Abstract group 2916.ea: $C_3^4:S_3^2$

• Label: 2916.ea
• Order: \( 2^{2} \cdot 3^{6} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $18$
• Trans deg.: $18$
• Rank: $2$

Group information

Description:	$C_3^4:S_3^2$
Order:	\(2916\)\(\medspace = 2^{2} \cdot 3^{6} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$\He_3^2:(S_3\times D_4)$, of order \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \)
Composition factors:	$C_2$ x 2, $C_3$ x 6
Derived length:	$3$

This group is nonabelian and supersolvable (hence solvable and monomial).

Group statistics

Order	1	2	3	6
Elements	1	135	728	2052	2916
Conjugacy classes	1	3	37	17	58
Divisions	1	3	24	11	39
Autjugacy classes	1	2	15	7	25

Dimension	1	2	4	8	12	18	24
Irr. complex chars.	12	18	15	0	9	4	0	58
Irr. rational chars.	4	10	9	6	3	4	3	39

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$18$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	12	12	12

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{6}=b^{6}=c^{3}=d^{3}=e^{3}=f^{3}=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(1,10)(2,11)(3,12)(4,13,6,15,5,14)(7,16,9,18,8,17), (1,2)(5,6)(7,9)(10,15,18)(11,13,16,12,14,17)\rangle$
Transitive group:	18T409	18T414	36T4060	36T4065	all 8
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^4$ $\,\rtimes\,$ $S_3^2$	$\He_3^2$ $\,\rtimes\,$ $C_2^2$	$(C_3^4:S_3)$ $\,\rtimes\,$ $S_3$ (2)	$(C_3^4:S_3)$ $\,\rtimes\,$ $C_6$ (2)	all 14
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^3$ . $(C_3:S_3^2)$	$C_3^3$ . $(C_3\times S_3^2)$ (3)	$C_3$ . $(C_3^3:S_3^2)$ (2)	$(C_3\times \He_3)$ . $S_3^2$ (2)	all 6
Aut. group:	$\Aut(C_2\times F_5\times \GL(2,3))$

Elements of the group are displayed as permutations of degree 18.

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 14367 subgroups in 641 conjugacy classes, 33 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^4:S_3^2$
Commutator:	$G' \simeq$ $C_3^4:C_3$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_3\wr C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $\He_3^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3^4:S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2$	$G/\operatorname{soc} \simeq$ $C_3\wr C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 2916: $C_3^4:S_3^2$

Order 1458: $\He_3\wr C_2$ x 2, $\He_3^2:C_2$

Order 972: $C_3^4:D_6$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$

Order 729: $\He_3^2$

Order 486: $C_3^4:S_3$ x 6, $C_3^4:C_6$ x 6, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$, $C_3^4:C_6$

Order 324: $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$, $C_3\wr C_2^2$

Order 243: $C_3^4:C_3$ x 13, $C_3^2\times \He_3$ x 3

Order 162: $C_3^3:C_6$ x 17, $C_3^3:C_6$ x 12, $C_3^2\wr C_2$ x 7, $C_3^3:S_3$ x 2, $S_3\times \He_3$ x 2, $C_3^3:S_3$, $C_3^3:C_6$

Order 108: $C_3:S_3^2$ x 4, $C_3\times S_3^2$ x 3, $C_3^2:D_6$ x 2, $C_3^2:D_6$ x 2, $C_3:S_3^2$

Order 81: $C_3\times \He_3$ x 60, $C_3^4$ x 7

Order 54: $C_3^2:C_6$ x 29, $C_3^2:C_6$ x 21, $S_3\times C_3^2$ x 15, $C_3^2:S_3$ x 8, $C_3^2:S_3$ x 2, $C_2\times \He_3$ x 2

Order 36: $S_3^2$ x 8, $C_6\times S_3$ x 3, $C_6:S_3$ x 2

Order 27: $\He_3$ x 78, $C_3^3$ x 58

Order 18: $C_3\times S_3$ x 43, $C_3:S_3$ x 25, $C_3\times C_6$ x 7

Order 12: $D_6$ x 5, $C_2\times C_6$

Order 9: $C_3^2$ x 108

Order 6: $S_3$ x 16, $C_6$ x 11

Order 4: $C_2^2$

Order 3: $C_3$ x 24

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2916: $C_3^4:S_3^2$

Order 1458: $\He_3^2:C_2$, $\He_3\wr C_2$

Order 972: $C_3^3:S_3^2$, $C_3^4:D_6$, $C_3^3:S_3^2$

Order 729: $\He_3^2$

Order 486: $C_3^4:S_3$ x 3, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 2, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4:S_3$

