Abstract group 1944.3577: $C_9:S_3^3$

• Label: 1944.3577
• Order: \( 2^{3} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{2} \)
• Perm deg.: $18$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_9:S_3^3$
Order:	\(1944\)\(\medspace = 2^{3} \cdot 3^{5} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^3:C_2^2.D_6\times D_9:C_3$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \)
Composition factors:	$C_2$ x 3, $C_3$ x 5
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	351	80	864	162	486	1944
Conjugacy classes	1	7	16	18	27	18	87
Divisions	1	7	16	18	9	6	57
Autjugacy classes	1	3	7	5	4	2	22

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	8	28	30	0	21	0	0	87
Irr. rational chars.	8	16	12	4	6	6	5	57

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$21504$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{6}=c^{6}=d^{3}=e^{9}=[a,c]=[a,e]=[b,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(2,4)(3,7)(5,6)(8,9)(11,12), (11,12)(13,14)(15,18)(16,17), (15,16)(17,18) \!\cdots\! \rangle$
Transitive group:	36T2862				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_9$ $\,\rtimes\,$ $S_3^3$	$(C_9:S_3^2)$ $\,\rtimes\,$ $S_3$	$(C_3:S_3^2)$ $\,\rtimes\,$ $D_9$	$(C_3:D_9)$ $\,\rtimes\,$ $S_3^2$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^2$ . $S_3^3$	$C_3^3$ . $(S_3\times D_6)$	$C_3$ . $(C_3:S_3^3)$	$C_3^4$ . $(C_2\times D_6)$	all 7

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 18968 subgroups in 786 conjugacy classes, 68 normal (11 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_9:S_3^3$
Commutator:	$G' \simeq$ $C_3^3\times C_9$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3:S_3^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^3\times C_9$	$G/\operatorname{Fit} \simeq$ $C_2^3$
Radical:	$R \simeq$ $C_9:S_3^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $C_2\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3\times C_9$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1944: $C_9:S_3^3$

Order 972: $C_3^3:D_{18}$ x 3, $C_3^3:D_{18}$ x 3, $C_9\times C_3:S_3^2$

Order 648: $D_9\times S_3^2$ x 3, $C_3:S_3^3$

Order 486: $C_3^3:C_{18}$ x 3, $C_3^3:D_9$ x 3, $C_3^3:D_9$

Order 324: $C_9:S_3^2$ x 9, $C_3^2:D_{18}$ x 6, $C_3^2:D_{18}$ x 6, $C_3\wr C_2^2$ x 4, $C_3^2:S_3^2$ x 3, $C_9:S_3^2$ x 3, $C_9\times S_3^2$ x 3

Order 243: $C_3^3\times C_9$

Order 216: $S_3^3$ x 4, $S_3\times D_{18}$ x 3

Order 162: $C_3^2:D_9$ x 15, $C_3^2:D_9$ x 9, $C_3^2:C_{18}$ x 9, $C_3^2\wr C_2$ x 6, $C_3^2:C_{18}$ x 6, $C_3^2\times D_9$ x 6, $C_3^3:S_3$

Order 108: $S_3\times D_9$ x 27, $C_3:S_3^2$ x 18, $C_3\times S_3^2$ x 12, $C_3:S_3^2$ x 4, $C_3:D_{18}$ x 3, $S_3\times C_{18}$ x 3, $C_3\times D_{18}$ x 3

Order 81: $C_3^2\times C_9$ x 15, $C_3^4$

Order 72: $S_3\times D_6$ x 6, $C_2\times D_{18}$

Order 54: $C_3:D_9$ x 42, $S_3\times C_9$ x 21, $C_3\times D_9$ x 21, $C_3^2:C_6$ x 18, $S_3\times C_3^2$ x 18, $C_3^2:S_3$ x 16, $C_3\times C_{18}$ x 3

Order 36: $S_3^2$ x 42, $C_6\times S_3$ x 12, $D_{18}$ x 9, $C_6:S_3$ x 6, $C_2\times C_{18}$

Order 27: $C_3\times C_9$ x 39, $C_3^3$ x 16

Order 24: $C_2\times D_6$ x 4

Order 18: $C_3\times S_3$ x 60, $C_3:S_3$ x 52, $D_9$ x 15, $C_3\times C_6$ x 6, $C_{18}$ x 6

Order 12: $D_6$ x 30, $C_2\times C_6$ x 4

Order 9: $C_3^2$ x 46, $C_9$ x 9

Order 8: $C_2^3$

Order 6: $S_3$ x 34, $C_6$ x 18

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 16

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1944: $C_9:S_3^3$

Order 972: $C_3^3:D_{18}$, $C_9\times C_3:S_3^2$, $C_3^3:D_{18}$

Order 648: $C_3:S_3^3$, $D_9\times S_3^2$

Order 486: $C_3^3:D_9$, $C_3^3:C_{18}$, $C_3^3:D_9$

Order 324: $C_3\wr C_2^2$ x 2, $C_9:S_3^2$ x 2, $C_3^2:D_{18}$ x 2, $C_3^2:S_3^2$, $C_9:S_3^2$, $C_9\times S_3^2$, $C_3^2:D_{18}$

Order 243: $C_3^3\times C_9$

Order 216: $S_3^3$ x 2, $S_3\times D_{18}$

Order 162: $C_3^2:D_9$ x 6, $C_3^2\wr C_2$ x 2, $C_3^2:C_{18}$ x 2, $C_3^2:D_9$ x 2, $C_3^2:C_{18}$ x 2, $C_3^2\times D_9$ x 2, $C_3^3:S_3$

