Abstract group 1944.3564: $C_3^2:S_3^3$

• Label: 1944.3564
• Order: \( 2^{3} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $15$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^2:S_3^3$
Order:	\(1944\)\(\medspace = 2^{3} \cdot 3^{5} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$C_3^4:(D_4\times D_6)$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \)
Composition factors:	$C_2$ x 3, $C_3$ x 5
Derived length:	$2$

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

Group statistics

Order	1	2	3	6
Elements	1	159	242	1542	1944
Conjugacy classes	1	7	27	55	90
Divisions	1	7	19	36	63
Autjugacy classes	1	5	14	22	42

Dimension	1	2	4	6	8	12	16	24
Irr. complex chars.	24	36	18	4	3	4	0	1	90
Irr. rational chars.	8	20	18	4	7	4	1	1	63

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$209664$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	10	10	10

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{6}=b^{3}=c^{6}=d^{6}=e^{3}=[a,d]=[a,e]=[b,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(2,4)(3,6)(5,7)(8,9), (10,11)(12,15)(13,14), (10,11)(12,14)(13,15), (2,5,9) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 61 \end{array}\right), \left(\begin{array}{rr} 31 & 10 \\ 60 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 45 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 71 & 0 \\ 72 & 89 \end{array}\right), \left(\begin{array}{rr} 73 & 54 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 46 & 45 \\ 45 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 30 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 46 & 45 \\ 45 & 26 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/90\Z)$
Transitive group:	36T2812	36T2856			more information
Direct product:	$S_3$ ${}^2$ $\, \times\, $ $(C_3^2:C_6)$
Semidirect product:	$C_3^2$ $\,\rtimes\,$ $S_3^3$	$(S_3\times C_3^3)$ $\,\rtimes\,$ $D_6$ (2)	$C_3^3$ $\,\rtimes\,$ $(S_3\times D_6)$ (2)	$(S_3\times \He_3)$ $\,\rtimes\,$ $D_6$ (2)	all 45
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3$ . $(C_3\times S_3^3)$	$C_3^3$ . $(C_6\times D_6)$	$C_3^2$ . $(C_6\times S_3^2)$ (2)	$(C_3\times S_3^2)$ . $(C_3\times S_3)$	all 7

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{90}\Z)$.

Homology

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

Subgroups

There are 12058 subgroups in 809 conjugacy classes, 86 normal (32 characteristic).

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

Special subgroups

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

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 1944: $C_3^2:S_3^3$

Order 972: $C_3^3:S_3^2$ x 2, $C_3^3:C_6^2$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $\He_3\times S_3^2$

Order 648: $S_3\times C_3^2:D_6$ x 2, $C_3:S_3^3$, $C_3\times S_3^3$

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

Order 324: $C_3^2:S_3^2$ x 13, $C_3^2:S_3^2$ x 5, $C_3^2\times S_3^2$ x 5, $C_3^2:S_3^2$ x 2, $C_3^3:D_6$ x 2, $D_6\times \He_3$ x 2, $C_3^2:C_6^2$ x 2, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3\wr C_2^2$

Order 243: $C_3^2\times \He_3$

Order 216: $C_6\times S_3^2$ x 3, $C_6:S_3^2$ x 2, $S_3^3$ x 2, $C_6^2:C_6$

Order 162: $S_3\times \He_3$ x 11, $C_3^3:C_6$ x 10, $C_3^3:C_6$ x 10, $S_3\times C_3^3$ x 7, $C_3^2\wr C_2$ x 6, $C_3^3:C_6$ x 3, $C_6\times \He_3$ x 2, $C_3^3:S_3$

Order 108: $C_3\times S_3^2$ x 24, $C_3:S_3^2$ x 19, $C_3^2\times D_6$ x 10, $C_3^2:D_6$ x 10, $C_3^2:D_6$ x 5, $C_3^2:D_6$ x 2, $C_3:S_3^2$ x 2, $C_2^2\times \He_3$

Order 81: $C_3\times \He_3$ x 12, $C_3^4$ x 3

Order 72: $S_3\times D_6$ x 5, $C_6\times D_6$ x 3, $C_6:D_6$

Order 54: $S_3\times C_3^2$ x 49, $C_3^2:C_6$ x 28, $C_3^2:C_6$ x 16, $C_3^2:S_3$ x 13, $C_2\times \He_3$ x 9, $C_3^2\times C_6$ x 7

