Abstract group 25015379558400.a: $\OmegaMinus(10,2)$

• Label: 250...400.a
• Order: \( 2^{20} \cdot 3^{6} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 17 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 17 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{21} \cdot 3^{6} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $495$
• Trans deg.: $495$
• Rank: $2$

Group information

Description:	$\OmegaMinus(10,2)$
Order:	\(25015379558400\)\(\medspace = 2^{20} \cdot 3^{6} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 17 \)
Exponent:	\(471240\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 17 \)
Automorphism group:	$\SOMinus(10,2)$, of order \(50030759116800\)\(\medspace = 2^{21} \cdot 3^{6} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 17 \)
Composition factors:	$\OmegaMinus(10,2)$
Derived length:	$0$

This group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	11	12	14	15	17	18	20	21	24	30	33	35
Elements	1	21999615	1234449152	24610037760	28043034624	279150009600	59560427520	1074879590400	694871654400	495096053760	1516083609600	2652817305600	893406412800	2029025230848	2942985830400	2084614963200	938076733440	1191208550400	1563461222400	2084614963200	3032167219200	1429450260480	25015379558400
Conjugacy classes	1	4	6	9	2	23	1	6	4	3	2	22	1	7	2	5	2	1	4	4	4	2	115
Divisions	1	4	5	9	2	18	1	6	3	3	1	17	1	5	1	3	2	1	3	3	1	1	91
Autjugacy classes	1	4	5	9	2	18	1	6	3	3	1	17	1	5	2	3	2	1	3	3	2	1	93

Minimal presentations

Permutation degree:	$495$
Transitive degree:	$495$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	154	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\OmegaMinus(10,2)$, $\POmegaMinus(10,2)$, $\SpinMinus(10,2)$
Permutation group:	Degree $495$ $\langle(1,3,9,19,38,67,115,93,53,29,13,5)(2,6,14,30,54,94,156,95,55,31,15,7)(4,10,21,41,72,125,199,139,81,47,25,11) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\OmegaMinus(10,2)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 2 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 $\OmegaMinus(10,2)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7.C_2^6.C_2^5.C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5:C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$

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 42301440 (order at least 591360) are shown, as well as all normal subgroups of any index.

Order 25015379558400: $\OmegaMinus(10,2)$

Order 50536120320: $C_2^8.\OmegaMinus(8,2)$

Order 47377612800: $\Sp(8,2)$

Order 1274019840: $C_2.C_2^{12}.S_3.\SU(4,2)$

Order 1045094400: $C_3.\OmegaPlus(8,2).C_2$

Order 990904320: $C_2^6.C_2^8.C_3.A_8$

Order 637009920: $C_2.C_2^{12}.C_3.\SU(4,2)$

Order 522547200: $C_3\times \OmegaPlus(8,2)$

Order 424673280: $C_2^8.C_2^6.\SU(4,2)$

Order 394813440: $\SOMinus(8,2)$

Order 371589120: $C_2^8.C(3,2)$

Order 348364800: $\SOPlus(8,2)$

Order 330301440: $C_2^{10}.C_2^4.A_8$, $C_2^3.C_2^{12}.A_5.\PSL(2,7)$

Order 239500800: $A_{12}$

Order 212336640: $C_2^7.C_2^6.\SU(4,2)$

Order 197406720: $\OmegaMinus(8,2)$

Order 185794560: $C_2^7.C(3,2)$

Order 174182400: $\OmegaPlus(8,2)$

Order 123863040: $C_2^6.C_2^8.C_3.A_7$

Order 66060288: $C_2^8.C_2^6.C_3.C_2^3.\PSL(2,7)$, $C_2^6.C_2^6.A_4.C_2^3.\PSL(2,7)$

Order 55050240: $C_2^3.C_2^{12}.D_5.\PSL(2,7)$

Order 47185920: $C_2^7.C_2^8.S_4.A_5$, $C_2^3.C_2^8.A_4.C_2^5.A_5$

Order 41287680: $C_2^{10}.C_2^4.A_7$, $C_2^3.C_2^{12}.C_7.C_3.A_5$

Order 41057280: $\Unitary(5,2)$

Order 35389440: $C_2^6.C_2^8.C_3.A_6.C_2$, $C_2^3.C_2^8.S_4.A_6.C_2$

Order 33030144: $C_2^3.C_2^6.C_3.C_2^6.C_2.\PSL(2,7)$

Order 31850496: $C_2.C_2^{12}.\He_3.C_6.A_4.C_2$, $C_2.C_2^{12}.C_3^3.(S_3\times S_4)$

