Abstract group 22932357120.f: $C_2^{12}.((C_2\times C_6^5).A_6)$

• Label: 22932357120.f
• Order: \( 2^{21} \cdot 3^{7} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{23} \cdot 3^{8} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_2^{12}.((C_2\times C_6^5).A_6)$
Order:	\(22932357120\)\(\medspace = 2^{21} \cdot 3^{7} \cdot 5 \)
Exponent:	\(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \)
Automorphism group:	Group of order \(275188285440\)\(\medspace = 2^{23} \cdot 3^{8} \cdot 5 \)
Composition factors:	$C_2$ x 18, $C_3$ x 5, $A_6$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	9	10	12	15	18	24	30
Elements	1	1921023	18632672	343388160	47775744	2144948256	1672151040	471121920	3009871872	4269957120	1528823808	2926264320	1911029760	4586471424	22932357120
Conjugacy classes	1	107	17	112	2	891	20	8	46	280	4	224	4	12	1728
Divisions	1	107	13	112	1	606	20	4	23	155	1	112	2	3	1160
Autjugacy classes	1	74	10	77	1	374	12	4	11	123	1	56	2	2	748

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Permutation group:	Degree $36$ $\langle(1,8,19,26,15,5,11,23,28,13,4,9,21,29,18)(2,7,20,25,16,6,12,24,27,14,3,10,22,30,17) \!\cdots\! \rangle$
Transitive group:	36T111396				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$C_2^{18}$ . $(C_3^5:A_6)$	$C_2^{17}$ . $(C_2\times C_3^5:A_6)$	$(C_2^{12}.(C_6^5.A_6))$ . $C_2$	$(C_2^{12}.C_3^5.C_2^6)$ . $A_6$	all 13

Elements of the group are displayed as permutations of degree 36.

Homology

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

Subgroups

There are 18 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	a subgroup isomorphic to $C_2^{12}.(C_6^5.A_6)$
Frattini:	a subgroup isomorphic to $C_2^5$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

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 324 (order at least 70778880) are shown, as well as all normal subgroups of any index.

Order 22932357120: $C_2^{12}.((C_2\times C_6^5).A_6)$

Order 11466178560: $C_2^{12}.(C_6^5.A_6)$

Order 3822059520: $C_2^{12}.C_3^5.C_2^6.A_5$, $C_2^{12}.((C_2\times C_6^5).A_5)$

Order 2293235712: $C_2^{12}.(C_6^4.(C_3\times C_6^2):C_4)$

Order 1911029760: $C_2^{12}.C_3^5.C_2^5.A_5$ x 3, $C_2^{12}.(C_6^5.A_5)$

Order 1528823808: $C_2^8.C_6^3.C_2^6.C_6^3.C_2$, $C_2^{12}.(C_3^5.C_2\wr S_4)$

Order 1274019840: $C_2^{10}.C_3^4.C_2^6.C_2^2.A_5$

Order 1146617856: $C_2^{12}.(C_2\times C_3^5:(A_4^2:C_4))$ x 2, $C_2^{12}.C_6^2.C_6^2.C_3^3.C_2^3$

Order 955514880: $C_2^{12}.C_3^5.C_2^4.A_5$, $C_2\times C_2^{10}.C_3^5.C_2^5.A_5$

Order 764411904: $C_2^8.C_6^3.C_2^6.C_3.C_6^2.C_2$ x 3, $C_2^8.C_3^3.C_2.C_2^6.A_4.C_6^2.C_2$ x 3, $C_2^{10}.A_4^2.C_6^3:S_4$ x 2, $C_2^8.C_6^3.C_2^6.C_6^3$, $C_2^{12}.(C_6^4.C_6^2:C_4)$, $C_2^{12}.(C_3^5.C_2\wr A_4)$

Order 716636160: $C_2\times C_2^{12}.C_3^5:A_6$

Order 637009920: $C_2^{10}.C_3^4.C_2^6.C_2.A_5$ x 7, $C_2^{10}.C_3^4.C_2^4.C_5.A_4.C_2^3$