Order 36: $C_6\times S_3$ x 37, $S_3^2$ x 26, $C_6:S_3$ x 15, $C_6^2$ x 5

Order 27: $C_3^3$ x 31, $\He_3$ x 17

Order 24: $C_2\times D_6$ x 4, $C_2^2\times C_6$

Order 18: $C_3\times S_3$ x 69, $C_3\times C_6$ x 38, $C_3:S_3$ x 36

Order 12: $D_6$ x 29, $C_2\times C_6$ x 14

Order 9: $C_3^2$ x 57

Order 8: $C_2^3$

Order 6: $C_6$ x 36, $S_3$ x 28

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 19

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1944: $C_3^2:S_3^3$

Order 972: $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $\He_3\times S_3^2$, $C_3^3:C_6^2$

Order 648: $C_3:S_3^3$, $C_3\times S_3^3$, $S_3\times C_3^2:D_6$

Order 486: $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4:C_6$

Order 324: $C_3^2:S_3^2$ x 7, $C_3^2\times S_3^2$ x 4, $C_3^2:S_3^2$ x 3, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3\wr C_2^2$, $C_3^3:D_6$, $D_6\times \He_3$, $C_3^2:C_6^2$

Order 243: $C_3^2\times \He_3$

Order 216: $C_6\times S_3^2$ x 2, $S_3^3$ x 2, $C_6:S_3^2$, $C_6^2:C_6$

Order 162: $C_3^3:C_6$ x 6, $S_3\times \He_3$ x 6, $C_3^3:C_6$ x 6, $C_3^2\wr C_2$ x 5, $S_3\times C_3^3$ x 4, $C_3^3:C_6$ x 2, $C_3^3:S_3$, $C_6\times \He_3$

Order 108: $C_3\times S_3^2$ x 15, $C_3:S_3^2$ x 11, $C_3^2:D_6$ x 6, $C_3^2\times D_6$ x 5, $C_3^2:D_6$ x 3, $C_3:S_3^2$ x 2, $C_3^2:D_6$, $C_2^2\times \He_3$

Order 81: $C_3\times \He_3$ x 8, $C_3^4$ x 3

Order 72: $S_3\times D_6$ x 3, $C_6\times D_6$ x 2, $C_6:D_6$

Order 54: $S_3\times C_3^2$ x 28, $C_3^2:C_6$ x 18, $C_3^2:C_6$ x 10, $C_3^2:S_3$ x 9, $C_2\times \He_3$ x 5, $C_3^2\times C_6$ x 4

Order 36: $C_6\times S_3$ x 20, $S_3^2$ x 15, $C_6:S_3$ x 9, $C_6^2$ x 4

Order 27: $C_3^3$ x 22, $\He_3$ x 11

Order 24: $C_2\times D_6$ x 3, $C_2^2\times C_6$

Order 18: $C_3\times S_3$ x 40, $C_3:S_3$ x 23, $C_3\times C_6$ x 22

Order 12: $D_6$ x 17, $C_2\times C_6$ x 10

Order 9: $C_3^2$ x 38

Order 8: $C_2^3$

Order 6: $C_6$ x 22, $S_3$ x 18

Order 4: $C_2^2$ x 5

Order 3: $C_3$ x 14

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1944: $C_3^2:S_3^3$ ($C_1$)

Order 972: $C_3^3:S_3^2$ ($C_2$) x 2, $C_3^3:C_6^2$ ($C_2$) x 2, $C_3^3:S_3^2$ ($C_2$), $C_3^3:S_3^2$ ($C_2$), $\He_3\times S_3^2$ ($C_2$)

Order 648: $C_3:S_3^3$ ($C_3$)

Order 486: $C_3^4:C_6$ ($C_2^2$) x 2, $C_3^4:C_6$ ($C_2^2$) x 2, $C_3^4:C_6$ ($C_2^2$), $C_3^4:C_6$ ($C_2^2$), $C_3^4:C_6$ ($C_2^2$)

Order 324: $C_3^2:S_3^2$ ($C_6$) x 2, $C_3^2:S_3^2$ ($C_6$) x 2, $C_3^2:S_3^2$ ($S_3$) x 2, $C_3^2:S_3^2$ ($C_6$), $C_3^2:S_3^2$ ($C_6$), $C_3^2\times S_3^2$ ($C_6$), $C_3^2\times S_3^2$ ($S_3$)

Order 243: $C_3^2\times \He_3$ ($C_2^3$)

Order 162: $C_3^3:C_6$ ($C_2\times C_6$) x 2, $C_3^2\wr C_2$ ($C_2\times C_6$) x 2, $S_3\times C_3^3$ ($C_2\times C_6$) x 2, $S_3\times C_3^3$ ($D_6$) x 2, $C_3^3:C_6$ ($D_6$) x 2, $S_3\times \He_3$ ($D_6$) x 2, $C_3^3:C_6$ ($D_6$) x 2, $C_3^3:S_3$ ($C_2\times C_6$), $C_3^2\wr C_2$ ($D_6$)