Order 30965760: $A_4.C_2^6.A_8.C_2$

Order 28311552: $C_2^5.C_2^5.C_2^4.C_2^4.C_3^3.C_2^2$

Order 27525120: $C_2^3.C_2^{12}.C_5.\PSL(2,7)$

Order 23592960: $C_2^7.C_2^8.A_4.A_5$, $C_2^3.C_2^8.A_4.C_2^4.A_5$

Order 22020096: $C_2^9.C_2^6.C_2^2.\PSL(2,7)$, $C_2^8.C_2^6.C_2^3.\PSL(2,7)$

Order 20643840: $C_2^{10}.A_8$

Order 19958400: $A_{11}$

Order 17694720: $C_2^3.C_2^8.A_4.A_6.C_2$ x 2, $C_2^6.C_2^8.C_3^2.S_5$, $C_2^6.C_2^8.C_3.A_6$, $C_2^3.C_2^8.S_4.A_6$

Order 16515072: $C_2^3.C_2^6.C_3.C_2^6.\PSL(2,7)$

Order 15925248: $C_2.C_2^{12}.C_3^4.A_4.C_2$, $C_2.C_2^{12}.C_3^3.A_4.S_3$, $C_2.C_2^{12}.C_3^3.A_4.C_6$, $C_2.C_2^6.C_2^6.\He_3.Q_8.C_3^2$

Order 15728640: $C_2^7.C_2^6.C_2^5.A_5$

Order 15482880: $A_4.C_2^6.A_8$

Order 14155776: $C_2^6.C_2^6.C_2^6.C_3^3.C_2$, $C_2^5.C_2^5.C_2^4.C_3.A_4^2.C_2$, $C_2^5.C_2^5.C_2^4.C_2^4.C_3^3.C_2$

Order 13762560: $C_2^3.C_2^{12}.C_7.A_5$

Order 13685760: $\SU(5,2)$

Order 13271040: $C_2^8.\SU(4,2).C_2$

Order 11796480: $C_2^{10}.C_2^4.A_6.C_2$, $C_2^7.C_2^6.C_2.A_6.C_2$, $C_2^3.C_2^8.C_3.C_2^5.A_5$

Order 11010048: $C_2^9.C_2^4.C_2^3.\PSL(2,7)$

Order 10616832: $C_2^7.C_2^6.\He_3.Q_8.C_6$, $C_2^7.C_2^6.C_3^3.S_4.C_2$, $C_2.C_2^{12}.\He_3.D_6.C_2^2$, $C_2.C_2^8.C_3^2.A_4^2.C_2^3.C_2$

Order 10321920: $C_2^8.A_8.C_2$

Order 9437184: $C_2^8.C_2^6.A_4\wr C_2.C_2$, $C_2^8.C_2^4.C_2^6.S_3^2$, $C_2^7.C_2^4.C_2^4.A_4^2.C_2$, $C_2^5.C_2^6.C_2.A_4.C_2^4.C_6.C_2$

Order 8847360: $C_2^6.C_2^4.C_2^4.C_3^2.A_5$, $C_2^3.C_2^8.S_3.A_6.C_2$, $C_2^3.C_2^8.A_4.A_6$

Order 8709120: $S_3.C(3,2)$

Order 8257536: $C_2^8.C_2^6.C_3.\PSL(2,7)$ x 2, $C_2^8.C_2^3.C_2^6.C_7.C_3^2$, $C_2^6.C_2^6.F_8.(C_3\times A_4)$, $C_2^6.C_2^6.A_4.\PSL(2,7)$

Order 7962624: $C_2.C_2^{12}.\He_3.(C_6\times S_3)$, $C_2.C_2^{12}.C_3^3.A_4.C_3$

Order 7864320: $C_2^7.C_2^6.C_2^4.A_5$ x 2, $C_2^7.C_2^5.C_2^5.A_5$, $C_2^7.C_2^2.C_2^8.C_5.D_6$, $C_2^3.C_2^8.C_2^6.C_5.D_6$