Order 573308928: $C_2^{12}.C_6^2.C_6^2.\He_3.C_2^2$ x 2, $C_2^{12}.C_6^2.C_6^2.C_3^3.C_2^2$ x 2, $C_2^{12}.C_3^5:(A_4^2:C_4)$ x 2, $C_2^{12}.C_3^4.C_2^4.C_3.C_6^2$

Order 509607936: $C_2^8.C_6^3.C_2^6.C_6^2.C_2^2$ x 3, $C_2^{12}.(C_3^4.C_2\wr S_4)$ x 3, $C_2^8.C_6^3.A_4.C_2^5.C_6.C_2^2$, $C_2^8.C_3^2.C_2^5.(C_2\times C_6^3).C_2^4$

Order 477757440: $C_2^5.(C_2^{10}.C_3\wr A_5)$ x 3

Order 382205952: $C_2^8.C_6^3.C_2^6.C_3.C_6^2$ x 5, $C_2^{12}.C_6^2.C_6^2.C_6^2.C_2$ x 2, $C_2^8.C_6^3.C_2^6.C_3^3.C_2^2$ x 2, $C_2^8.C_3^3.C_2.C_2^6.(C_6\times C_3:S_4)$ x 2, $C_2^{12}.C_2^5:(\He_3^2:C_4)$ x 2, $C_2^8.C_6^3.C_2^6.C_3^3.C_4$, $C_2^8.C_3^3.C_2^3.C_2^6.C_3^3.C_4$, $C_2^6:C_3.C_2^6.C_6^3.C_6^2.C_2^2$, $C_2^{12}.(C_6^4.C_3:S_4)$, $C_2^9.A_4^2.C_6^3:S_4$, $C_2^{10}.A_4^2.C_6^3:A_4$

Order 358318080: $C_2^{12}.C_3^5:A_6$

Order 318504960: $C_2^{10}.C_3^4.C_2^6.A_5$ x 10, $C_2^{10}.C_3^4.C_2^4.C_5.A_4.C_2^2$ x 6, $C_2\times C_2^{10}.C_3^4.C_2^5.A_5$ x 3, $C_2^{10}.C_3^4.C_2^6.C_{30}.C_2$

Order 286654464: $C_2^{12}.C_6^2.C_6^2.C_3^3.C_2$ x 2, $C_2^{12}.C_3^4.C_2^4.C_3^3.C_2$ x 2, $C_2^{12}.C_6^2.C_6^2.\He_3.C_2$, $C_2^{12}.C_6^2.C_3^3.C_6^2.C_2$

Order 283115520: $C_2^{12}.(C_3.C_2\wr A_6)$ x 6, $C_2^{10}.(A_4\times C_2\wr A_6)$ x 3

Order 254803968: $C_2^8.C_6^3.C_2^6.C_6^2.C_2$ x 22, $C_2^{12}.C_2^5:C_3\wr S_4$ x 6, $C_2^8.C_3^2.C_2^5.(C_2\times C_6^3).C_2^3$ x 4, $C_2^8.C_6^3.A_4.C_2^5.C_6.C_2$ x 3, $C_2^{12}.(C_3^4.C_2\wr A_4)$ x 3, $C_2^8.C_6^3.A_4.C_2^4.C_6.C_2^2$ x 2, $C_2^8.C_3^3.C_2^5.C_2^4.C_6^2.C_2$ x 2, $C_2^8.C_3^2.C_2^5.C_6^3.C_2^4$ x 2, $C_2^8.C_3^2.C_2^5.C_6^3.C_2.C_2^3$ x 2, $C_2^{12}.C_6^3.C_6^2.C_2^3$, $C_2^{12}.C_3^4.C_2^4.C_6.C_2^3$, $C_2^8.C_6^3.C_2^4.C_6^2.C_2^3$, $C_2^8.C_6^3.A_4^2.C_2^5$, $C_2^8.C_3^3.Q_8.C_2^6.C_6^2.C_2$, $C_2^8.C_3^3.C_2^3.A_4.C_2^5.C_6.C_2$, $C_2^8.C_3^2.C_2^4.C_6^2.C_2^2.C_6.C_2^3$, $C_2^8.C_3^2.C_2^4.(C_2\times C_6^3).C_2^4$, $C_2^8.C_3^2.C_2^4.(C_2\times C_6^3).C_2^3.C_2$, $C_2^8.(C_2\times C_6^3).C_6.C_2^3.C_6.C_2^3$

