Abstract group 68797071360.d: $C_2^{12}.(C_6^6.A_6)$

• Label: 68797071360.d
• Order: \( 2^{21} \cdot 3^{8} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{23} \cdot 3^{8} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_2^{12}.(C_6^6.A_6)$
Order:	\(68797071360\)\(\medspace = 2^{21} \cdot 3^{8} \cdot 5 \)
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 6, $A_6$
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	30598640	343388160	47775744	4765293072	1672151040	1196052480	3009871872	13496647680	3822059520	10694799360	9077391360	20639121408	68797071360
Conjugacy classes	1	93	39	94	2	2547	12	30	30	1010	16	722	28	80	4704
Divisions	1	93	23	94	1	1403	12	15	15	520	4	361	14	20	2576
Autjugacy classes	1	74	22	77	1	965	12	13	11	407	4	179	14	14	1794

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 $36$ $\langle(1,20,32,30,6,21,34,27,4,24,36,26,2,19,31,29,5,22,33,28,3,23,35,25)(7,15,11,17,9,14) \!\cdots\! \rangle$
Transitive group:	36T115299				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$C_2^{18}$ . $(C_3\wr A_6)$	$C_2^{12}$ . $(C_6^6.A_6)$	$C_2^{17}$ . $(C_2\times C_3\wr A_6)$	$C_2^6$ . $(C_2^{12}.C_3\wr A_6)$	all 18

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

Homology

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

Subgroups

There are 24 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 256 (order at least 268738560) are shown, as well as all normal subgroups of any index.

Order 68797071360: $C_2^{12}.(C_6^6.A_6)$

Order 34398535680: $C_2^{12}.((C_3\times C_6^5).A_6)$

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

Order 11466178560: $C_2^{12}.C_3^5.C_6.C_2^5.A_5$, $C_2^{12}.(C_6^5.A_6)$, $C_2^{12}.(C_6^6.A_5)$

Order 6879707136: $C_2^{12}.(C_6^4.(C_3\times C_6^2):C_{12})$

Order 5733089280: $C_2^{12}.C_3^5.C_3.C_2^5.A_5$ x 3, $C_2^{12}.((C_3\times C_6^5).A_5)$

Order 4586471424: $C_2^8.C_6^3.C_2^6.C_3.C_6^3.C_2$, $C_2^{12}.C_6\wr A_4.C_2$

Order 3822059520: $C_2^{12}.C_3^5.C_2^6.A_5$ x 3, $C_2^{10}.C_3^5.C_2^6.C_2^2.A_5$, $C_2^{12}.((C_2\times C_6^5).A_5)$

Order 3439853568: $C_2^{12}.(C_2\times C_6^4.C_3\wr C_4)$ x 2, $C_2^{12}.C_6^2.C_6^2.C_3^4.C_2^3$

Order 2866544640: $C_2^{12}.C_3.C_3^5.C_2^4.A_5$, $C_2^{10}.C_3^5.C_6.C_2^5.A_5$

Order 2293235712: $C_2^8.C_6^3.C_2^6.C_3.C_6.C_6^2$ x 2, $C_2^8.C_3^3.C_2.C_2^6.A_4.C_6.C_6^2$ x 2, $C_2^5.(C_2^{12}.C_3\wr S_4)$ x 2, $C_2^8.C_6^3.C_2^6.C_3^4.C_2^3$, $C_2^8.C_6^3.C_2^6.C_3.C_6^3$, $C_2^8.C_3^3.C_2.C_2^6.A_4.C_3.C_6^2.C_2$, $C_2^{12}.(C_6^4.(C_3\times C_6^2):C_4)$, $C_2^{12}.C_6\wr A_4$, $C_2^{12}.(C_6^4.C_6^2:C_{12})$

Order 2149908480: $C_2^{12}.(C_2\times C_3\wr A_6)$

Order 1911029760: $C_2^{12}.C_3^5.C_2^5.A_5$ x 9, $C_2^2\times C_2^{10}.C_3^5.C_2^5.A_5$ x 4, $C_2^{10}.C_3^5.C_2^6.C_2.A_5$ x 3, $C_2^{10}.C_3^4.C_2^4.C_5.A_4.C_6.C_2^2$, $C_2^{12}.(C_6^5.A_5)$

