Abstract group 1944.2339: $S_3\times D_9^2$

• Label: 1944.2339
• Order: \( 2^{3} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3^{2} \)
• Perm deg.: $21$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$S_3\times D_9^2$
Order:	\(1944\)\(\medspace = 2^{3} \cdot 3^{5} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$D_9^2:(C_3\times S_3^2)$, of order \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \)
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	399	26	438	216	864	1944
Conjugacy classes	1	7	7	12	42	39	108
Divisions	1	7	7	12	14	13	54
Autjugacy classes	1	5	5	7	6	6	30

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	8	36	48	0	16	0	0	108
Irr. rational chars.	8	12	6	8	1	14	5	54

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$48384$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	24
Arbitrary	6	6	14

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{6}=c^{18}=d^{9}=[a,c]=[a,d]=[b,d]=1, b^{a}=b^{5}, c^{b}=c^{17}, d^{c}=d^{8} \rangle$
Permutation group:	Degree $21$ $\langle(2,4)(3,7)(5,6)(8,9), (14,16)(15,19)(17,18)(20,21), (11,12), (13,14,17,19,21,20,15,18,16) \!\cdots\! \rangle$
Transitive group:	36T2937				more information
Direct product:	$S_3$ $\, \times\, $ $D_9$ ${}^2$
Semidirect product:	$(C_9\times D_9)$ $\,\rtimes\,$ $D_6$ (2)	$(C_9:D_9)$ $\,\rtimes\,$ $D_6$	$(S_3\times C_9)$ $\,\rtimes\,$ $D_{18}$ (2)	$C_9$ $\,\rtimes\,$ $(S_3\times D_{18})$ (2)	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^2$ . $S_3^3$	$(C_3\times D_9)$ . $S_3^2$ (2)	$C_3^3$ . $(S_3\times D_6)$	$C_3$ . $(D_9\times S_3^2)$ (2)	all 13

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 9926 subgroups in 446 conjugacy classes, 61 normal (17 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_3\times D_9^2$
Commutator:	$G' \simeq$ $C_3\times C_9^2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $S_3^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_9^2$	$G/\operatorname{Fit} \simeq$ $C_2^3$
Radical:	$R \simeq$ $S_3\times D_9^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^3$	$G/\operatorname{soc} \simeq$ $S_3\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9^2$

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 1944: $S_3\times D_9^2$

Order 972: $C_3:D_9^2$ x 2, $C_9^2:D_6$ x 2, $C_3:D_9^2$, $C_9^2:D_6$, $C_3\times D_9^2$

Order 648: $D_9\times S_3^2$ x 2, $D_9\times D_{18}$

Order 486: $C_9^2:C_6$ x 2, $C_9^2:S_3$ x 2, $S_3\times C_9^2$, $C_9^2:S_3$, $C_9^2:C_6$

Order 324: $D_9^2$ x 6, $C_9:S_3^2$ x 5, $C_3^2:D_{18}$ x 4, $C_9\times D_{18}$ x 2, $C_9:S_3^2$ x 2, $C_3^2:D_{18}$ x 2, $C_9\times S_3^2$ x 2, $C_9:D_{18}$

Order 243: $C_3\times C_9^2$

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

Order 162: $C_9\times D_9$ x 8, $C_3^2:D_9$ x 5, $C_3^2:C_{18}$ x 5, $C_9:D_9$ x 5, $C_3^2:D_9$ x 3, $C_3^2:C_{18}$ x 2, $C_3^2\times D_9$ x 2, $C_9\times C_{18}$

Order 108: $S_3\times D_9$ x 25, $C_3:D_{18}$ x 5, $S_3\times C_{18}$ x 4, $C_3\times D_{18}$ x 4, $C_3:S_3^2$ x 3, $C_3\times S_3^2$ x 3, $C_3:S_3^2$

Order 81: $C_9^2$ x 4, $C_3^2\times C_9$ x 3

Order 72: $S_3\times D_6$ x 3, $C_2\times D_{18}$ x 2

Order 54: $C_3:D_9$ x 25, $S_3\times C_9$ x 17, $C_3\times D_9$ x 17, $C_3\times C_{18}$ x 5, $C_3^2:C_6$ x 3, $S_3\times C_3^2$ x 3, $C_3^2:S_3$

Order 36: $D_{18}$ x 19, $S_3^2$ x 15, $C_6\times S_3$ x 6, $C_6:S_3$ x 3, $C_2\times C_{18}$ x 2

Order 27: $C_3\times C_9$ x 20, $C_3^3$

Order 24: $C_2\times D_6$ x 3

Order 18: $D_9$ x 27, $C_3\times S_3$ x 18, $C_{18}$ x 13, $C_3:S_3$ x 10, $C_3\times C_6$ x 3

Order 12: $D_6$ x 21, $C_2\times C_6$ x 3

Order 9: $C_9$ x 14, $C_3^2$ x 7

Order 8: $C_2^3$

Order 6: $S_3$ x 19, $C_6$ x 12

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1944: $S_3\times D_9^2$

Order 972: $C_3:D_9^2$, $C_9^2:D_6$, $C_3:D_9^2$, $C_9^2:D_6$, $C_3\times D_9^2$

Order 648: $D_9\times S_3^2$, $D_9\times D_{18}$

Order 486: $S_3\times C_9^2$, $C_9^2:C_6$, $C_9^2:S_3$, $C_9^2:C_6$, $C_9^2:S_3$

Order 324: $D_9^2$ x 4, $C_9:S_3^2$ x 3, $C_3^2:D_{18}$ x 2, $C_9:D_{18}$, $C_9\times D_{18}$, $C_9:S_3^2$, $C_3^2:D_{18}$, $C_9\times S_3^2$

