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

• Label: 2916.et
• Order: \( 2^{2} \cdot 3^{6} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $27$
• Trans deg.: $27$
• Rank: $3$

Group information

Description:	$C_3^4:S_3^2$
Order:	\(2916\)\(\medspace = 2^{2} \cdot 3^{6} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^3.S_3^3$, of order \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \)
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	9
Elements	1	135	512	2052	216	2916
Conjugacy classes	1	3	19	17	2	42
Divisions	1	3	15	12	2	33
Autjugacy classes	1	3	14	11	1	30

Dimension	1	2	3	4	6	12	18	24	36
Irr. complex chars.	4	10	8	4	10	3	2	0	1	42
Irr. rational chars.	4	10	0	4	6	5	2	1	1	33

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$27$
Rank:	$3$
Inequivalent generating triples:	$1469664$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	18	18
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{6}=c^{3}=d^{3}=e^{3}=f^{9}=[c,e]=[d,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $27$ $\langle(1,20,17,3,19,16,2,21,18)(4,24,10,6,23,12,5,22,11)(7,25,15,9,27,14,8,26,13) \!\cdots\! \rangle$
Transitive group:	27T466	27T471	27T494		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^4$ $\,\rtimes\,$ $S_3^2$	$(C_3^4:S_3)$ $\,\rtimes\,$ $S_3$	$(C_3^4:S_3)$ $\,\rtimes\,$ $S_3$	$(C_3^4:C_3)$ $\,\rtimes\,$ $D_6$ (2)	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^3$ . $(C_3:S_3^2)$	$C_3$ . $(C_3^3:S_3^2)$	$(C_3\times \He_3)$ . $S_3^2$	$C_3^2$ . $(C_3^2:S_3^2)$	all 7
Aut. group:	$\Aut(C_3^3:(C_3\times C_6))$

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 10500 subgroups in 459 conjugacy classes, 29 normal (23 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:	$G' \simeq$ $C_3^4:C_3^2$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_3^3$	$G/\Phi \simeq$ $C_3:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^4:C_3^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3^4:S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3$	$G/\operatorname{soc} \simeq$ $C_3^3:S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4:C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

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

Order 1458: $C_3^4:(C_3\times S_3)$, $C_3^4:C_3:S_3$, $C_3^4:(C_3\times S_3)$

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

Order 729: $C_3^4:C_3^2$

Order 486: $C_3\wr C_3:C_6$ x 3, $C_3^3:(C_3\times C_6)$ x 2, $(C_3\times \He_3):S_3$ x 2, $C_3^4:S_3$ x 2, $C_3^3:(C_3\times C_6)$ x 2, $C_3\wr C_3:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^3:(C_3\times S_3)$, $C_3^4:S_3$, $C_3^4:S_3$, $C_3^4:S_3$

Order 324: $C_3^3:D_6$ x 3, $C_3^3:D_6$ x 2, $C_3^2:S_3^2$ x 2, $C_3\wr C_2^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$

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

Order 162: $C_3^3:C_6$ x 12, $C_3^3:S_3$ x 7, $C_3^3:C_6$ x 6, $S_3\times \He_3$ x 4, $C_3^3:S_3$ x 4, $C_3\wr S_3$ x 4, $C_3^2\wr C_2$ x 3, $C_3^3:S_3$ x 2, $D_9:C_3^2$ x 2, $\He_3:S_3$ x 2, $C_3^3:C_6$ x 2

Order 108: $C_3\times S_3^2$ x 6, $C_3^2:D_6$ x 4, $C_3:S_3^2$ 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 16, $C_3\wr C_3$ x 8, $\He_3:C_3$ x 4, $C_9:C_3^2$ x 2, $C_3^4$

Order 54: $C_3^2:C_6$ x 22, $S_3\times C_3^2$ x 21, $C_3^2:C_6$ x 14, $C_3^2:S_3$ x 9, $C_9:C_6$ x 6, $C_3\times D_9$ x 2, $C_3^2:S_3$ x 2, $C_2\times \He_3$ x 2

Order 36: $S_3^2$ x 8, $C_6\times S_3$ x 6, $C_6:S_3$

Order 27: $\He_3$ x 28, $C_3^3$ x 19, $C_9:C_3$ x 6, $C_3\times C_9$ x 2

Order 18: $C_3\times S_3$ x 47, $C_3:S_3$ x 12, $C_3\times C_6$ x 8, $D_9$ x 2

Order 12: $D_6$ x 7, $C_2\times C_6$

Order 9: $C_3^2$ x 39, $C_9$ x 2

Order 6: $S_3$ x 18, $C_6$ x 12

Order 4: $C_2^2$

Order 3: $C_3$ x 15

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: $C_3^4:(C_3\times S_3)$, $C_3^4:C_3:S_3$, $C_3^4:(C_3\times S_3)$