Order 1719926784: $C_2^{12}.(C_2\times C_6^3).C_3^3.C_6^2$ x 2, $C_2^{12}.(C_6^4.C_3\wr C_4)$ x 2, $C_2^{12}.C_6^2.C_6^2.\He_3.C_6.C_2$, $C_2^{12}.C_6^2.C_6^2.C_3^4.C_2^2$, $C_2^{12}.C_3^4.C_2^4.C_3^4.C_2^2$

Order 1528823808: $C_2^8.C_6^3.C_2^6.C_6^3.C_2$ x 5, $C_2^8.C_6^3.A_4.C_2^5.C_6^2.C_2$, $C_2^8.C_3^3.C_2^5.C_2^4.C_6^3.C_2$, $C_2^8.C_3^2.C_2^5.C_6^2.C_6^2.C_2^4$, $C_2^{12}.(C_3^5.C_2\wr S_4)$

Order 1433272320: $C_2^{10}.C_3^5.C_3.C_2^5.A_5$ x 3

Order 1274019840: $C_2^{10}.C_3^4.C_2^3.C_2^5.A_5$

Order 1146617856: $C_2^8.C_6^3.C_2^6.C_3^4.C_2^2$ x 7, $C_2^{12}.C_6^2.C_6^2.\He_3.C_2^3$ x 2, $C_2^{12}.(C_2\times C_3^5:(A_4^2:C_4))$ x 2, $C_2^{12}.(C_2^5.\He_3^2:C_{12})$ x 2, $C_2^{12}.C_6^2.C_6^2.C_3^3.C_2^3$, $C_2^{12}.(C_2\times C_6^3).C_3.\He_3.C_2^3$, $C_2^8.C_6^3.C_2^6.C_3^4.C_4$, $C_2^8.C_3^3.C_2^5.C_2^4.C_3^4.C_4$, $C_2^8.C_3^3.C_2^3.C_2^6.C_3^4.C_2^2$, $C_2^8.C_3^3.C_2.C_2^6.A_4.C_3.C_6^2$, $C_2^6:C_3.C_2^6.C_6^3.C_6^3.C_2$, $C_2^{12}.(C_6^3.C_6\wr S_3)$, $C_2^{12}.(C_6^3.C_6\wr S_3)$, $C_2^5.(C_2^{12}.C_3\wr A_4)$

Order 1074954240: $C_2^{12}.C_3\wr A_6$

Order 955514880: $C_2^{10}.C_3^5.C_2^6.A_5$ x 16, $C_2^{12}.C_3^5.C_2^4.A_5$ x 3, $C_2^{10}.C_3^4.C_2^4.C_5.A_4.C_6.C_2$ x 3, $C_2^{10}.C_3^4.C_2^4.C_5.C_6\times C_2\times A_4$ x 2, $C_2^{10}.C_3^4.C_2^4.C_{30}\times C_2\times A_4$, $C_2^{10}.C_3^4.C_2^4.C_{30}.C_2\times A_4$

Order 859963392: $C_2^{12}.C_3^4.C_2^4.C_3^4.C_2$ x 2, $C_2^{12}.(C_2\times C_6^3).C_3^4.C_6$ x 2, $C_2^{12}.C_6^2.C_6^2.C_3^4.C_2$, $C_2^{12}.C_6^2.C_3^4.C_6^2.C_2$

