times D_9:C_3)$, $C_3\times \GL(2,8)$

Order 9408: $C_2^6.C_7^2.C_3$ x 23, $F_8.D_7:A_4$, $F_8.C_7:A_4.C_2$, $C_7:D_7.A_4.C_2^3$, $(C_7\times F_8).A_4.C_2$

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

Order 9072: $S_3\times {}^2G(2,3)$ x 8, $D_9.\SL(2,8)$ x 6, $\SL(2,8)\times D_9$ x 3, $C_{18}.\SL(2,8)$ x 3, $C_6.\SL(2,8).C_3$ x 3, $C_6\times {}^2G(2,3)$ x 3, $F_8.D_9:C_9$ x 2, $F_8.C_3^3.C_6$ x 2, $F_8.(C_3\times C_9).C_6$ x 2, $(C_9\times F_8).C_{18}$ x 2, $C_3\times S_3\times \SL(2,8)$ x 2, $F_8.C_9:C_3.C_6$, $C_9^2.F_8.C_2$, $C_9^2.(D_4\times D_7)$, $(C_9\times F_8).C_3.C_6$, $C_3:S_3\times \SL(2,8)$

Order 8748: $C_9^3:(C_2\times C_6)$ x 3, $C_9^3.C_6.C_2$ x 2, $C_9^3:(C_2\times C_6)$ x 2, $C_9^3:A_4$ x 2, $C_9^3:C_{12}$, $C_9^3:C_{12}$, $C_9^3:D_6$, $C_9^2.(C_3\times S_3^2)$

Order 8232: $D_7^3:C_3$ x 2, $D_7^2:F_7$ x 2, $D_7\wr C_3$ x 2, $C_2^3:(C_7^3:C_3)$, $C_7^3:(C_2\times C_{12})$, $C_7^3:(C_3\times D_4)$, $(C_7\times D_7^2):C_6$, $C_7^3:S_4$, $C_7^3:S_4$

Order 8064: $C_2^4.\SL(2,8)$ x 8, $C_2^6.C_{21}.C_6$ x 5, $C_9.C_2^6:C_7.C_2$ x 3, $C_9.C_2^3:F_8.C_2$ x 2, $C_2^2:C_4.\SL(2,8)$ x 2, $C_2\times D_4\times \SL(2,8)$ x 2, $C_2^6:C_{21}:C_6$ x 2, $F_8.(C_6\times A_4).C_2$, $C_2^6.D_{63}$, $C_2^6.D_7:C_9$, $C_2^6.C_{63}.C_2$, $C_2^6.C_{126}$, $(C_9\times F_8).C_2^4$, $(C_2^2\times C_4).\SL(2,8)$, $(C_2^5\times C_6):F_7$, $C_3\times C_2^6:F_7$, $C_2^6:C_{21}:C_6$

Order 7776: $C_9^2.(D_4\times A_4)$ x 4, $C_9^2.A_4.C_2^3$ x 2, $C_{18}^2.C_6.C_2^2$

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

Order 7056: $D_7.\SL(2,8)$ x 8, $C_{14}.\SL(2,8)$ x 3, $(C_7\times F_8).C_3.C_6$ x 2, $F_8.D_7:C_9$, $F_8.C_{21}.C_6$, $C_{63}.F_8.C_2$, $C_7^2.(D_4\times D_9)$, $(C_9\times F_8).C_{14}$, $(C_7\times F_8).C_{18}$, $D_7^2:C_6\times S_3$

Order 6912: $(C_2^4\times C_{18}).C_6.C_2^2$

Order 6804: $(C_9\times C_{63}).C_6.C_2$ x 7, $(C_9\times C_{63}).C_{12}$ x 2, $(C_3\times C_{63}).C_6^2$ x 2

Order 6561: $C_9^3:C_3^2$

Order 6272: $F_8\wr C_2$ x 8, $C_7.C_2^6:C_7.C_2$ x 3, $C_7.C_2^3:F_8.C_2$ x 2, $C_2^6.C_7^2.C_2$ x 2, $C_2^6:C_7.C_{14}$, $C_2^6.C_7:D_7$, $C_2^3\times (C_7\times F_8).C_2$, $C_2^3:F_8.C_{14}$

Order 6174: $C_7^3:(C_3\times S_3)$, $C_7^3:(C_3\times S_3)$, $C_7^3:(C_3\times C_6)$

Order 6048: $\SL(2,8):A_4$ x 5, $D_6\times \SL(2,8)$ x 4, $F_8.D_9:C_3.C_2$ x 3, $C_2^2\times {}^2G(2,3)$ x 3, $F_8:C_3\times S_3^2$ x 2, $F_8.C_9:A_4$, $C_{63}.A_4.C_2^3$, $C_9.(C_6\times F_8).C_2$, $C_3^2.(S_3\times F_8).C_2$, $(C_2\times C_6).\SL(2,8)$, $\SL(2,8):C_{12}$, $(C_3^2\times F_8):C_{12}$