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

Order 729: $C_3^4:C_3^2$

Order 486: $C_3^3:(C_3\times C_6)$ x 2, $C_3\wr C_3:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^3:(C_3\times S_3)$, $C_3^4:S_3$, $(C_3\times \He_3):S_3$, $C_3^4:S_3$, $C_3^3:(C_3\times C_6)$, $C_3\wr C_3:C_6$, $C_3^4:S_3$

Order 324: $C_3^3:D_6$ x 2, $C_3^2:S_3^2$ x 2, $C_3^3:D_6$, $C_3\wr C_2^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$

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

Order 162: $C_3^3:C_6$ x 10, $C_3^3:S_3$ x 6, $C_3^3:C_6$ x 6, $S_3\times \He_3$ x 4, $C_3^2\wr C_2$ x 3, $C_3^3:S_3$ x 3, $C_3^3:S_3$ x 2, $C_3\wr S_3$ x 2, $D_9:C_3^2$, $\He_3:S_3$, $C_3^3:C_6$

Order 108: $C_3\times S_3^2$ x 5, $C_3^2:D_6$ x 3, $C_3:S_3^2$ 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 15, $C_3\wr C_3$ x 6, $\He_3:C_3$ x 2, $C_3^4$, $C_9:C_3^2$

Order 54: $S_3\times C_3^2$ x 19, $C_3^2:C_6$ x 18, $C_3^2:C_6$ x 13, $C_3^2:S_3$ x 8, $C_9:C_6$ x 3, $C_3^2:S_3$ x 2, $C_2\times \He_3$ x 2, $C_3\times D_9$

Order 36: $S_3^2$ x 7, $C_6\times S_3$ x 5, $C_6:S_3$

Order 27: $\He_3$ x 25, $C_3^3$ x 18, $C_9:C_3$ x 3, $C_3\times C_9$

Order 18: $C_3\times S_3$ x 42, $C_3:S_3$ x 11, $C_3\times C_6$ x 7, $D_9$

Order 12: $D_6$ x 6, $C_2\times C_6$

Order 9: $C_3^2$ x 36, $C_9$

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

Order 4: $C_2^2$

Order 3: $C_3$ x 14

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 1458: $C_3^4:(C_3\times S_3)$ ($C_2$), $C_3^4:C_3:S_3$ ($C_2$), $C_3^4:(C_3\times S_3)$ ($C_2$)

Order 729: $C_3^4:C_3^2$ ($C_2^2$)

Order 486: $C_3^3:(C_3\times C_6)$ ($S_3$) x 2, $C_3^3:(C_3\times C_6)$ ($S_3$), $C_3^4:S_3$ ($S_3$), $C_3^4:S_3$ ($S_3$)

Order 243: $C_3^3:C_3^2$ ($D_6$) x 2, $C_3^4:C_3$ ($D_6$) x 2, $C_3^3:C_3^2$ ($D_6$)

Order 162: $C_3^3:C_6$ ($C_3:S_3$)

Order 81: $C_3\times \He_3$ ($S_3^2$) x 3, $C_3^4$ ($S_3^2$), $C_3\times \He_3$ ($C_6:S_3$)

Order 54: $C_3^2:S_3$ ($C_3^2:S_3$)

Order 27: $C_3^3$ ($C_3:S_3^2$), $C_3^3$ ($C_3^2:D_6$), $C_3^3$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_3^2:S_3^2$), $C_3^2$ ($C_3^2:S_3^2$)

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

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: $C_3^4:(C_3\times S_3)$ ($C_2$), $C_3^4:C_3:S_3$ ($C_2$), $C_3^4:(C_3\times S_3)$ ($C_2$)

Order 729: $C_3^4:C_3^2$ ($C_2^2$)

Order 486: $C_3^3:(C_3\times C_6)$ ($S_3$), $C_3^4:S_3$ ($S_3$), $C_3^3:(C_3\times C_6)$ ($S_3$), $C_3^4:S_3$ ($S_3$)

Order 243: $C_3^4:C_3$ ($D_6$) x 2, $C_3^3:C_3^2$ ($D_6$), $C_3^3:C_3^2$ ($D_6$)

Order 162: $C_3^3:C_6$ ($C_3:S_3$)

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

Order 54: $C_3^2:S_3$ ($C_3^2:S_3$)

Order 27: $C_3^3$ ($C_3:S_3^2$), $C_3^3$ ($C_3^2:D_6$), $C_3^3$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_3^2:S_3^2$), $C_3^2$ ($C_3^2:S_3^2$)

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

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

Series

Derived series

$C_3^4:S_3^2$

$\rhd$

$C_3^4:C_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$C_3^4:S_3^2$

$\rhd$

$C_3^4:(C_3\times S_3)$

$\rhd$

$C_3^4:C_3^2$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_3^3:C_3^2$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^4:S_3^2$

$\rhd$

$C_3^4:C_3^2$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

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

Character theory

Complex character table

See the $42 \times 42$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $33 \times 33$ rational character table.