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

• Label: 2916.md
• Order: \( 2^{2} \cdot 3^{6} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $30$
• Trans deg.: $54$
• Rank: $2$

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:	$\He_3.C_3^3.C_2^3$, 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	18
Elements	1	63	566	1638	162	486	2916
Conjugacy classes	1	3	38	45	9	9	105
Divisions	1	3	21	24	5	5	59
Autjugacy classes	1	3	17	19	4	4	48

Dimension	1	2	4	6	8	12	18	36
Irr. complex chars.	36	36	9	18	0	0	6	0	105
Irr. rational chars.	4	20	17	2	4	8	2	2	59

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$54$
Rank:	$2$
Inequivalent generating pairs:	$144$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	36	36
Arbitrary	18	20	20

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{6}=b^{6}=c^{3}=d^{3}=e^{9}=[a,c]=[b,c]=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(2,5)(3,9)(4,11,21,23,10,22)(6,12)(7,13)(8,17,16,25,15,26)(14,20,24,27,18,19) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $(C_3^3.S_3^2)$
Semidirect product:	$(C_3^4:S_3)$ $\,\rtimes\,$ $S_3$	$(C_3^4.S_3)$ $\,\rtimes\,$ $S_3$	$(C_9:C_3^3)$ $\,\rtimes\,$ $D_6$	$(C_3^3.S_3^2)$ $\,\rtimes\,$ $C_3$	all 29
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^4$ . $S_3^2$	$C_3$ . $(C_3^4:D_6)$	$C_3^3$ . $(C_3\times S_3^2)$ (4)	$C_3^3$ . $(C_3^2:D_6)$	all 9
Aut. group:	$\Aut(C_9.\He_3)$	$\Aut(C_3^3.(C_3\times S_3))$	$\Aut(C_3^3.(C_3\times S_3))$	$\Aut(C_3^3.(C_3\times S_3))$	all 6

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

Homology

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

Subgroups

There are 7752 subgroups in 649 conjugacy classes, 68 normal (35 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3^3.S_3^2$
Commutator:	$G' \simeq$ $\He_3:C_3$	$G/G' \simeq$ $C_6^2$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_3^2\times S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $\He_3:C_3^3$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3^4.S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2$	$G/\operatorname{soc} \simeq$ $C_3^3:D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3:C_3^3$

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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Subgroup information

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Classes of subgroups up to conjugation

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

Order 1458: $\He_3:(S_3\times C_3^2)$, $C_3^4.(C_3\times S_3)$, $C_3^4.(C_3\times S_3)$

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

Order 729: $\He_3:C_3^3$

Order 486: $C_3^3.(C_3\times S_3)$ x 6, $C_3^3.(C_3\times S_3)$ x 6, $C_3^4:C_6$ x 3, $(C_3\times \He_3):S_3$ x 3, $(C_3^2\times C_9):C_6$ x 2, $(C_3^2\times C_9):S_3$ x 2, $C_3^4.S_3$, $C_3^4:S_3$, $C_3^4.C_6$, $D_9:C_3^3$, $C_3\times \He_3:S_3$

Order 324: $C_3^3:D_6$ x 4, $C_3^2.S_3^2$ x 3, $C_3^2\times S_3^2$ x 2, $C_3^2.S_3^2$, $C_9:C_6^2$, $C_3^2:D_{18}$

Order 243: $\He_3:C_3^2$ x 12, $\He_3:C_3^2$ x 4, $C_3^2\times \He_3$ x 2, $C_9:C_3^3$

Order 162: $C_3^3:C_6$ x 28, $D_9:C_3^2$ x 9, $C_3^3.C_6$ x 6, $\He_3.S_3$ x 5, $\He_3.C_6$ x 5, $S_3\times C_3^3$ x 4, $C_3^3:S_3$ x 4, $C_3^2\wr C_2$ x 3, $C_3^3.S_3$ x 3, $C_3^2:C_{18}$ x 2, $C_{18}:C_3^2$, $C_3^2:D_9$, $C_3^2\times D_9$, $\He_3:S_3$

Order 108: $C_3\times S_3^2$ x 8, $C_3^2\times D_6$ x 3, $C_{18}:C_6$ x 3, $C_3\times D_{18}$, $C_3^2:D_6$, $S_3\times D_9$

Order 81: $C_3\times \He_3$ x 28, $\He_3:C_3$ x 25, $C_9:C_3^2$ x 11, $C_3^4$ x 3, $C_3^2\times C_9$ x 2

Order 54: $S_3\times C_3^2$ x 36, $C_3^2:C_6$ x 19, $C_3^2:C_6$ x 12, $C_9:C_6$ x 9, $C_9:C_6$ x 6, $S_3\times C_9$ x 5, $C_3\times D_9$ x 4, $C_3^2\times C_6$ x 3, $C_3\times C_{18}$ x 2, $C_3^2:S_3$, $C_3:D_9$