Order 7741440: $(C_2^5\times C_6).A_8.C_2$

Order 7077888: $C_2^5.C_2^5.C_2^6.C_2^2.C_3^3$

Order 6881280: $C_2^3.C_2^{12}.C_{35}.C_6$

Order 6635520: $C_2^7.\SU(4,2).C_2$ x 2, $C_2^8.\SU(4,2)$

Order 5898240: $C_2^7.C_2^6.A_6.C_2$ x 6, $C_2^6.C_2^8.C_3.S_5$ x 2, $C_2^{10}.C_2^4.C_3.S_5$, $C_2^{10}.C_2^4.A_6$, $C_2^7.C_2^6.C_2.A_6$, $C_2^3.C_2^8.S_4.S_5$, $C_2^3.C_2^8.C_3.C_2^4.A_5$, $C_2.C_2^8.C_2^4.S_3.S_5$

Order 5505024: $C_2^9.C_2^6.\PSL(2,7)$

Order 5308416: $C_2^7.C_2^6.C_3^3.S_4$ x 3, $C_2.C_2^8.C_3^2.A_4^2.D_4$ x 3, $Q_8:C_2^2.C_2^4.C_2^4.C_3^4.C_2^3$, $C_2^7.C_2^6.\He_3.Q_8.C_3$, $C_2^7.C_2^6.C_3^3.A_4.C_2$, $C_2.C_2^{12}.\He_3.C_3.Q_8$, $C_2.C_2^{12}.C_3.C_3^3.C_2^2.C_2$, $C_2.C_2^8.C_3^2.C_2^4.C_6^2.C_2$, $C_2.C_2^8.C_2^4.C_3^4.C_2^3$, $C_2.C_2^8.A_4^2.C_3:S_3.C_2^2$, $C_2.C_2^6.C_2^6.\He_3.Q_8.C_3$, $C_2.C_2^6.C_2^6.\He_3.C_{12}.C_2$

Order 5160960: $C_2^7.A_8.C_2$ x 2, $C_2^8.A_8$

Order 4718592: $C_2^8.C_2^6.C_3.C_2^4.C_6$, $C_2^8.C_2^6.C_2^4.C_3.C_6$, $C_2^8.C_2^6.A_4^2.C_2$, $C_2^8.C_2^6.A_4\wr C_2$, $C_2^8.C_2^5.C_2^6.C_3^2$, $C_2^8.C_2^4.C_2^6.C_3.S_3$, $C_2^8.C_2^4.C_2^6.C_3.C_6$, $C_2^7.C_2^6.C_2^4.S_3^2$, $C_2^6.C_2^6.A_4.C_2^4.C_6$, $C_2^6.C_2^4.C_2^6.C_6^2.C_2$, $C_2^6.C_2^4.C_2^4.C_3.A_4.C_2^3$, $C_2^6.C_2^4.C_2^4.A_4\wr C_2$, $C_2^5.C_2^6.C_2.A_4.C_2^4.C_6$, $C_2^5.C_2^5.C_2^6.A_4.C_6$, $C_2.C_2^6.C_2^6.A_4.C_6.C_2^3$

Order 4423680: $C_2^3.C_2^8.C_3.A_6.C_2$ x 2, $C_2^3.C_2^8.S_3.A_6$

Order 4354560: $C_3\times C(3,2)$ x 2

Order 4128768: $C_2^3.C_2^6.C_3.C_2^4.\PSL(2,7)$ x 2, $C_2^6.C_2^6.F_8.C_3.C_6$

Order 3981312: $C_2.C_2^{12}.\He_3.C_3.C_6$, $C_2.C_2^{12}.C_3^4.C_6$, $C_2.C_2^6.C_2^6.\He_3.C_3.C_6$

Order 3932160: $C_2^7.C_2^4.C_2^4.(D_5\times A_4)$ x 2, $C_2^7.C_2^8.A_4.C_{10}$, $C_2^7.C_2^5.C_2^4.A_5$, $C_2^7.C_2^2.C_2^8.D_{15}$, $C_2^3.C_2^8.C_2^6.D_{15}$, $C_2^3.C_2^8.A_4.C_2^4.C_{10}$

Order 3870720: $(C_2^5\times C_6).A_8$ x 2, $(C_2^6\times A_4).S_7$

Order 3628800: $S_{10}$

Order 3538944: $C_2^7.C_2^6.\He_3.Q_8.C_2$, $C_2^6.C_2^8.C_3:S_3.C_6.C_2$, $C_2^6.C_2^4.C_2^4.C_3.A_4.C_6$, $C_2^5.C_2^5.C_2^6.C_3^3.C_2$, $C_2^5.A_4^2.C_2^4.D_6.C_2^2$, $C_2.C_2^8.C_6^2.(D_4\times S_4)$