Order 764411904: $C_2^8.C_6^3.C_2^6.C_6^3$ x 22, $C_2^8.C_6^3.C_2^6.C_3.C_6^2.C_2$ x 14, $C_2^8.C_6^3.C_2^6.C_6.C_6^2$ x 6, $C_2^8.C_6^3.C_2^6.C_3^3.C_2^3$ x 4, $C_2^8.C_6^3.A_4.C_2^5.C_6^2$ x 4, $C_2^8.C_3^3.C_2^5.C_2^4.C_3.C_6^2.C_2$ x 4, $C_2^8.C_3^2.C_2^5.C_6^3.C_6.C_2^3$ x 4, $C_2^8.C_3^2.C_2^5.C_6^2.C_6^2.C_2^3$ x 4, $C_2^8.C_6^3.C_2^4.C_6^3.C_2^2$ x 2, $C_2^8.C_6^3.A_4.C_2^4.C_6^2.C_2$ x 2, $C_2^8.C_3^3.Q_8.C_2^6.C_3.C_6^2.C_2$ x 2, $C_2^8.C_3^3.C_2^5.C_2^4.C_6.C_6^2$ x 2, $C_2^8.C_3^3.C_2.C_2^6.A_4.C_6^2.C_2$ x 2, $C_2^8.C_3^2.C_2^4.C_6^2.C_6^2.C_2^4$ x 2, $C_2^8.C_3^2.C_2^4.C_6^2.C_6^2.C_2^3.C_2$ x 2, $C_2^{10}.A_4^2.C_6^3:S_4$ x 2, $C_2^{12}.C_6^3.C_6^3.C_2^2$, $C_2^{12}.C_3.C_6^3.C_6^2.C_2^3$, $C_2^8.C_3^3.C_2^5.C_2^4.C_3^3.C_2^3$, $C_2^8.C_3^3.C_2^3.C_2^6.C_3.C_6^2.C_2$, $C_2^{12}.(C_6^4.C_6^2:C_4)$, $C_2^{12}.(C_3^5.C_2\wr A_4)$, $C_2^{12}.(C_3^5.C_2\wr A_4)$

Order 716636160: $C_2^{12}.C_3^4.C_6^2.A_5$, $C_2^{10}.C_3^5.C_3.C_2^4.A_5$, $C_2\times C_2^{12}.C_3^5:A_6$

Order 637009920: $C_2^{10}.C_3^4.C_2^6.C_2.A_5$ x 6, $C_2^{10}.C_3^4.C_2^4.C_5.A_4.C_2^3$ x 3, $C_2^{10}.C_3^4.C_2^4.C_{30}.C_2^4$, $C_2^{10}.C_3^4.C_2.C_2^6.A_5$

Order 573308928: $C_2^6:C_3.C_2^6.C_6^3.C_6^3$ x 12, $C_2^{12}.C_3^4.C_2^4.C_3.C_6^2$ x 10, $C_2^{12}.C_6^2.C_6^2.C_3^3.C_2^2$ x 8, $C_2^8.C_6^3.C_2^6.C_3^4.C_2$ x 7, $C_2^{12}.C_6^2.C_6^2.C_3.C_6^2$ x 6, $C_2^{12}.(C_2\times C_6^3).C_3^3.C_6.C_2$ x 5, $C_2^3.(C_2^{12}.C_3\wr S_4)$ x 4, $C_2^3.(C_2^{12}.C_3\wr S_4)$ x 4, $C_2^{12}.C_3.C_6^2.C_6^2.C_6^2$ x 2, $C_2^6:C_3.C_2^6.C_3.C_6^2.C_6^3.C_2$ x 2, $C_2^3.(C_2^{12}.C_3\wr S_4)$ x 2, $C_2^3.(C_2^{12}.C_3\wr S_4)$ x 2, $C_2^{12}.C_3^5:(A_4^2:C_4)$ x 2, $C_2^{12}.(C_6^4.C_3^2:C_{12})$ x 2, $C_2^{12}.C_6^2.C_6^2.\He_3.C_2^2$, $C_2^8.C_6^3.C_2^4.C_3.C_6^3$, $C_2^8.C_3^3.C_2^6.C_3.C_6^3.C_2$, $C_2^8.C_3^3.C_2^5.(C_6\times A_4).C_6^2$, $C_2^6.C_6^2.C_2^6.C_3^5.C_2^4$, $C_2^{12}.(C_6^3.C_6\wr C_3)$, $C_2^{12}.(C_6^3.C_6\wr C_3)$, $C_2^{12}.(C_6^3.C_6\wr C_3)$

