Abstract group 279936.jm: $C_3^6.C_2^5:A_4$

• Label: 279936.jm
• Order: \( 2^{7} \cdot 3^{7} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3^{2} \)
• Perm deg.: $24$
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$C_3^6.C_2^5:A_4$
Order:	\(279936\)\(\medspace = 2^{7} \cdot 3^{7} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_6^6.A_4^2.C_6.C_2^3$, of order \(322486272\)\(\medspace = 2^{14} \cdot 3^{9} \)
Composition factors:	$C_2$ x 7, $C_3$ x 7
Derived length:	$3$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	6271	11096	96680	82944	82944	279936
Conjugacy classes	1	43	34	538	16	16	648
Divisions	1	43	25	465	8	8	550
Autjugacy classes	1	7	10	46	2	2	68

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$36$
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid c^{6}=d^{6}=e^{6}=f^{6}=g^{6}=h^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(3,6,9)(13,15)(18,23,21), (2,4)(5,7)(10,11)(13,15)(16,17,20)(18,21,23)(19,22,24) \!\cdots\! \rangle$
Transitive group:	36T25491				more information
Direct product:	$C_2$ $\, \times\, $ $(C_3^6.C_2^4:A_4)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_3^6.C_2^5)$ . $A_4$ (3)	$C_3^6$ . $(C_2^5:A_4)$	$(C_3^6.C_2^6)$ . $C_6$	$(C_3^6.C_2^4:A_4)$ . $C_2$	all 21

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

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$2$

Subgroups

There are 122 normal subgroups (12 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^6.C_2^4:A_4$
Commutator:	$G' \simeq$ $C_3^6.C_2^6$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^6.C_2^5:A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^3\times C_6^3$	$G/\operatorname{Fit} \simeq$ $C_2^2:A_4$
Radical:	$R \simeq$ $C_3^6.C_2^5:A_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times C_6^3$	$G/\operatorname{soc} \simeq$ $C_2^2:A_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\wr C_3$

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 9 (order at least 31104) are shown, as well as all normal subgroups of any index.

Order 279936: $C_3^6.C_2^5:A_4$

Order 139968: $C_3^6.C_2^4:A_4$

Order 93312: $C_3^6.C_2^6.C_2$

Order 69984: $C_3^3.C_6^3:A_4$ x 16, $C_3^3.C_6^3:A_4$ x 3, $C_2^3.C_3\wr A_4$ x 2

Order 46656: $C_3^6.C_2^6$ x 38, $C_3^6.C_2^6$ x 3, $C_3^6.C_2^6$ x 2

Order 34992: $C_3^6:(C_2^2:A_4)$ x 16, $C_3^5:D_6.A_4$ x 3, $C_6^2.C_3\wr A_4$ x 2

Order 31104: $C_3^5.C_2^6.C_2$ x 2

Order 23328: $C_3^6.C_2^5$ x 16, $C_2\times C_3^6.C_2^4$ x 3, $C_3^6.C_2^5$ x 2

Order 11664: $C_3^6.C_2^4$ x 16, $C_3^6.C_2^4$ x 3, $C_3^6.C_2^4$ x 2

Order 10368: $(C_2^2\times C_6^3).A_4$ x 2

Order 5832: $C_3^6.C_2^3$ x 12, $C_3^6.C_2^3$ x 8, $C_3^3\times C_6^3$

Order 2916: $C_3^5:D_6$ x 12, $C_3^3\times C_3^3:C_2^2$ x 8, $C_3^4\times C_6^2$

Order 2187: $C_3^2\wr C_3$

Order 1458: $C_3^5\times C_6$

Order 864: $C_6:D_6^2$ x 2

Order 729: $C_3^6$

Order 432: $C_3:D_6^2$ x 2

Order 216: $C_6:S_3^2$ x 8, $C_6^3$ x 2

Order 128: $C_2^7$

Order 108: $C_3:S_3^2$ x 8, $C_3\times C_6^2$ x 2

Order 54: $C_3^2\times C_6$ x 2

Order 27: $C_3^3$ x 2

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 9 (order at least 31104) are shown, as well as all normal subgroups of any index.

Order 279936: $C_3^6.C_2^5:A_4$

Order 139968: $C_3^6.C_2^4:A_4$

Order 93312: $C_3^6.C_2^6.C_2$

Order 69984: $C_3^3.C_6^3:A_4$, $C_3^3.C_6^3:A_4$, $C_2^3.C_3\wr A_4$

Order 46656: $C_3^6.C_2^6$ x 5, $C_3^6.C_2^6$, $C_3^6.C_2^6$

Order 34992: $C_3^5:D_6.A_4$, $C_6^2.C_3\wr A_4$, $C_3^6:(C_2^2:A_4)$

Order 31104: $C_3^5.C_2^6.C_2$

Order 23328: $C_3^6.C_2^5$, $C_2\times C_3^6.C_2^4$, $C_3^6.C_2^5$

Order 11664: $C_3^6.C_2^4$, $C_3^6.C_2^4$, $C_3^6.C_2^4$

Order 10368: $(C_2^2\times C_6^3).A_4$

Order 5832: $C_3^6.C_2^3$, $C_3^3\times C_6^3$, $C_3^6.C_2^3$

Order 2916: $C_3^4\times C_6^2$, $C_3^3\times C_3^3:C_2^2$, $C_3^5:D_6$

Order 2187: $C_3^2\wr C_3$

Order 1458: $C_3^5\times C_6$

Order 864: $C_6:D_6^2$

Order 729: $C_3^6$

Order 432: $C_3:D_6^2$

Order 216: $C_6^3$, $C_6:S_3^2$

Order 128: $C_2^7$

Order 108: $C_3\times C_6^2$, $C_3:S_3^2$

Order 54: $C_3^2\times C_6$

Order 27: $C_3^3$

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 279936: $C_3^6.C_2^5:A_4$ ($C_1$)