Order 3440640: $C_2^3.C_2^{12}.C_{35}.C_3$

Order 3317760: $C_2\times C_2^6.\SU(4,2)$ x 2, $C_2^6.\SU(4,2).C_2$

Order 3145728: $C_2^8.C_2^6.C_2^4.D_6$, $C_2^7.C_2^6.C_6.C_2^3.C_2^3$, $C_2^7.C_2^6.C_2^4.D_6.C_2$, $C_2^7.C_2^5.C_2^5.C_6.C_2^2$

Order 3096576: $C_2^8.\SU(3,3).C_2$

Order 2949120: $C_2^7.C_2^5.A_6.C_2$ x 6, $C_2^7.C_2^6.A_6$ x 3, $C_2^3.C_2^8.A_4.S_5$ x 2, $Q_8:C_2^2.C_2^8.S_3.A_5$, $C_2^{10}.C_2^4.C_3.A_5$, $C_2^7.C_2^6.C_3.S_5$, $C_2^6.C_2^8.C_{15}.C_{12}$, $C_2^6.C_2^8.C_3.A_5$, $C_2^6.C_2^4.C_3.C_2^4.A_5$, $C_2^6.C_2^4.C_2^4.C_3.A_5$, $C_2^3.C_2^8.S_4.A_5$, $C_2.C_2^8.C_2^4.C_3.S_5$

Order 2903040: $C_2\times C(3,2)$

Order 2752512: $C_2^8.C_2^6.F_8.C_3$ x 4, $C_2^6.C_2^6.F_8.A_4$ x 2, $C_2^{10}.C_2^4.\PSL(2,7)$, $C_2^9.C_2^6.C_7.C_6.C_2$, $C_2^9.C_2^5.\PSL(2,7)$, $C_2^7.C_2^6.C_2.\PSL(2,7)$, $C_2^7.C_2^4.C_2^3.\PSL(2,7)$, $C_2^6.C_2^6.C_2^5.C_{21}$, $C_2^5.C_2^6.C_2^3.\PSL(2,7)$

Order 2654208: $Q_8:C_2^2.C_2^4.C_2^4.C_3^4.C_2^2$ x 3, $Q_8:C_2^2.C_2^8.C_3^4.C_2^2$ x 2, $C_2^7.C_2^6.C_3^3.A_4$ x 2, $Q_8:C_2^2.C_2^8.C_3^4.C_4$, $Q_8:C_2^2.C_2^6.A_4.C_3^2.D_6$, $Q_8.C_2^8.A_4.C_3^3.C_4$, $C_2^7.C_2^6.\He_3.C_6.C_2$, $C_2.C_2^{12}.\He_3.D_6$, $C_2.C_2^{12}.C_3.C_3^3.C_4$, $C_2.C_2^8.C_6^2.A_4.C_6.C_2$, $C_2.C_2^6.C_2^6.\He_3.C_{12}$

Order 2621440: $C_2^7.C_2^4.C_2^4.(D_4\times D_5)$

Order 2580480: $C_2^{10}.A_7$, $C_2^7.A_8$, $C_2^6.A_8.C_2$

Order 2419200: $A_5.S_8$

Order 2359296: $C_2^8.C_2^6.C_2^4.C_3^2$ x 2, $C_2^6.C_2^6.C_2^6.C_3^2$ x 2, $C_2^6.C_2^6.A_4^2.C_2^2$ x 2, $C_2.C_2^6.C_2^4.A_4^2.D_4$ x 2, $C_2^8.C_2^6.(S_3\times A_4).C_2$, $C_2^8.C_2^4.A_4^2.C_2^2$, $C_2^7.C_2^6.C_2^4.C_3.S_3$, $C_2^7.C_2^6.C_2^4.C_3.C_6$, $C_2^7.C_2^4.A_4^2.C_2^3$, $C_2^6.C_2^6.C_3.C_2^4.C_6.C_2$, $C_2^6.C_2^4.C_2^6.C_3.C_{12}$, $C_2^6.C_2^4.C_2^6.(C_6\times S_3)$, $C_2^6.C_2^4.C_2^4.(C_6\times A_4).C_2$, $C_2^5.C_2^6.C_3.C_2^6.C_6$, $C_2^5.C_2^5.C_2^4.A_4^2$, $C_2^5.C_2^5.C_2^4.(S_3\times A_4).C_2$, $C_2^3.C_2^6.C_2^6.S_3^2.C_2$, $C_2^3.C_2^6.A_4.C_2^5.C_6.C_2$, $C_2.C_2^6.C_2^6.A_4.C_3.D_4$, $C_2.C_2^6.C_2^6.(C_3:C_4\times S_4)$, $C_2.C_2^6.C_2^4.A_4^2.C_2^3$