Order 36: $C_6\times S_3$ x 12, $S_3^2$ x 2, $D_{18}$, $C_6^2$

Order 27: $C_3^3$ x 37, $\He_3$ x 34, $C_9:C_3$ x 12, $C_3\times C_9$ x 9

Order 18: $C_3\times S_3$ x 45, $C_3\times C_6$ x 24, $C_{18}$ x 5, $C_3:S_3$ x 3, $D_9$ x 2

Order 12: $C_2\times C_6$ x 4, $D_6$ x 3

Order 9: $C_3^2$ x 67, $C_9$ x 5

Order 6: $C_6$ x 24, $S_3$ x 7

Order 4: $C_2^2$

Order 3: $C_3$ x 21

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: $\He_3:(S_3\times C_3^2)$, $C_3^4.(C_3\times S_3)$, $C_3^4.(C_3\times S_3)$

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

Order 729: $\He_3:C_3^3$

Order 486: $C_3^4:C_6$ x 3, $C_3^3.(C_3\times S_3)$ x 2, $C_3^3.(C_3\times S_3)$ x 2, $(C_3^2\times C_9):C_6$ x 2, $(C_3^2\times C_9):S_3$ x 2, $C_3^4.S_3$, $C_3^4:S_3$, $C_3^4.C_6$, $D_9:C_3^3$, $(C_3\times \He_3):S_3$, $C_3\times \He_3:S_3$

Order 324: $C_3^2\times S_3^2$ x 2, $C_3^3:D_6$ x 2, $C_3^2.S_3^2$, $C_9:C_6^2$, $C_3^2.S_3^2$, $C_3^2:D_{18}$

Order 243: $\He_3:C_3^2$ x 4, $\He_3:C_3^2$ x 4, $C_3^2\times \He_3$ x 2, $C_9:C_3^3$

Order 162: $C_3^3:C_6$ x 13, $D_9:C_3^2$ x 5, $S_3\times C_3^3$ x 4, $C_3^2\wr C_2$ x 3, $\He_3.S_3$ x 3, $\He_3.C_6$ x 3, $C_3^3:S_3$ x 2, $C_3^3.C_6$ x 2, $C_3^2:C_{18}$ x 2, $C_{18}:C_3^2$, $C_3^3.S_3$, $C_3^2:D_9$, $C_3^2\times D_9$, $\He_3:S_3$

Order 108: $C_3\times S_3^2$ x 4, $C_3^2\times D_6$ x 3, $C_{18}:C_6$, $C_3\times D_{18}$, $C_3^2:D_6$, $S_3\times D_9$

Order 81: $C_3\times \He_3$ x 13, $\He_3:C_3$ x 10, $C_9:C_3^2$ x 5, $C_3^4$ x 3, $C_3^2\times C_9$ x 2

Order 54: $S_3\times C_3^2$ x 22, $C_3^2:C_6$ x 10, $C_3^2:C_6$ x 6, $C_3\times D_9$ x 4, $C_9:C_6$ x 3, $S_3\times C_9$ x 3, $C_3^2\times C_6$ x 3, $C_3\times C_{18}$ x 2, $C_9:C_6$ x 2, $C_3^2:S_3$, $C_3:D_9$

Order 36: $C_6\times S_3$ x 6, $S_3^2$ x 2, $D_{18}$, $C_6^2$

Order 27: $C_3^3$ x 21, $\He_3$ x 16, $C_3\times C_9$ x 7, $C_9:C_3$ x 4

Order 18: $C_3\times S_3$ x 25, $C_3\times C_6$ x 14, $C_3:S_3$ x 3, $C_{18}$ x 3, $D_9$ x 2

Order 12: $D_6$ x 3, $C_2\times C_6$ x 2

Order 9: $C_3^2$ x 35, $C_9$ x 3

Order 6: $C_6$ x 14, $S_3$ x 7

Order 4: $C_2^2$

Order 3: $C_3$ x 13

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: $\He_3:(S_3\times C_3^2)$ ($C_2$), $C_3^4.(C_3\times S_3)$ ($C_2$), $C_3^4.(C_3\times S_3)$ ($C_2$)

Order 972: $C_3^3.S_3^2$ ($C_3$) x 3, $C_3^3.S_3^2$ ($C_3$)

Order 729: $\He_3:C_3^3$ ($C_2^2$)