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

Order 81: $C_3\times \He_3$ ($C_2\times D_6$) x 2, $C_3^4$ ($C_2^2\times C_6$), $C_3^4$ ($C_2\times D_6$)

Order 54: $S_3\times C_3^2$ ($C_6\times S_3$) x 4, $C_3^2:C_6$ ($C_6\times S_3$) x 3, $C_3^2:S_3$ ($C_6\times S_3$) x 2, $S_3\times C_3^2$ ($S_3^2$) x 2, $C_3^2:C_6$ ($S_3^2$)

Order 36: $S_3^2$ ($C_3^2:C_6$)

Order 27: $C_3^3$ ($C_6\times D_6$) x 3, $C_3^3$ ($S_3\times D_6$) x 2, $\He_3$ ($S_3\times D_6$)

Order 18: $C_3\times S_3$ ($C_3\times S_3^2$) x 2, $C_3\times S_3$ ($C_3^2:D_6$) x 2, $C_3:S_3$ ($C_3\times S_3^2$), $C_3:S_3$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_6\times S_3^2$) x 3, $C_3^2$ ($S_3^3$), $C_3^2$ ($C_6^2:C_6$)

Order 6: $S_3$ ($C_3^2:S_3^2$) x 2

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

Order 1: $C_1$ ($C_3^2:S_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1944: $C_3^2:S_3^3$ ($C_1$)

Order 972: $C_3^3:S_3^2$ ($C_2$), $C_3^3:S_3^2$ ($C_2$), $C_3^3:S_3^2$ ($C_2$), $\He_3\times S_3^2$ ($C_2$), $C_3^3:C_6^2$ ($C_2$)

Order 648: $C_3:S_3^3$ ($C_3$)

Order 486: $C_3^4:C_6$ ($C_2^2$), $C_3^4:C_6$ ($C_2^2$), $C_3^4:C_6$ ($C_2^2$), $C_3^4:C_6$ ($C_2^2$), $C_3^4:C_6$ ($C_2^2$)

Order 324: $C_3^2:S_3^2$ ($C_6$), $C_3^2:S_3^2$ ($C_6$), $C_3^2:S_3^2$ ($C_6$), $C_3^2:S_3^2$ ($C_6$), $C_3^2\times S_3^2$ ($C_6$), $C_3^2\times S_3^2$ ($S_3$), $C_3^2:S_3^2$ ($S_3$)

Order 243: $C_3^2\times \He_3$ ($C_2^3$)

Order 162: $C_3^2\wr C_2$ ($C_2\times C_6$) x 2, $C_3^3:S_3$ ($C_2\times C_6$), $C_3^3:C_6$ ($C_2\times C_6$), $C_3^2\wr C_2$ ($D_6$), $S_3\times C_3^3$ ($C_2\times C_6$), $S_3\times C_3^3$ ($D_6$), $C_3^3:C_6$ ($D_6$), $S_3\times \He_3$ ($D_6$), $C_3^3:C_6$ ($D_6$)

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

Order 81: $C_3^4$ ($C_2^2\times C_6$), $C_3^4$ ($C_2\times D_6$), $C_3\times \He_3$ ($C_2\times D_6$)

Order 54: $C_3^2:C_6$ ($C_6\times S_3$) x 2, $S_3\times C_3^2$ ($C_6\times S_3$) x 2, $C_3^2:C_6$ ($S_3^2$), $C_3^2:S_3$ ($C_6\times S_3$), $S_3\times C_3^2$ ($S_3^2$)

Order 36: $S_3^2$ ($C_3^2:C_6$)

Order 27: $C_3^3$ ($C_6\times D_6$) x 2, $C_3^3$ ($S_3\times D_6$), $\He_3$ ($S_3\times D_6$)

Order 18: $C_3:S_3$ ($C_3\times S_3^2$), $C_3:S_3$ ($C_3^2:D_6$), $C_3\times S_3$ ($C_3\times S_3^2$), $C_3\times S_3$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_6\times S_3^2$) x 2, $C_3^2$ ($S_3^3$), $C_3^2$ ($C_6^2:C_6$)

Order 6: $S_3$ ($C_3^2:S_3^2$)

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

Order 1: $C_1$ ($C_3^2:S_3^3$)

Series

Derived series

$C_3^2:S_3^3$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

Chief series

$C_3^2:S_3^3$

$\rhd$

$\He_3\times S_3^2$

$\rhd$

$C_3^2\times S_3^2$

$\rhd$

$C_3\times S_3^2$

$\rhd$

$S_3\times C_3^2$

$\rhd$

$C_3\times S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^2:S_3^3$

$\rhd$

$C_3^4$

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 5 larger groups in the database.

Character theory

Complex character table

See the $90 \times 90$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $63 \times 63$ rational character table.