Order 5832: $C_3^3.S_3^3$ x 3, $D_9^3$ x 2, $C_9^3:D_4$ x 2, $(C_3\times D_9^2):C_6$ x 2, $D_9^2:C_{18}$, $D_9^2:C_{18}$, $C_9^2:C_4\times D_9$, $C_9^2:D_{36}$, $S_3\times C_9^2:C_{12}$, $C_9^2:(C_3\times D_{12})$, $C_3\times D_9^2:C_6$

Order 5488: $F_8\times C_7\times D_7$, $C_7^2.F_8.C_2$, $(C_7\times F_8).C_{14}$, $D_7^3:C_2$

Order 5376: $F_8.C_2^4.C_6$ x 3, $C_2^3.F_8:A_4$ x 2, $C_2^3.F_8:A_4$ x 2, $F_8.(D_4\times A_4)$, $C_{14}.C_2^4:C_3.C_2^3$, $C_3.C_2^6:C_7.C_2^2$, $S_3\times C_2^6:D_7$, $C_2^4:F_8:C_6$, $C_2\wr F_7$

Order 5292: $C_7^2:(C_2\times D_9:C_3)$ x 7, $C_7^2:(C_2^2\times C_9:C_3)$ x 2, $C_7^2:(C_{18}.C_6)$, $C_7^2:(C_4\times C_9:C_3)$

Order 5184: $D_9^2.C_2^4$

Order 4704: $C_7^2.(D_4\times A_4)$ x 4, $(C_7\times F_8).C_6.C_2$ x 3, $C_7^2.A_4.C_2^3$ x 2, $C_{21}.F_8.C_2^2$, $C_{14}^2.C_6.C_2^2$, $(C_7\times F_8).A_4$

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

Order 4536: $C_3\times {}^2G(2,3)$ x 6, $C_9\times \SL(2,8)$ x 6, $C_3.\SL(2,8).C_3$ x 4, $F_8.C_9:C_9$ x 2, $(C_3\times F_8).C_3^3$ x 2, $F_8.C_9^2$, $C_{63}.C_6^2.C_2$, $C_{21}.C_3^3.C_2^3$, $C_9^2.D_{28}$, $C_9^2.D_{14}.C_2$, $C_7\times C_9^2:D_4$, $C_7.D_9^2.C_2$, $C_7.C_9:D_9.C_2^2$, $(C_9\times C_{63}).D_4$, $(C_9\times C_{63}).C_2^3$, $C_3^2\times \SL(2,8)$

Order 4374: $C_3^2.C_3^4.C_6$ x 3, $C_9^3:C_6$ x 2, $C_9^3.C_6$, $(C_3\times C_9).C_3^3.C_6$, $C_9^2.C_9:C_6$, $C_9^2.(S_3\times C_3^2)$, $C_9^2:C_3^2:S_3$, $C_9\wr S_3$, $C_9^2.(S_3\times C_3^2)$, $C_9^2.(S_3\times C_9)$, $C_9^3:S_3$

Order 4116: $C_7^3:(C_2\times C_6)$ x 3, $C_7^3:(C_2\times C_6)$ x 2, $C_7^3:A_4$ x 2, $C_7^3:(C_2\times C_6)$ x 2, $C_7^3:C_{12}$, $C_7^3:C_{12}$, $C_7^3:D_6$

Order 4032: $C_2^3\times \SL(2,8)$ x 17, $C_3\times C_2^6:C_7.C_3$ x 2, $C_2^6:C_7.C_9$ x 2, $C_2^6.C_{63}$ x 2, $C_2^3:F_8.C_9$ x 2, $(C_9\times F_8).C_2^3$ x 2, $C_2^6:C_{63}$ x 2, $S_3\times F_8:A_4$ x 2, $C_2^6:C_{21}:C_3$ x 2, $D_4\times \SL(2,8)$ x 2, $C_2^6:C_{21}:C_3$ x 2, $F_8.C_{18}.C_2^2$, $F_8.C_2^2:C_9.C_2$, $C_3\times F_8.A_4.C_2$, $F_8\times \SOPlus(4,2)$, $C_2\times C_4\times \SL(2,8)$

Order 3888: $C_{18}^2.C_6.C_2$ x 6, $C_{18}^2.C_{12}$ x 2, $C_2\times C_{18}^2.C_6$ x 2, $(C_6\times C_{18}).C_6^2$, $D_9^2:D_6$, $(D_9\times S_3^2):C_6$, $C_2\times D_9^2:C_6$, $C_3\times S_3\wr S_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 2304433152: $\SL(2,8)\wr C_3.C_6$ ($C_1$)

Order 1152216576: $\SL(2,8)^3.C_3^2$ ($C_2$)

Order 768144384: $\SL(2,8)\wr S_3$ ($C_3$)

Order 384072192: $\SL(2,8).\SL(2,8).\SL(2,8).C_3$ ($S_3$), $\SL(2,8)\wr C_3$ ($C_6$)

Order 128024064: $\SL(2,8).\SL(2,8).\SL(2,8)$ ($C_3\times S_3$)

Order 1: $C_1$ ($\SL(2,8)\wr C_3.C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 1 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 $149 \times 149$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $107 \times 107$ rational character table.