Order 238878720: $C_2^{12}.C_3^5.C_2^2.A_5$, $C_2^{10}.C_6^4:\GL(2,4)$

Order 212336640: $C_2^3\times C_2^{10}.C_3^4.C_2^4.D_{10}$

Order 191102976: $C_2^6:C_3.C_2^6.C_6^3.C_6^2.C_2$ x 12, $C_2^{12}.C_6^2.C_6^2.(C_6\times S_3)$ x 6, $C_2^8.C_6^3.C_2^6.C_3^3.C_2$ x 6, $C_2^{12}.C_6^2.C_6^2.C_3.D_6$ x 4, $C_2^{12}.((C_3^2\times C_6^3).S_4)$ x 4, $C_2^{12}.(C_3^2\times C_6^3):S_4$ x 4, $C_2^{12}.C_3^3.C_2^4.C_3.C_6^2$ x 3, $C_2^6:C_3.C_2^6.C_3.C_6^2.C_6^2.C_2^2$ x 2, $C_2^{12}.(C_3^2\times C_6^3):S_4$ x 2, $C_2^{12}.((C_3^2\times C_6^3).S_4)$ x 2, $C_2^{12}.C_3^4:(A_4^2:C_4)$ x 2, $C_2^{12}.C_6^2.C_6^2.C_6^2$, $C_2^8.C_6^3.C_2^4.C_6^3$, $C_2^8.C_3^3.Q_8.C_2^4.C_3.C_6^2.C_2$, $C_2^8.C_3^3.C_2^6.C_6^3.C_2$, $C_2^8.C_3^3.C_2^3.C_2^6.C_3^3.C_2$, $C_2^6.C_6^2.C_2^6.C_3^4.C_2^4$, $C_2^{12}.(C_6^4.(C_3\times A_4))$, $C_2^{12}.(C_6^4.(C_3\times A_4))$, $C_2^{12}.(C_6^4.(C_3\times A_4))$

Order 169869312: $C_2^8.C_6^3.C_2^6.C_6.C_2^3$ x 3, $C_2^{12}.A_4^3:S_4$ x 3, $C_2^8.C_6^3.A_4.C_2^6.C_2^2$, $C_2^8.C_3^2.C_2^5.C_6^2.C_2^5.C_2$, $C_2^8.C_3^2.C_2^5.(C_6^2\times D_4).C_2^3$

Order 159252480: $C_2\times C_2^{10}.C_6^4:A_5$ x 7, $C_2^{10}.C_3^4.C_2^4.(D_5\times A_4)$ x 4, $C_2^{10}.C_3^4.C_2^6.C_{30}$ x 3, $C_2\times C_2^{10}.C_3^4.C_2^4.C_{30}.C_2$

Order 143327232: $C_2^{12}.C_6^2.C_3^4.C_6.C_2$ x 6, $C_2^{12}.C_3^4.C_2^4.C_3^3$, $C_2^{12}.C_3^3.C_6^2.C_6^2$, $C_2^{12}.C_3^5:(C_6^2:C_4)$

Order 141557760: $C_2^{12}.((C_2^4\times C_6).A_6)$ x 6, $C_2^{10}.(A_4\times C_2^5:A_6)$ x 3