Order 2293760: $C_2^3.C_2^{12}.C_{35}.C_2$

Order 2211840: $C_2^3.C_2^8.C_3.A_6$

Order 2088960: $C_2^8.\SL(2,16).C_2$

Order 2064384: $C_2^3.C_2^6.C_3.C_2^3.\PSL(2,7)$ x 4, $C_2^6.C_2^3.C_2^6.C_7.C_3^2$

Order 1990656: $C_2.C_2^6.C_2^6.\He_3.C_3^2$

Order 1966080: $C_2^{10}.C_2^4.S_5$ x 2, $C_2^7.C_2^6.C_2.S_5$ x 2, $C_2^7.C_2^6.C_2^4.C_{15}$, $C_2^7.C_2^4.C_2^6.C_{15}$, $C_2^7.C_2^4.C_2^4.A_5$, $C_2^3.C_2^8.C_2^4.C_5.D_6$

Order 1958400: $\PSigmaSp(4,4)$

Order 1935360: $A_4\times C_2^6:A_7$

Order 1866240: $S_3^2.\SU(4,2).C_2$

Order 1814400: $A_{10}$

Order 1769472: $C_2^5.A_4^2.C_2^4.C_3.D_4$ x 3, $C_2.C_2^8.C_6^2.A_4.D_4$ x 3, $C_2^7.C_2^6.\He_3.Q_8$ x 2, $C_2^6.C_2^6.A_4^2.C_3$ x 2, $C_2^7.C_2^6.\He_3.C_4.C_2$, $C_2^6.C_2^8.C_3^3.C_4$, $C_2^6.C_2^8.C_3^3.C_2^2$, $C_2^6.C_2^4.C_2^4.C_3^3.C_2^2$, $C_2^5.C_2^6.C_6^2.D_6.C_2$, $C_2^5.C_2^6.A_4.C_3^2.C_4.C_2$, $C_2^5.C_2^4.A_4^2.C_6.C_2^2$, $C_2^5.A_4^2.C_2^4.D_6.C_2$, $C_2^5.A_4^2.C_2^4.C_6.C_2^2$, $C_2.C_2^8.C_6^2.C_2^4:S_3$, $C_2.C_2^8.C_3^2.C_2^5.C_6.C_2$, $C_2.C_2^8.C_2^4.S_3^3$, $C_2.C_2^8.C_2^4.C_3^2.D_6.C_2$, $C_2.C_2^6.C_2^6.S_3^3$

Order 1658880: $C_2^6.\SU(4,2)$

Order 1572864: $C_2^8.C_2^6.A_4.C_2^3$ x 3, $C_2^8.C_2^6.C_2^4.S_3$ x 2, $C_2^8.C_2^6.A_4.C_4.C_2$ x 2, $C_2^7.C_2^6.C_2^4.D_6$ x 2, $C_2^8.C_2^6.C_2^4.C_6$, $C_2^8.C_2^6.A_4.D_4$, $C_2^8.C_2^6.(C_6\times D_4).C_2$, $C_2^8.C_2^5.C_2^6.C_3$, $C_2^8.C_2^5.A_4.C_2^4$, $C_2^8.C_2^5.(C_2\times D_4\times A_4)$, $C_2^7.C_2^6.C_6.C_2^5$, $C_2^7.C_2^6.C_6.C_2^4.C_2$, $C_2^7.C_2^6.C_6.C_2^3.C_2^2$, $C_2^7.C_2^6.C_6.C_2^2\wr C_2$, $C_2^7.C_2^6.C_6.C_2.D_4.C_2$, $C_2^7.C_2^6.C_6.C_2.C_2^4$, $C_2^7.C_2^6.C_2^4.C_6.C_2$, $C_2^7.C_2^5.C_2^4.C_6.C_2^2$, $C_2^6.C_2^6.C_2^6.C_6$, $C_2^6.C_2^4.C_6.C_2^4.C_2^4$

Order 1548288: $C_2^7.\SU(3,3).C_2$ x 2, $C_2^8.\SU(3,3)$

Order 1474560: $C_2^7.C_2^4.A_6.C_2$ x 3, $C_2^7.C_2^5.A_6$ x 2, $C_2^7.C_2^6.C_3.A_5$, $C_2^6.C_2^4.F_{16}.C_6$, $C_2^3.C_2^8.S_3.S_5$, $C_2^3.C_2^8.A_4.A_5$, $C_2.C_2^8.S_4.S_5$