Order 509607936: $C_2^8.C_6^3.C_2^6.C_6^2.C_2^2$ x 7, $C_2^8.C_3^2.C_2^5.(C_2\times C_6^3).C_2^4$ x 4, $C_2^8.C_6^3.A_4.C_2^5.C_6.C_2^2$ x 3, $C_2^8.C_3^3.C_2.C_2^6.A_4.C_6.C_2^3$ x 3, $C_2^8.C_6^3.A_4.C_2^3.C_6.C_2^4$, $C_2^8.C_6^2.C_2^3.(C_2\times C_6^3).C_2^4$, $C_2^{12}.(C_3^4.C_2\wr S_4)$

Order 477757440: $C_2^5.(C_2^{10}.C_3\wr A_5)$ x 13, $C_2^{10}.C_3^4.C_2^4.C_5.A_4.C_6$ x 4, $C_2^{10}.C_3^5.C_2^5.A_5$ x 3, $C_2^{10}.C_3^4.C_2^6.C_{15}.C_6$, $C_2^{10}.C_3^4.C_2^4.C_{30}\times A_4$, $C_2^{10}.C_3^4.C_2^4.C_{15}\times C_2\times A_4$, $C_2^{10}.C_3^4.C_2^4.(C_6^2\times D_5)$

Order 429981696: $C_2^{12}.C_6^2.C_3^4.C_6^2$ x 6, $C_2^{12}.C_3^4.C_6^2.C_6^2$, $C_2^{12}.C_3^4.C_2^4.C_3^4$, $C_2^{12}.(C_3^5.C_6^2:C_{12})$

Order 382205952: $C_2^8.C_6^3.C_2^6.C_3.C_6^2$ x 139, $C_2^8.C_6^3.C_2^4.C_6^3.C_2$ x 22, $C_2^8.C_6^3.C_2^6.C_3^3.C_2^2$ x 12, $C_2^6:C_3.C_2^6.C_6^3.C_6^2.C_2^2$ x 12, $C_2^8.C_6^3.C_2^6.C_3^3.C_4$ x 8, $C_2^8.C_3^2.C_2^5.C_6^3.C_6.C_2^2$ x 8, $C_2^8.C_3^2.C_2^4.C_6^2.C_6^2.C_2^3$ x 8, $C_2^{12}.C_6^3.C_6^3.C_2$ x 6, $C_2^{12}.C_6^2.C_6^2.C_6^2.C_2$ x 6, $C_2^8.C_6^3.A_4.C_2^4.C_6^2$ x 4, $C_2^8.C_3^3.C_2^3.C_2^6.C_3.C_6^2$ x 4, $C_2^8.(C_2\times C_6^3).C_6^3.C_2^3.C_2$ x 4, $C_2^{12}.C_3^3.C_6^3.C_2^3.C_2$ x 3, $C_2^{12}.C_3.C_6^3.C_6^2.C_2^2$ x 3, $C_2^8.C_3^2.C_2^4.C_6^3.C_6.C_2^3$ x 3, $C_2^{12}.C_3^3.C_6^3.C_2^4$ x 2, $C_2^8.C_6^3.A_4.C_2^4.C_3.C_{12}$ x 2, $C_2^8.C_3^3.C_2^6.C_6^3.C_2^2$ x 2, $C_2^8.C_3^3.C_2^5.C_2^4.C_3.C_6^2$ x 2, $C_2^8.C_3^3.C_2^5.A_4^2.C_6.C_2$ x 2, $C_2^8.C_3^3.C_2^3.C_2^6.C_3^3.C_4$ x 2, $C_2^8.C_3^3.C_2.C_2^6.A_4.(C_6\times S_3)$ x 2, $C_2^8.(C_2\times C_6^3).C_6^3.C_2^4$ x 2, $C_2^8.(C_2\times C_6^3).C_6.C_6^2.C_4^2$ x 2, $C_2^8.(C_2\times C_6^3).C_6.C_6^2.C_2^4$ x 2, $C_2^{12}.C_2^5:(\He_3^2:C_4)$ x 2, $C_2^{12}.C_6^2.A_4.\He_3.C_2^3$, $C_2^8.C_6^3.C_2^2.C_6^3.C_2^3$, $C_2^8.C_6^3.A_4.C_2^3.C_6^2.C_2$, $C_2^8.C_6^2.C_2^4.C_3^4.C_2^5$, $C_2^8.C_3^3.Q_8.C_2^6.C_3^3.C_4$, $C_2^8.C_3^3.Q_8.C_2^6.C_3^3.C_2^2$, $C_2^8.C_3^3.C_2^5.C_2^4.C_3^3.C_4$, $C_2^8.C_3^3.C_2^3.C_2^6.C_3^3.C_2^2$, $C_2^8.C_3^3.C_2.C_2^6.A_4.C_6^2$, $C_2^8.C_3^2.C_2^4.C_6^3.C_3.C_4^2$, $C_2^8.C_3.C_6^2.C_2^5.C_6^3.C_2$, $C_2^8.(C_2\times C_6^3).C_6^3.C_4^2$, $C_2^8.(C_2\times C_6^3).C_6^3.C_2.C_2^3$, $C_2^6:C_3.C_2^6.C_6^2.C_6^3.C_2^2$, $C_2^6:C_3.C_2^6.C_3^2.C_6^3.C_2^4$, $C_2^6:C_3.C_2^6.C_3^2.C_2^3.C_6^3.C_2$, $C_2^6:C_3.C_2^4.C_2^4.C_3.C_6^2.C_6^2.C_2$, $C_2^6.C_6^2.C_3.C_2^5.C_2^2.C_6^3.C_2$, $C_2^{12}.(C_6^4.C_3:S_4)$, $C_2^9.A_4^2.C_6^3:S_4$, $C_2^{10}.A_4^2.C_6^3:A_4$, $C_2^{10}.A_4^2.C_6^3:A_4$