Order 127401984: $C_2^8.C_6^3.C_2^6.C_6^2$ x 41, $C_2^8.C_6^3.C_2^4.C_6^2.C_2^2$ x 15, $C_2^8.C_3^2.C_2^5.C_6^3.C_2^3$ x 8, $C_2^{12}.C_6^3.C_6^2.C_2^2$ x 6, $C_2^{12}.(C_2\times C_6^3).C_6^2.C_2$ x 6, $C_2^{12}.C_3.C_6^3.C_6.C_2^3$ x 5, $C_2^8.C_6^3.C_2^6.(C_6\times S_3)$ x 5, $C_2^8.C_6^3.A_4^2.C_2^4$ x 5, $C_2^8.C_6^3.C_2^6.C_3.D_6$ x 4, $C_2^8.C_3^2.C_2^4.C_6^3.C_2^4$ x 4, $C_2^8.C_3^2.C_2^4.C_6^3.C_2.C_2^3$ x 4, $C_2^8.C_3^2.C_2^4.C_6^2.C_6.C_2^4$ x 4, $C_2^{12}.C_2^4:\He_3.C_6^2.C_2$ x 3, $C_2^8.C_6^3.A_4.C_2^4.C_6.C_2$ x 3, $C_2^8.C_3^3.C_2.C_2^6.(C_6\times S_4)$ x 3, $C_2^8.C_3^2.C_2^4.C_6^2.C_6.C_2.C_2^3$ x 3, $C_2^6:C_3.C_2^6.C_6^3.C_6.C_2^3$ x 3, $C_2^{12}.C_6^4:S_4$ x 3, $C_2^{12}.(C_6^4.S_4)$ x 3, $C_2^{12}.C_2^5:C_3\wr A_4$ x 3, $C_2^{12}.C_3^2.C_6^3.C_2^4$ x 2, $C_2^8.C_6^3.C_2^6.C_3^2.C_4$ x 2, $C_2^8.C_6^3.C_2^6.C_3.C_{12}$ x 2, $C_2^8.C_6^3.A_4.C_2^4.C_{12}$ x 2, $C_2^8.C_3^3.C_2^6.C_6^2.C_2^3$ x 2, $C_2^8.C_3^3.C_2.C_2^6.A_4.C_{12}$ x 2, $C_2^8.C_3^2.C_2^4.C_6^3.C_4^2$ x 2, $C_2^8.C_3^2.C_2^4.C_6^3.C_2^3.C_2$ x 2, $C_2^8.C_3^2.C_2^4.C_6^2.C_6.C_4^2$ x 2, $C_2^8.(C_2\times C_6^3).(C_6^2\times D_4).C_2^2$ x 2, $C_2^6:C_3.C_2^6.C_3.(C_6^2\times A_4).C_2^3$ x 2, $C_2^{12}.C_3^2.C_6^3.C_2^3.C_2$, $C_2^8.C_6^3.A_4^2.C_2^3.C_2$, $C_2^8.C_6^3.A_4.C_2.A_4.C_2^3$, $C_2^8.C_6^3.(C_6\times A_4).C_2^5$, $C_2^8.C_6^2.C_2^4.C_6^3.C_2^2$, $C_2^8.C_3^3.Q_8.A_4.C_2^4.C_6.C_2$, $C_2^8.C_3^2.C_2^4.C_6^2.C_6.C_2^3.C_2$, $C_2^8.C_3.C_6^2.C_2^5.C_6^2.C_2^2$, $C_2^6:C_3.C_2^6.C_6^2.C_3.A_4.C_2^3$, $C_2^6:C_3.C_2^6.C_3.C_6^3.C_2^4$, $C_2^6.C_6^2.C_3.C_2^3.A_4^2.C_2^4$, $C_2\times C_2^8.C_6^3.A_4^2.C_2^3$

Order 119439360: $C_2^{12}.C_3^5.C_2.A_5$ x 3, $C_2^{12}.(C_2\times C_3^5:A_5)$ x 2

Order 106168320: $C_2^2\times C_2^{10}.C_3^4.C_2^4.D_{10}$ x 9, $C_2\times C_2^{10}.C_3^4.C_2^4.C_5.C_2^3$ x 3, $C_2^3\times C_2^{10}.C_3^4.C_2^4.D_5$ x 2, $C_2^3\times C_2^{10}.C_3^4.C_2^4.C_{10}$