Order 1451520: $\Sp(6,2)$ x 2

Order 1376256: $C_2^9.C_2^4.\PSL(2,7)$ x 3, $C_2^9.C_2^6.C_7.C_6$, $C_2^7.C_2^6.\PSL(2,7)$, $C_2^6.C_2^6.C_2.\PSL(2,7)$, $C_2^3.C_2^6.C_3.C_2^6.C_{14}$

Order 1327104: $C_2^7.C_2^6.\He_3.C_6$ x 3, $Q_8:C_2^2.C_2^4.C_2^4.C_3^4.C_2$ x 2, $C_2.C_2^6.C_2^6.\He_3.C_6$ x 2, $Q_8:C_2^2.C_2^4.C_6^2.A_4.C_6$, $Q_8:C_2^2.C_2^4.C_3.C_2^4.C_3^3.C_2$, $Q_8.C_2^6.C_2^4.C_3^4.C_2$, $C_2^7.C_2^6.C_3^3.C_6$, $C_2.C_2^{12}.\He_3:S_3$, $C_2.C_2^{12}.C_3^3.C_6$, $C_2.C_2^{12}.C_3.C_9.C_6$, $C_2.C_2^8.C_6^2.C_3^2.D_4$

Order 1310720: $C_2^7.C_2^4.C_2^4.D_{10}.C_2$ x 3, $C_2^7.C_2^5.C_2^4.C_{10}.C_2$, $C_2^7.C_2^4.C_2^4.D_{20}$, $C_2^7.C_2^4.C_2^4.C_5.D_4$, $C_2.C_2^6.C_2^4.C_2^4.C_5.D_4$

Order 1290240: $C_2^6.A_8$ x 2, $C_2^8.S_7$

Order 1209600: $A_5\times A_8$

Order 1179648: $C_2^8.C_2^4.A_4^2.C_2$ x 3, $C_2^5.C_2^6.A_4^2.C_2^2$ x 3, $C_2^8.C_2^6.A_4.C_6$ x 2, $C_2^8.C_2^6.(S_3\times A_4)$ x 2, $C_2^7.C_2^6.(S_3\times S_4)$ x 2, $C_2^6.C_2^6.C_3.C_2^4.C_6$ x 2, $C_2^6.C_2^6.A_4^2.C_2$ x 2, $C_2^6.C_2^4.C_2^4.(S_3\times A_4)$ x 2, $C_2^3.C_2^6.C_2^6.S_3^2$ x 2, $C_2^{10}.A_4^2.D_4$, $C_2^9.C_2^6.S_3^2$, $C_2^9.C_2^6.C_3.D_6$, $C_2^9.C_2^4.(S_3\times A_4).C_2$, $C_2^8.C_2^6.S_3^2.C_2$, $C_2^8.C_2^6.C_3:S_4$, $C_2^8.C_2^4.A_4\wr C_2$, $C_2^7.C_2^6.C_2^4.C_3^2$, $C_2^7.C_2^6.A_4^2$, $C_2^7.C_2^6.A_4.D_6$, $C_2^7.C_2^4.A_4^2.C_2^2$, $C_2^7.C_2^3.C_2^4.(S_3\times A_4)$, $C_2^6.C_2^6.A_4\wr C_2$, $C_2^6.C_2^6.(S_3\times S_4).C_2$, $C_2^6.C_2^4.C_2^4.C_6.C_6.C_2$, $C_2^5.C_2^6.C_6.C_2^4.C_6$, $C_2^5.C_2^6.(C_6\times A_4).C_2^3$, $C_2^5.C_2^5.C_2^2.A_4^2.C_2$, $C_2^5.C_2^4.(C_2\times A_4^2).C_2^3$, $C_2^5.A_4^2.C_2^5.C_2^3$, $C_2^3.C_2^6.C_2^6.C_6^2$, $C_2^3.C_2^6.C_2^6.C_3.D_6$, $C_2^3.C_2^6.A_4^2.C_2^3.C_2$, $C_2.C_2^6.C_2^5.A_4^2.C_2$, $C_2.C_2^6.C_2^4.A_4^2.C_4$, $C_2.C_2^6.C_2^4.A_4^2.C_2^2$, $C_2.C_2^6.A_4.C_2^6.C_{12}$, $C_2.C_2^6.A_4.C_2^6.C_6.C_2$