Order 108: $C_3:S_3^2$ x 5, $S_3\times D_9$ x 5, $C_3\times S_3^2$ x 3, $C_3:S_3^2$ x 2, $C_3:D_{18}$, $S_3\times C_{18}$, $C_3\times D_{18}$

Order 81: $C_3^2\times C_9$ x 6, $C_3^4$

Order 72: $S_3\times D_6$ x 2, $C_2\times D_{18}$

Order 54: $C_3:D_9$ x 11, $C_3^2:S_3$ x 7, $C_3^2:C_6$ x 5, $S_3\times C_3^2$ x 5, $S_3\times C_9$ x 4, $C_3\times D_9$ x 4, $C_3\times C_{18}$

Order 36: $S_3^2$ x 9, $D_{18}$ x 3, $C_6\times S_3$ x 3, $C_6:S_3$ x 2, $C_2\times C_{18}$

Order 27: $C_3\times C_9$ x 10, $C_3^3$ x 7

Order 24: $C_2\times D_6$ x 2

Order 18: $C_3:S_3$ x 15, $C_3\times S_3$ x 12, $D_9$ x 6, $C_3\times C_6$ x 2, $C_{18}$ x 2

Order 12: $D_6$ x 8, $C_2\times C_6$ x 2

Order 9: $C_3^2$ x 13, $C_9$ x 4

Order 8: $C_2^3$

Order 6: $S_3$ x 12, $C_6$ x 5

Order 4: $C_2^2$ x 3

Order 3: $C_3$ x 7

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1944: $C_9:S_3^3$ ($C_1$)

Order 972: $C_3^3:D_{18}$ ($C_2$) x 3, $C_3^3:D_{18}$ ($C_2$) x 3, $C_9\times C_3:S_3^2$ ($C_2$)

Order 486: $C_3^3:C_{18}$ ($C_2^2$) x 3, $C_3^3:D_9$ ($C_2^2$) x 3, $C_3^3:D_9$ ($C_2^2$)

Order 324: $C_9:S_3^2$ ($S_3$) x 3, $C_3\wr C_2^2$ ($S_3$)

Order 243: $C_3^3\times C_9$ ($C_2^3$)

Order 162: $C_3^2:D_9$ ($D_6$) x 6, $C_3^2\wr C_2$ ($D_6$) x 3, $C_3^2:C_{18}$ ($D_6$) x 3

Order 108: $C_3:S_3^2$ ($D_9$)

Order 81: $C_3^2\times C_9$ ($C_2\times D_6$) x 3, $C_3^4$ ($C_2\times D_6$)

Order 54: $C_3:D_9$ ($S_3^2$) x 3, $C_3^2:C_6$ ($D_{18}$) x 3, $C_3^2:C_6$ ($S_3^2$) x 3

Order 27: $C_3^3$ ($S_3\times D_6$) x 3, $C_3\times C_9$ ($S_3\times D_6$) x 3, $C_3^3$ ($C_2\times D_{18}$)

Order 18: $C_3:S_3$ ($S_3\times D_9$) x 3

Order 9: $C_3^2$ ($S_3^3$) x 3, $C_3^2$ ($S_3\times D_{18}$) x 3, $C_9$ ($S_3^3$)

Order 3: $C_3$ ($D_9\times S_3^2$) x 3, $C_3$ ($C_3:S_3^3$)

Order 1: $C_1$ ($C_9:S_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1944: $C_9:S_3^3$ ($C_1$)

Order 972: $C_3^3:D_{18}$ ($C_2$), $C_9\times C_3:S_3^2$ ($C_2$), $C_3^3:D_{18}$ ($C_2$)

Order 486: $C_3^3:D_9$ ($C_2^2$), $C_3^3:C_{18}$ ($C_2^2$), $C_3^3:D_9$ ($C_2^2$)

Order 324: $C_3\wr C_2^2$ ($S_3$), $C_9:S_3^2$ ($S_3$)

Order 243: $C_3^3\times C_9$ ($C_2^3$)

Order 162: $C_3^2\wr C_2$ ($D_6$), $C_3^2:C_{18}$ ($D_6$), $C_3^2:D_9$ ($D_6$)

Order 108: $C_3:S_3^2$ ($D_9$)

Order 81: $C_3^4$ ($C_2\times D_6$), $C_3^2\times C_9$ ($C_2\times D_6$)

Order 54: $C_3:D_9$ ($S_3^2$), $C_3^2:C_6$ ($D_{18}$), $C_3^2:C_6$ ($S_3^2$)

Order 27: $C_3^3$ ($S_3\times D_6$), $C_3^3$ ($C_2\times D_{18}$), $C_3\times C_9$ ($S_3\times D_6$)

Order 18: $C_3:S_3$ ($S_3\times D_9$)

Order 9: $C_3^2$ ($S_3^3$), $C_3^2$ ($S_3\times D_{18}$), $C_9$ ($S_3^3$)

Order 3: $C_3$ ($C_3:S_3^3$), $C_3$ ($D_9\times S_3^2$)

Order 1: $C_1$ ($C_9:S_3^3$)

Series

Derived series

$C_9:S_3^3$

$\rhd$

$C_3^3\times C_9$

$\rhd$

$C_1$

Chief series

$C_9:S_3^3$

$\rhd$

$C_3^3:D_{18}$

$\rhd$

$C_3^3:D_9$

$\rhd$

$C_3^2:D_9$

$\rhd$

$C_3^2\times C_9$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_9:S_3^3$

$\rhd$

$C_3^3\times C_9$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 6 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 $87 \times 87$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $57 \times 57$ rational character table.