Order 95551488: $C_2^6:C_3.C_2^6.C_6^3.C_6^2$ x 16, $C_2^6:C_3.C_2^6.C_3.C_6^2.C_6^2.C_2$ x 16, $C_2^{12}.C_3.C_6^2.C_6^2.C_6$ x 8, $C_2^8.C_6^3.C_2^4.C_3.C_6^2$ x 8, $C_2^6.C_6^2.C_2^6.C_3.C_6^3$ x 7, $C_2^{12}.C_6^2.C_6^2.C_3.C_6$ x 6, $C_2^8.C_3^3.C_2^6.C_3.C_6^2.C_2$ x 5, $C_2^{12}.((C_3^3\times C_6^2).S_4)$ x 4, $C_2^{12}.(C_3^3\times C_6^2):S_4$ x 4, $C_2^{12}.C_6^2.C_6^2.C_3.S_3$ x 3, $C_2^{12}.C_6^2.C_3.\He_3.C_2^3$ x 2, $C_2^{12}:C_3^2.C_6^3:A_4$ x 2, $C_2^{12}.((C_3^2\times C_6^3).A_4)$ x 2, $C_2^{12}.C_6^2.C_3^4.C_2^3$, $C_2^{12}.C_6^2.C_3^2.C_6^2.C_2$, $C_2^{12}.C_3^3.C_2^4.C_3^3.C_2$, $C_2^8.C_6^3.C_2^6.C_3^3$, $C_2^8.C_6^3.C_2^2.C_6^3.C_2$, $C_2^8.C_3^3.Q_8.C_2^4.C_3^3.C_4$, $C_2^8.C_3^3.C_2^6.C_6^3$, $C_2^8.C_3^3.C_2^6.C_3^3.C_2^3$, $C_2^8.C_3^3.C_2^5.(C_6\times C_3:S_4)$, $C_2^6:C_3.C_2^6.C_3^2.C_6^3.C_2^2$, $C_2^6:C_3.C_2^4.C_6^3.C_6^2.C_2^2$, $C_2^{12}.((C_3^2\times C_6^3).A_4)$, $C_2^{12}.(C_3^2\times C_6^3):A_4$, $C_2^{12}.((C_3^2\times C_6^3).A_4)$

Order 94371840: $C_2^{18}.A_6$ x 3

Order 84934656: $C_2^8.C_6^3.A_4.C_2^6.C_2$ x 14, $C_2^8.C_6^3.C_2^6.C_6.C_2^2$ x 10, $C_2^8.C_6^3.C_2^6.C_3.D_4$ x 8, $C_2^8.C_6^3.C_2^5.C_6.C_2^3$ x 6, $C_2^8.C_3^2.C_2^5.C_6^2.C_2^4.C_2$ x 6, $C_2^8.C_3^2.C_2^5.(C_6^2\times D_4).C_2^2$ x 6, $C_2^{12}.(C_2^2\times C_6^3):S_4$ x 6, $C_2^5.(C_2^{12}.(C_6\times \He_3):C_4)$ x 6, $C_2^8.C_3^2.C_2^4.(C_6^2\times D_4).C_2^3$ x 4, $C_2^8.C_6^3.C_2^4.C_6.C_2^4$ x 3, $C_2^{12}.A_4^3:A_4$ x 3, $C_2^{10}.A_4\wr C_4$ x 3, $C_2^{12}.C_6^3.C_6.C_2^4$ x 2, $C_2^{12}.C_6^3.C_6.C_2^3.C_2$ x 2, $C_2^8.C_6^2.C_2^5.C_6^2.C_2^3$ x 2, $C_2^8.C_6^2.C_2^3.C_6^2.C_2^4.C_2$ x 2, $C_2^8.C_6^2.A_4.C_6.C_2^6.C_2$ x 2, $C_2^8.C_3^2.C_2^5.C_6.C_2.C_6.C_2^4$ x 2, $C_2^{12}.C_3^4.C_2^4.C_2^3.C_2$, $C_2^8.C_6^3.A_4.C_2^5.C_2^2$, $C_2^8.C_6^2.C_2^2.C_6^2.C_2^4.C_2^2$, $C_2^8.C_6^2.A_4.C_6.C_2^5.C_2^2$, $C_2^8.C_3^3.Q_8.A_4.C_2^6.C_2$, $C_2^8.C_3^3.C_2^6.C_6.C_2^5$, $C_2^8.C_3^3.C_2^6.A_4.C_2^4$, $C_2^8.C_3^3.C_2^2.C_2^6.C_6.C_2^3$, $C_2^8.C_3^3.C_2.C_2^5.A_4.C_2^4$, $C_2^8.(C_2\times C_6^3).C_2^5.C_6.C_2^2$