Order 1146880: $C_2^3.C_2^{12}.C_{35}$

Order 1105920: $A_4^2.C_2^6.S_5$, $A_4.C_2^6.C_2.A_6.C_2$

Order 1088640: $C_3:S_9$

Order 1048576: $C_2^7.C_2^6.C_2^5.C_2^2$

Order 1044480: $C_2^8.\SL(2,16)$

Order 1036800: $S_6\wr C_2$

Order 1032192: $C_2^6.C_2^6.C_7.A_4.C_3$, $(C_2^3\times A_4).C_2^6.\PSL(2,7)$

Order 983040: $C_2^7.C_2^6.S_5$ x 9, $C_2^{10}.C_2^4.A_5$ x 2, $C_2^7.C_2^6.C_2.A_5$ x 2, $D_4.C_2^5.C_2^5.S_5$, $C_2^{10}.F_{16}.C_4$, $C_2^7.C_2^8.D_{15}$, $C_2^7.C_2^4.C_2^4.C_5.C_6$, $C_2^6.C_2^8.C_{15}.C_4$, $C_2^6.C_2^6.C_2.S_5$, $C_2^5.C_2^6.C_2^2.S_5$, $C_2^3.C_2^8.C_3.C_2^4.C_{10}$, $C_2.C_2^8.C_2^4.(S_3\times F_5)$

Order 979200: $\Sp(4,4)$

Order 967680: $(C_2^5\times C_6).S_7$

Order 933120: $C_3:S_3.\SU(4,2).C_2$ x 2, $S_3^2.\SU(4,2)$

Order 917504: $C_2^9.C_2^6.C_{14}.C_2$, $C_2^8.C_2^6.F_8$

Order 884736: $C_2.C_2^8.C_6^2.C_6.C_2^3$ x 4, $C_2^5.A_4^2.C_2^4.C_6.C_2$ x 3, $C_2.C_2^6.C_2^6.C_3^2.D_6$ x 3, $Q_8:C_2^2.C_2^8.C_3^3.C_2^2$ x 2, $Q_8:C_2^2.C_2^4.A_4^2.C_6.C_2$ x 2, $Q_8.C_2^6.C_2^4.C_3^2.D_6$ x 2, $C_2^7.C_2^6.\He_3.C_4$ x 2, $C_2^7.C_2^6.C_3^3.C_4$ x 2, $C_2^7.C_2^6.C_3^2.D_6$ x 2, $C_2.C_2^8.C_6^2.(C_6.D_4)$ x 2, $Q_8:C_2^2.C_2^8.C_3^2.D_6$, $Q_8:C_2^2.C_2^4.C_6^2.A_4.C_2^2$, $Q_8:C_2^2.A_4^2.C_2^4.C_6.C_2$, $Q_8.C_2^8.C_3^3.C_2^3.C_2$, $Q_8.C_2^4.C_2^4.C_3^3.C_2^3.C_2$, $C_2^6.C_2^4.C_2^4.C_3^3.C_2$, $C_2^6.C_2^4.A_4^2.C_6$, $C_2^5.C_2^5.C_3.A_4^2.C_2$, $C_2^5.C_2^4.C_3.C_2^4.C_6^2$, $C_2^5.C_2^4.C_3.A_4^2.C_2^2$, $C_2^5.C_2^4.C_2^4.C_3^3.C_2^2$, $C_2^5.C_2^4.C_2^4.C_3.C_6^2$, $C_2^5.C_2^4.A_4^2.C_6.C_2$, $C_2^5.A_4^2.A_4.C_2^3.C_2$, $C_2^3.C_2^8.A_4.C_3^2.C_4$, $C_2^3.C_2^6.C_2^4.C_3^3.C_2^2$, $C_2^3.C_2^6.A_4^2.C_6.C_2$, $C_2.C_2^8.C_6^2.A_4.C_4$, $C_2.C_2^8.C_6^2.A_4.C_2^2$, $C_2.C_2^6.C_2^6.C_3^3.C_2^2$