Order 324: $C_3^2:S_3^2$, $C_3\wr C_2^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$

Order 243: $C_3^4:C_3$ x 9, $C_3^2\times \He_3$ x 3

Order 162: $C_3^3:C_6$ x 8, $C_3^3:C_6$ x 7, $C_3^2\wr C_2$ x 4, $C_3^3:S_3$, $C_3^3:C_6$, $C_3^3:S_3$, $S_3\times \He_3$

Order 108: $C_3:S_3^2$ x 2, $C_3\times S_3^2$ x 2, $C_3:S_3^2$, $C_3^2:D_6$, $C_3^2:D_6$

Order 81: $C_3\times \He_3$ x 30, $C_3^4$ x 6

Order 54: $C_3^2:C_6$ x 13, $C_3^2:C_6$ x 12, $S_3\times C_3^2$ x 8, $C_3^2:S_3$ x 4, $C_3^2:S_3$, $C_2\times \He_3$

Order 36: $S_3^2$ x 4, $C_6\times S_3$ x 2, $C_6:S_3$

Order 27: $C_3^3$ x 31, $\He_3$ x 28

Order 18: $C_3\times S_3$ x 19, $C_3:S_3$ x 12, $C_3\times C_6$ x 4

Order 12: $D_6$ x 3, $C_2\times C_6$

Order 9: $C_3^2$ x 50

Order 6: $S_3$ x 8, $C_6$ x 7

Order 4: $C_2^2$

Order 3: $C_3$ x 15

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2916: $C_3^4:S_3^2$ ($C_1$)

Order 1458: $\He_3\wr C_2$ ($C_2$) x 2, $\He_3^2:C_2$ ($C_2$)

Order 972: $C_3^3:S_3^2$ ($C_3$)

Order 729: $\He_3^2$ ($C_2^2$)

Order 486: $C_3^4:S_3$ ($C_6$) x 2, $C_3^4:S_3$ ($S_3$) x 2, $C_3^4:C_6$ ($C_6$), $C_3^4:C_6$ ($S_3$)

Order 243: $C_3^4:C_3$ ($D_6$) x 3, $C_3^4:C_3$ ($C_2\times C_6$)

Order 162: $C_3^3:S_3$ ($C_3\times S_3$) x 2, $C_3^3:S_3$ ($C_3\times S_3$)

Order 81: $C_3\times \He_3$ ($C_6\times S_3$) x 2, $C_3\times \He_3$ ($S_3^2$) x 2, $C_3^4$ ($C_6\times S_3$), $C_3^4$ ($S_3^2$)

Order 27: $C_3^3$ ($C_3\times S_3^2$) x 3, $C_3^3$ ($C_3:S_3^2$)

Order 9: $C_3^2$ ($C_3\wr C_2^2$)

Order 3: $C_3$ ($C_3^3:S_3^2$) x 2

Order 1: $C_1$ ($C_3^4:S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2916: $C_3^4:S_3^2$ ($C_1$)

Order 1458: $\He_3^2:C_2$ ($C_2$), $\He_3\wr C_2$ ($C_2$)

Order 972: $C_3^3:S_3^2$ ($C_3$)

Order 729: $\He_3^2$ ($C_2^2$)

Order 486: $C_3^4:C_6$ ($C_6$), $C_3^4:C_6$ ($S_3$), $C_3^4:S_3$ ($C_6$), $C_3^4:S_3$ ($S_3$)

Order 243: $C_3^4:C_3$ ($D_6$) x 2, $C_3^4:C_3$ ($C_2\times C_6$)

Order 162: $C_3^3:S_3$ ($C_3\times S_3$), $C_3^3:S_3$ ($C_3\times S_3$)

Order 81: $C_3^4$ ($C_6\times S_3$), $C_3^4$ ($S_3^2$), $C_3\times \He_3$ ($C_6\times S_3$), $C_3\times \He_3$ ($S_3^2$)

Order 27: $C_3^3$ ($C_3\times S_3^2$) x 2, $C_3^3$ ($C_3:S_3^2$)

Order 9: $C_3^2$ ($C_3\wr C_2^2$)

Order 3: $C_3$ ($C_3^3:S_3^2$)

Order 1: $C_1$ ($C_3^4:S_3^2$)

Series

Derived series

$C_3^4:S_3^2$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_3^4:S_3^2$

$\rhd$

$\He_3^2:C_2$

$\rhd$

$\He_3^2$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^4:S_3^2$

$\rhd$

$C_3^4:C_3$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 12 larger groups in the database.

Character theory

Complex character table

See the $58 \times 58$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $39 \times 39$ rational character table.