Order 79626240: $C_2^{10}.C_6\wr D_5$ x 6, $C_2^{10}.C_3^4.C_2^6.C_{15}$, $C_2^{10}.C_3^4.C_2^4.A_5$, $C_2\times C_2^{10}.C_3^4.C_2^4.C_{30}$, $C_2^{10}.C_6^4:A_5$

Order 71663616: $C_2^{12}.C_6^2.C_3^4.C_6$ x 4, $C_2^{12}.C_3^4.C_6^2.C_6$ x 2, $C_2^{12}.\He_3^2:(C_6:C_4)$ x 2, $C_2^{12}.C_3^4.C_3^3.C_2^3$, $C_2^{12}.C_3^3.C_6^2.C_3.C_6$

Order 70778880: $C_2^{10}.(C_3\times C_2\wr A_6)$ x 3

Order 63700992: $C_2^{12}.C_3^5.C_2^6$

Order 31850496: $C_2^{12}.C_3^5.C_2^5$

Order 1990656: $C_2^{12}.C_3^5.C_2$

Order 995328: $C_2^{12}.C_3^5$

Order 786432: $C_2^{12}.C_6.C_2^5$

Order 393216: $C_2^{12}.C_6.C_2^4$

Order 262144: $C_2^{18}$

Order 131072: $C_2^{17}$

Order 24576: $C_2^{12}.C_6$

Order 12288: $C_2^{12}.C_3$

Order 8192: $C_2^{13}$

Order 4096: $C_2^{12}$

Order 64: $C_2^6$

Order 32: $C_2^5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 22932357120: $C_2^{12}.((C_2\times C_6^5).A_6)$ ($C_1$)

Order 11466178560: $C_2^{12}.(C_6^5.A_6)$ ($C_2$)

Order 63700992: $C_2^{12}.C_3^5.C_2^6$ ($A_6$)

Order 31850496: $C_2^{12}.C_3^5.C_2^5$ ($C_2\times A_6$)

Order 1990656: $C_2^{12}.C_3^5.C_2$ ($C_2^5:A_6$)

Order 995328: $C_2^{12}.C_3^5$ ($C_2\wr A_6$)

Order 786432: $C_2^{12}.C_6.C_2^5$ ($C_3^4:A_6$)

Order 393216: $C_2^{12}.C_6.C_2^4$ ($(C_3^3\times C_6).A_6$)

Order 262144: $C_2^{18}$ ($C_3^5:A_6$)

Order 131072: $C_2^{17}$ ($C_2\times C_3^5:A_6$)

Order 24576: $C_2^{12}.C_6$ ($(C_2\times C_6^4).A_6$)

Order 12288: $C_2^{12}.C_3$ ($(C_2^2\times C_6^4).A_6$)

Order 8192: $C_2^{13}$ ($C_6^5.A_6$)

Order 4096: $C_2^{12}$ ($(C_2\times C_6^5).A_6$)

Order 64: $C_2^6$ ($C_2^{12}.C_3^5:A_6$)

Order 32: $C_2^5$ ($C_2\times C_2^{12}.C_3^5:A_6$)

Order 2: $C_2$ ($C_2^{12}.C_3^5.C_2^5.A_6$)

Order 1: $C_1$ ($C_2^{12}.((C_2\times C_6^5).A_6)$)

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 4 larger groups in the database.

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

Character theory

Complex character table

The $1728 \times 1728$ character table is not available for this group.

Rational character table

The $1160 \times 1160$ rational character table is not available for this group.