Order 486: $(C_3\times \He_3):S_3$ ($C_6$) x 3, $C_3^3.(C_3\times S_3)$ ($C_6$) x 3, $C_3^3.(C_3\times S_3)$ ($C_6$) x 3, $C_3^4.S_3$ ($S_3$), $C_3^4:S_3$ ($S_3$), $C_3\times \He_3:S_3$ ($C_6$), $(C_3^2\times C_9):C_6$ ($C_6$), $(C_3^2\times C_9):S_3$ ($C_6$)

Order 324: $C_3^2.S_3^2$ ($C_3^2$)

Order 243: $\He_3:C_3^2$ ($C_2\times C_6$) x 3, $C_9:C_3^3$ ($D_6$), $C_3^2\times \He_3$ ($D_6$), $\He_3:C_3^2$ ($C_2\times C_6$)

Order 162: $C_3^3:S_3$ ($C_3\times S_3$) x 4, $C_3^3.S_3$ ($C_3\times S_3$) x 3, $C_3^2:D_9$ ($C_3\times S_3$), $\He_3:S_3$ ($C_3\times C_6$), $\He_3.S_3$ ($C_3\times C_6$), $\He_3.C_6$ ($C_3\times C_6$)

Order 81: $C_3\times \He_3$ ($C_6\times S_3$) x 4, $C_9:C_3^2$ ($C_6\times S_3$) x 3, $\He_3:C_3$ ($C_6^2$), $C_3^4$ ($S_3^2$), $C_3^2\times C_9$ ($C_6\times S_3$)

Order 54: $C_3^2:S_3$ ($S_3\times C_3^2$), $C_3:D_9$ ($S_3\times C_3^2$)

Order 27: $C_3^3$ ($C_3\times S_3^2$) x 4, $C_3^3$ ($C_3^2:D_6$), $\He_3$ ($C_3^2\times D_6$), $C_3\times C_9$ ($C_3^2\times D_6$)

Order 9: $C_3^2$ ($C_3^3:D_6$) x 4, $C_3^2$ ($C_3^2\times S_3^2$)

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

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: $\He_3:(S_3\times C_3^2)$ ($C_2$), $C_3^4.(C_3\times S_3)$ ($C_2$), $C_3^4.(C_3\times S_3)$ ($C_2$)

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

Order 729: $\He_3:C_3^3$ ($C_2^2$)

Order 486: $C_3^4.S_3$ ($S_3$), $C_3^4:S_3$ ($S_3$), $(C_3\times \He_3):S_3$ ($C_6$), $C_3\times \He_3:S_3$ ($C_6$), $C_3^3.(C_3\times S_3)$ ($C_6$), $C_3^3.(C_3\times S_3)$ ($C_6$), $(C_3^2\times C_9):C_6$ ($C_6$), $(C_3^2\times C_9):S_3$ ($C_6$)

Order 324: $C_3^2.S_3^2$ ($C_3^2$)

Order 243: $C_9:C_3^3$ ($D_6$), $C_3^2\times \He_3$ ($D_6$), $\He_3:C_3^2$ ($C_2\times C_6$), $\He_3:C_3^2$ ($C_2\times C_6$)

Order 162: $C_3^3:S_3$ ($C_3\times S_3$) x 2, $C_3^3.S_3$ ($C_3\times S_3$), $C_3^2:D_9$ ($C_3\times S_3$), $\He_3:S_3$ ($C_3\times C_6$), $\He_3.S_3$ ($C_3\times C_6$), $\He_3.C_6$ ($C_3\times C_6$)

Order 81: $C_3\times \He_3$ ($C_6\times S_3$) x 2, $\He_3:C_3$ ($C_6^2$), $C_3^4$ ($S_3^2$), $C_9:C_3^2$ ($C_6\times S_3$), $C_3^2\times C_9$ ($C_6\times S_3$)

Order 54: $C_3^2:S_3$ ($S_3\times C_3^2$), $C_3:D_9$ ($S_3\times C_3^2$)

Order 27: $C_3^3$ ($C_3\times S_3^2$) x 2, $C_3^3$ ($C_3^2:D_6$), $\He_3$ ($C_3^2\times D_6$), $C_3\times C_9$ ($C_3^2\times D_6$)

Order 9: $C_3^2$ ($C_3^3:D_6$) x 2, $C_3^2$ ($C_3^2\times S_3^2$)

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

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

Series

Derived series

$C_3^4.S_3^2$

$\rhd$

$\He_3:C_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_3^4.S_3^2$

$\rhd$

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

$\rhd$

$(C_3^2\times C_9):C_6$

$\rhd$

$\He_3:C_3^2$

$\rhd$

$C_3^2\times C_9$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^4.S_3^2$

$\rhd$

$\He_3:C_3$

Upper central series

$C_1$

$\lhd$

$C_3$

Supergroups

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

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $105 \times 105$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $59 \times 59$ rational character table.