Order 786432: $C_2^8.C_2^6.A_4.C_2^2$ x 5, $C_2^8.C_2^5.A_4.C_2^3$ x 5, $C_2^7.C_2^6.C_6.C_2^4$ x 5, $C_2^8.C_2^6.C_6.C_2^3$ x 3, $C_2^6.C_2^4.C_6.C_2^4.C_2^3$ x 3, $C_2^5.C_2^5.C_6.C_2^4.C_2^3$ x 3, $C_2^9.C_2^6.C_6.C_2^2$ x 2, $C_2^8.C_2^5.A_4.C_2^2.C_2$ x 2, $C_2^7.C_2^6.C_6.C_2^2:C_4$ x 2, $C_2^7.C_2^6.C_2^4.S_3$ x 2, $C_2^7.C_2^6.C_2^2:D_{12}$ x 2, $C_2^7.C_2^5.A_4.C_2^4$ x 2, $C_2^5.C_2^4.C_6.C_2^5.C_2^3$ x 2, $C_2^9.C_2^4.D_6.C_2^3$, $C_2^8.C_2^6.C_6.C_2^2.C_2$, $C_2^8.C_2^6.C_2^4.C_3$, $C_2^8.C_2^6.A_4.C_4$, $C_2^8.C_2^5.A_4.C_4.C_2$, $C_2^7.C_2^6.C_6.C_4.C_2^2$, $C_2^7.C_2^6.C_6.C_2^3.C_2$, $C_2^7.C_2^6.C_2^4.C_6$, $C_2^7.C_2^6.C_2^3:D_6$, $C_2^7.C_2^6.A_4.D_4$, $C_2^7.C_2^4.C_6.C_2^4.C_2^2$, $C_2^6.C_2^6.C_2^6.C_3$, $C_2^5.C_2^6.A_4.C_2^4.C_2$, $C_2^5.C_2^6.A_4.C_2^3.C_2^2$, $C_2^5.C_2^6.A_4.C_2.C_2^4$, $C_2^5.C_2^5.C_6.C_2^6.C_2$, $C_2^5.C_2^5.C_6.C_2^5.C_2^2$, $C_2^5.C_2^5.C_2^4.D_6.C_2^2$, $C_2^5.C_2^4.C_6.C_2^4.C_2^4$, $C_2^3.C_2^5.A_4.C_2^5.C_2^3$, $C_2.C_2^6.C_2^6.C_2^3:D_6$, $C_2.C_2^6.C_2^6.A_4.D_4$, $C_2.C_2^6.C_2^5.A_4.C_2^4$, $C_2.C_2^6.A_4.C_2^5.C_2^4$, $C_2.C_2^6.A_4.C_2^4.C_2^5$

Order 774144: $C_2^7.\SU(3,3)$

Order 737280: $C_2^3.C_2^8.S_3.A_5$ x 2, $C_2^3.C_2^8.C_3.S_5$ x 2, $C_2.C_2^8.A_4.S_5$ x 2, $C_2^7.C_2^4.C_3.S_5$, $C_2^7.C_2^4.A_6$, $C_2^6.C_2^4.F_{16}.C_3$, $C_2.C_2^8.S_4.A_5$, $C_2^{10}.S_6$, $C_2^{10}.S_6$

Order 688128: $C_2^9.C_2^3.\PSL(2,7)$ x 3, $C_2^9.C_2^6.C_7.C_3$, $C_2^6.C_2^6.\PSL(2,7)$, $C_2^3.C_2^6.C_2^6.C_7.C_3$, $C_2^2.D_4:C_2.C_2^6.F_8.C_3$

Order 663552: $C_2^7.C_2^6.C_3.\He_3$ x 2, $C_2.C_2^6.C_2^6.C_3^3.C_3$ x 2, $C_2.C_2^6.C_2^6.C_3.\He_3$ x 2, $Q_8.C_2^8.C_3^4.C_4$, $Q_8.C_2^6.A_4^2.C_3^2$, $C_2.C_2^8.C_6^2.C_6^2$, $C_2.C_2^8.C_3^4.C_2^3.C_2$, $C_2.C_2^6.C_2^4.C_3^3.D_6$

Order 655360: $C_2^7.C_2^4.C_2^4.D_{10}$ x 3, $C_2^7.C_2^4.C_2^4.C_{10}.C_2$ x 2, $C_2^7.C_2^4.C_2^4.C_{20}$, $C_2^7.C_2^4.C_2^4.C_5.C_4$

Order 645120: $C_2^8.A_7$, $C_2^6.A_7:C_4$, $C_2^7.S_7$

Order 466560: $C_3:S_3^4:S_5$

Order 729: $C_3^5:C_3$

Order 25: $C_5^2$

Order 17: $C_{17}$

Order 11: $C_{11}$

Order 7: $C_7$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 25015379558400: $\OmegaMinus(10,2)$ ($C_1$)

Order 1: $C_1$ ($\OmegaMinus(10,2)$)

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

Character theory

Complex character table

The $115 \times 115$ character table is not available for this group.

Rational character table

The $91 \times 91$ rational character table is not available for this group.