Order 358318080: $C_2^{12}.C_3.C_3^5.C_2.A_5$ x 2, $C_2^{12}.C_3^5.C_6.A_5$ x 2, $C_2^{12}.C_3^5.C_6.A_5$, $C_2^{12}.C_3^5:A_6$

Order 318504960: $C_2^{10}.C_3^4.C_2^4.C_5.A_4.C_2^2$ x 15, $C_2^{10}.C_3^4.C_2^6.A_5$ x 13, $C_2^2\times C_2^{10}.C_3^4.C_2^4.C_{30}.C_2$ x 9, $C_2\times C_2^{10}.C_3^4.C_2^4.C_{30}.C_2^2$ x 3, $C_2^{10}.C_3^4.C_2^6.C_{30}.C_2$ x 2, $C_2^{10}.C_3^4.C_2^4.D_5\times C_2\times A_4$ x 2, $C_2^3\times C_2^{10}.C_3^4.C_2^4.C_5.C_6$ x 2, $C_2^{10}.C_3^4.C_2^4.D_{10}\times A_4$, $C_2^{10}.C_3^4.C_2^4.C_{10}\times C_2\times A_4$, $C_2^3\times C_2^{10}.C_3^4.C_2^4.C_{30}$

Order 286654464: $C_2^{12}.C_3^4.C_2^4.C_3^3.C_2$ x 28, $C_2^6:C_3.C_2^6.C_6^3.C_3.C_6^2$ x 16, $C_2^6:C_3.C_2^6.C_3.C_6^2.C_6^3$ x 10, $C_2^{12}.C_6^2.C_3^3.C_6^2.C_2$ x 7, $C_2^6.C_6^2.C_2^6.C_3^5.C_2^3$ x 7, $C_2^{12}.C_3^3.C_2^4.C_3^4.C_2$ x 6, $C_2^6:C_3.C_2^6.C_3^2.C_6^2.C_6^2.C_2$ x 6, $C_2^{12}.C_6^2.C_6^2.C_3^3.C_2$ x 5, $C_2^{12}.(C_2\times C_6^3).\He_3.C_6$ x 4, $C_2^8.C_3^3.C_2^6.C_3.C_6.C_6^2$ x 4, $C_2^{10}.A_4^2.C_3\wr S_4$ x 4, $C_2^{10}.A_4^2.C_3\wr S_4$ x 4, $C_2^{12}.(C_2\times C_6^3).C_3^3.C_6$ x 3, $C_2^8.C_6^3.C_2^4.C_3^4.C_2^2$ x 3, $C_2\times C_2^8.C_6^3.C_2^4.C_3^4.C_2$ x 3, $C_3\times C_2^8.C_6^3.C_2^4.C_3.C_6^2$ x 2, $C_2^{12}.C_6^2.C_6^2.\He_3.C_2$ x 2, $C_2^3.(C_2^{12}.C_3\wr A_4)$ x 2, $C_2^3.(C_2^{12}.C_3\wr A_4)$ x 2, $C_2^8.C_6^3.C_2^6.C_3^4$, $C_2^8.C_6^3.(C_6\times A_4).C_6^2.C_2$, $C_2^8.C_3^3.Q_8.C_2^4.C_3^4.C_2^2$, $C_2^8.C_3^3.C_2^6.C_3^4.C_2^3$, $C_2^8.C_3^3.C_2^6.C_3^4.C_2^2.C_2$, $C_2^8.C_3^3.C_2^6.C_3.C_6^3$, $C_2^8.C_3^3.C_2.C_2^6.C_3^4.C_4$, $C_2^6:C_3.C_2^6.C_3^3.C_6^3.C_2^2$, $C_2^6:C_3.C_2^4.C_6^3.C_6^3.C_2$, $C_2^3.(C_2^{12}.C_3\wr A_4)$, $C_2^3.(C_2^{12}.C_3\wr A_4)$, $C_2^3.(C_2^{12}.C_3\wr A_4)$

Order 283115520: $C_2^{12}.(C_3.C_2\wr A_6)$ x 2, $C_2^{10}.(A_4\times C_2\wr A_6)$

Order 191102976: $C_2^{12}.C_3^5.C_6.C_2^5$