Order 243: $C_3\times C_9^2$

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

Order 162: $C_9\times D_9$ x 4, $C_9:D_9$ x 4, $C_3^2:D_9$ x 3, $C_3^2:C_{18}$ x 3, $C_3^2:D_9$ x 2, $C_3^2:C_{18}$, $C_3^2\times D_9$, $C_9\times C_{18}$

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

Order 81: $C_9^2$ x 3, $C_3^2\times C_9$ x 2

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

Order 54: $C_3:D_9$ x 13, $S_3\times C_9$ x 8, $C_3\times D_9$ x 8, $C_3\times C_{18}$ x 3, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2:S_3$

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

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

Order 24: $C_2\times D_6$ x 2

Order 18: $D_9$ x 12, $C_3\times S_3$ x 10, $C_3:S_3$ x 7, $C_{18}$ x 6, $C_3\times C_6$ x 2

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

Order 9: $C_9$ x 6, $C_3^2$ x 5

Order 8: $C_2^3$

Order 6: $S_3$ x 12, $C_6$ x 7

Order 4: $C_2^2$ x 5

Order 3: $C_3$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1944: $S_3\times D_9^2$ ($C_1$)

Order 972: $C_3:D_9^2$ ($C_2$) x 2, $C_9^2:D_6$ ($C_2$) x 2, $C_3:D_9^2$ ($C_2$), $C_9^2:D_6$ ($C_2$), $C_3\times D_9^2$ ($C_2$)

Order 486: $C_9^2:C_6$ ($C_2^2$) x 2, $C_9^2:S_3$ ($C_2^2$) x 2, $S_3\times C_9^2$ ($C_2^2$), $C_9^2:S_3$ ($C_2^2$), $C_9^2:C_6$ ($C_2^2$)

Order 324: $C_3^2:D_{18}$ ($S_3$) x 2, $D_9^2$ ($S_3$)

Order 243: $C_3\times C_9^2$ ($C_2^3$)

Order 162: $C_3^2:D_9$ ($D_6$) x 2, $C_3^2:C_{18}$ ($D_6$) x 2, $C_3^2\times D_9$ ($D_6$) x 2, $C_9\times D_9$ ($D_6$) x 2, $C_9:D_9$ ($D_6$)

Order 108: $S_3\times D_9$ ($D_9$) x 2

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

Order 54: $C_3:D_9$ ($D_{18}$) x 2, $S_3\times C_9$ ($D_{18}$) x 2, $C_3\times D_9$ ($D_{18}$) x 2, $C_3\times D_9$ ($S_3^2$) x 2, $S_3\times C_3^2$ ($S_3^2$)

Order 27: $C_3\times C_9$ ($S_3\times D_6$) x 2, $C_3\times C_9$ ($C_2\times D_{18}$) x 2, $C_3^3$ ($S_3\times D_6$)

Order 18: $C_3\times S_3$ ($S_3\times D_9$) x 2, $D_9$ ($S_3\times D_9$) x 2

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

Order 6: $S_3$ ($D_9^2$)

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

Order 1: $C_1$ ($S_3\times D_9^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1944: $S_3\times D_9^2$ ($C_1$)

Order 972: $C_3:D_9^2$ ($C_2$), $C_9^2:D_6$ ($C_2$), $C_3:D_9^2$ ($C_2$), $C_9^2:D_6$ ($C_2$), $C_3\times D_9^2$ ($C_2$)

Order 486: $S_3\times C_9^2$ ($C_2^2$), $C_9^2:C_6$ ($C_2^2$), $C_9^2:S_3$ ($C_2^2$), $C_9^2:C_6$ ($C_2^2$), $C_9^2:S_3$ ($C_2^2$)

Order 324: $D_9^2$ ($S_3$), $C_3^2:D_{18}$ ($S_3$)

Order 243: $C_3\times C_9^2$ ($C_2^3$)

Order 162: $C_3^2:D_9$ ($D_6$), $C_3^2:C_{18}$ ($D_6$), $C_3^2\times D_9$ ($D_6$), $C_9\times D_9$ ($D_6$), $C_9:D_9$ ($D_6$)

Order 108: $S_3\times D_9$ ($D_9$)

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

Order 54: $C_3:D_9$ ($D_{18}$), $S_3\times C_9$ ($D_{18}$), $C_3\times D_9$ ($D_{18}$), $C_3\times D_9$ ($S_3^2$), $S_3\times C_3^2$ ($S_3^2$)

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

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

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

Order 6: $S_3$ ($D_9^2$)

Order 3: $C_3$ ($D_9\times S_3^2$), $C_3$ ($D_9\times D_{18}$)

Order 1: $C_1$ ($S_3\times D_9^2$)

Series

Derived series

$S_3\times D_9^2$

$\rhd$

$C_3\times C_9^2$

$\rhd$

$C_1$

Chief series

$S_3\times D_9^2$

$\rhd$

$C_9^2:D_6$

$\rhd$

$C_9^2:C_6$

$\rhd$

$C_9\times D_9$

$\rhd$

$C_9^2$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$S_3\times D_9^2$

$\rhd$

$C_3\times C_9^2$

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 $108 \times 108$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $54 \times 54$ rational character table.