Order 139968: $C_3^6.C_2^4:A_4$ ($C_2$)

Order 93312: $C_3^6.C_2^6.C_2$ ($C_3$)

Order 46656: $C_3^6.C_2^6$ ($C_6$)

Order 23328: $C_3^6.C_2^5$ ($A_4$) x 16, $C_2\times C_3^6.C_2^4$ ($A_4$) x 3, $C_3^6.C_2^5$ ($A_4$) x 2

Order 11664: $C_3^6.C_2^4$ ($C_2\times A_4$) x 16, $C_3^6.C_2^4$ ($C_2\times A_4$) x 3, $C_3^6.C_2^4$ ($C_2\times A_4$) x 2

Order 5832: $C_3^6.C_2^3$ ($C_2^2:A_4$) x 12, $C_3^6.C_2^3$ ($C_2^2:A_4$) x 8, $C_3^3\times C_6^3$ ($C_2^2:A_4$)

Order 2916: $C_3^5:D_6$ ($C_2^3:A_4$) x 12, $C_3^3\times C_3^3:C_2^2$ ($C_2^3:A_4$) x 8, $C_3^4\times C_6^2$ ($C_2^3:A_4$)

Order 1458: $C_3^5\times C_6$ ($C_2^4:A_4$)

Order 864: $C_6:D_6^2$ ($C_3^3:A_4$) x 2

Order 729: $C_3^6$ ($C_2^5:A_4$)

Order 432: $C_3:D_6^2$ ($C_2\times C_3^3:A_4$) x 2

Order 216: $C_6:S_3^2$ ($(C_3\times C_6^2):A_4$) x 8, $C_6^3$ ($(C_3\times C_6^2):A_4$) x 2

Order 108: $C_3:S_3^2$ ($C_6^3:A_4$) x 8, $C_3\times C_6^2$ ($C_6^3:A_4$) x 2

Order 54: $C_3^2\times C_6$ ($(C_2\times C_6^3).A_4$) x 2

Order 27: $C_3^3$ ($(C_2^2\times C_6^3).A_4$) x 2

Order 8: $C_2^3$ ($C_3^6:(C_2^2:A_4)$)

Order 4: $C_2^2$ ($C_3^3.C_6^3:A_4$)

Order 2: $C_2$ ($C_3^6.C_2^4:A_4$)

Order 1: $C_1$ ($C_3^6.C_2^5:A_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 279936: $C_3^6.C_2^5:A_4$ ($C_1$)

Order 139968: $C_3^6.C_2^4:A_4$ ($C_2$)

Order 93312: $C_3^6.C_2^6.C_2$ ($C_3$)

Order 46656: $C_3^6.C_2^6$ ($C_6$)

Order 23328: $C_3^6.C_2^5$ ($A_4$), $C_2\times C_3^6.C_2^4$ ($A_4$), $C_3^6.C_2^5$ ($A_4$)

Order 11664: $C_3^6.C_2^4$ ($C_2\times A_4$), $C_3^6.C_2^4$ ($C_2\times A_4$), $C_3^6.C_2^4$ ($C_2\times A_4$)

Order 5832: $C_3^6.C_2^3$ ($C_2^2:A_4$), $C_3^3\times C_6^3$ ($C_2^2:A_4$), $C_3^6.C_2^3$ ($C_2^2:A_4$)

Order 2916: $C_3^4\times C_6^2$ ($C_2^3:A_4$), $C_3^3\times C_3^3:C_2^2$ ($C_2^3:A_4$), $C_3^5:D_6$ ($C_2^3:A_4$)

Order 1458: $C_3^5\times C_6$ ($C_2^4:A_4$)

Order 864: $C_6:D_6^2$ ($C_3^3:A_4$)

Order 729: $C_3^6$ ($C_2^5:A_4$)

Order 432: $C_3:D_6^2$ ($C_2\times C_3^3:A_4$)

Order 216: $C_6^3$ ($(C_3\times C_6^2):A_4$), $C_6:S_3^2$ ($(C_3\times C_6^2):A_4$)

Order 108: $C_3\times C_6^2$ ($C_6^3:A_4$), $C_3:S_3^2$ ($C_6^3:A_4$)

Order 54: $C_3^2\times C_6$ ($(C_2\times C_6^3).A_4$)

Order 27: $C_3^3$ ($(C_2^2\times C_6^3).A_4$)

Order 8: $C_2^3$ ($C_3^6:(C_2^2:A_4)$)

Order 4: $C_2^2$ ($C_3^3.C_6^3:A_4$)

Order 2: $C_2$ ($C_3^6.C_2^4:A_4$)

Order 1: $C_1$ ($C_3^6.C_2^5:A_4$)

Series

Derived series

$C_3^6.C_2^5:A_4$

$\rhd$

$C_3^6.C_2^6$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$C_3^6.C_2^5:A_4$

$\rhd$

$C_3^6.C_2^4:A_4$

$\rhd$

$C_3^6.C_2^6$

$\rhd$

$C_3^6.C_2^4$

$\rhd$

$C_3^3\times C_3^3:C_2^2$

$\rhd$

$C_3^6$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Lower central series

$C_3^6.C_2^5:A_4$

$\rhd$

$C_3^6.C_2^6$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 15 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

The $648 \times 648$ character table is not available for this group.

Rational character table

The $550 \times 550$ rational character table is not available for this group.