Order 95551488: $C_2^{12}.C_3^5.C_6.C_2^4$

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

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

Order 5971968: $C_2^{12}.C_3^5.C_6$

Order 2985984: $C_2^{12}.C_3^5.C_3$

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 68797071360: $C_2^{12}.(C_6^6.A_6)$ ($C_1$)

Order 34398535680: $C_2^{12}.((C_3\times C_6^5).A_6)$ ($C_2$)

Order 22932357120: $C_2^{12}.((C_2\times C_6^5).A_6)$ ($C_3$)

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

Order 191102976: $C_2^{12}.C_3^5.C_6.C_2^5$ ($A_6$)

Order 95551488: $C_2^{12}.C_3^5.C_6.C_2^4$ ($C_2\times A_6$)

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

Order 31850496: $C_2^{12}.C_3^5.C_2^5$ ($C_6\times A_6$)

Order 5971968: $C_2^{12}.C_3^5.C_6$ ($C_2^5:A_6$)

Order 2985984: $C_2^{12}.C_3^5.C_3$ ($C_2\wr A_6$)

Order 1990656: $C_2^{12}.C_3^5.C_2$ ($(C_2^4\times C_6).A_6$)

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

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

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

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

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

Order 24576: $C_2^{12}.C_6$ ($C_6^5.A_6$)

Order 12288: $C_2^{12}.C_3$ ($(C_2\times C_6^5).A_6$)

Order 8192: $C_2^{13}$ ($(C_3\times C_6^5).A_6$)

Order 4096: $C_2^{12}$ ($C_6^6.A_6$)

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

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

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

Order 1: $C_1$ ($C_2^{12}.(C_6^6.A_6)$)

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 3 larger groups in the database.

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

Character theory

Complex character table

The $4704 \times 4704$ character table is not available for this group.

Rational character table

The $2576 \times 2576$ rational character table is not available for this group.