Abstract group 1944.2345: $C_3^2.S_3^3$

• Label: 1944.2345
• Order: \( 2^{3} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 3 \)
• Perm deg.: $30$
• Trans deg.: $54$
• Rank: $3$

Group information

Description:	$C_3^2.S_3^3$
Order:	\(1944\)\(\medspace = 2^{3} \cdot 3^{5} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$(C_3\times C_9):C_3^2:C_2^2\times S_3$, of order \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \)
Composition factors:	$C_2$ x 3, $C_3$ x 5
Derived length:	$3$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	255	80	960	162	486	1944
Conjugacy classes	1	7	7	15	8	13	51
Divisions	1	7	7	15	4	5	39
Autjugacy classes	1	7	7	15	4	5	39

Dimension	1	2	4	6	8	12	18	36
Irr. complex chars.	8	12	6	16	1	8	0	0	51
Irr. rational chars.	8	12	6	4	1	2	4	2	39

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$54$
Rank:	$3$
Inequivalent generating triples:	$290304$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	36
Arbitrary	8	8	20

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{6}=c^{6}=d^{9}=e^{3}=[a,c]=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(2,5)(3,10)(4,13)(6,14)(7,9)(8,21)(11,24)(12,25)(15,22)(16,23)(17,20)(18,19) \!\cdots\! \rangle$
Direct product:	$S_3$ $\, \times\, $ $(C_3^2.S_3^2)$
Semidirect product:	$(C_9:S_3^2)$ $\,\rtimes\,$ $S_3$	$(C_3:D_9)$ $\,\rtimes\,$ $S_3^2$	$(\He_3.S_3)$ $\,\rtimes\,$ $D_6$	$(\He_3.S_3)$ $\,\rtimes\,$ $D_6$	all 27
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^2$ . $S_3^3$	$(C_3^3:S_3)$ . $D_6$	$(C_3^3:S_3)$ . $D_6$	$C_3^3$ . $(S_3\times D_6)$	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 9008 subgroups in 418 conjugacy classes, 44 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

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

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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Classes of subgroups up to conjugation

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

Order 972: $C_3^3.S_3^2$, $\He_3.S_3^2$, $C_3^3.S_3^2$, $C_3^3.S_3^2$, $\He_3.S_3^2$, $(C_3\times C_9):S_3^2$, $C_3^3.S_3^2$

Order 648: $C_3.S_3^3$, $D_9\times S_3^2$, $(C_3\times C_{18}):D_6$

Order 486: $(C_3^2\times C_9):S_3$, $(C_3^2\times C_9):C_6$, $(C_3^2\times C_9):S_3$, $(C_3^2\times C_9):S_3$, $(C_3^2\times C_9):C_6$, $(C_3^2\times C_9):C_6$, $(C_3^2\times C_9):S_3$

Order 324: $C_3^2.S_3^2$ x 6, $C_3^2:S_3^2$ x 2, $C_9:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^2:D_{18}$ x 2, $\He_3.D_6$, $\He_3.D_6$, $(C_3\times C_{18}):S_3$, $C_3^2:S_3^2$, $C_9:S_3^2$, $C_3^2:S_3^2$, $C_3^2:D_{18}$, $C_3^2.S_3^2$, $C_3^3:D_6$, $C_9\times S_3^2$

Order 243: $C_3^2.C_3^3$

Order 216: $S_3^3$ x 2, $S_3\times D_{18}$ x 2, $C_6.S_3^2$

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

Order 108: $S_3\times D_9$ x 11, $C_3:S_3^2$ x 6, $C_3\times S_3^2$ x 6, $C_3^2:D_6$ x 6, $C_3:S_3^2$ x 2, $C_3:D_{18}$ x 2, $C_3^2:D_6$ x 2, $S_3\times C_{18}$ x 2, $C_3\times D_{18}$ x 2, $C_3^2:D_6$, $C_{18}:C_6$

Order 81: $C_3.\He_3$ x 4, $C_9:C_3^2$, $C_3\times \He_3$, $C_3^2\times C_9$

Order 72: $S_3\times D_6$ x 5, $C_2\times D_{18}$

Order 54: $C_3^2:C_6$ x 8, $C_3:D_9$ x 7, $S_3\times C_9$ x 7, $C_3\times D_9$ x 7, $C_3^2:C_6$ x 6, $S_3\times C_3^2$ x 6, $C_3^2:S_3$ x 4, $C_9:C_6$ x 4, $C_3\times C_{18}$ x 2, $C_3^2:S_3$ x 2, $C_9:C_6$, $C_2\times \He_3$

Order 36: $S_3^2$ x 24, $C_6\times S_3$ x 10, $D_{18}$ x 8, $C_6:S_3$ x 5, $C_2\times C_{18}$

Order 27: $C_3\times C_9$ x 5, $C_9:C_3$ x 3, $\He_3$ x 3, $C_3^3$ x 2

Order 24: $C_2\times D_6$ x 4

Order 18: $C_3\times S_3$ x 28, $C_3:S_3$ x 14, $D_9$ x 9, $C_3\times C_6$ x 5, $C_{18}$ x 5

Order 12: $D_6$ x 27, $C_2\times C_6$ x 4

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

Order 8: $C_2^3$

Order 6: $S_3$ x 22, $C_6$ x 15

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: $C_3^2.S_3^3$

Order 972: $C_3^3.S_3^2$, $\He_3.S_3^2$, $C_3^3.S_3^2$, $C_3^3.S_3^2$, $\He_3.S_3^2$, $(C_3\times C_9):S_3^2$, $C_3^3.S_3^2$

Order 648: $C_3.S_3^3$, $D_9\times S_3^2$, $(C_3\times C_{18}):D_6$

Order 486: $(C_3^2\times C_9):S_3$, $(C_3^2\times C_9):C_6$, $(C_3^2\times C_9):S_3$, $(C_3^2\times C_9):S_3$, $(C_3^2\times C_9):C_6$, $(C_3^2\times C_9):C_6$, $(C_3^2\times C_9):S_3$

Order 324: $C_3^2.S_3^2$ x 6, $C_3^2:S_3^2$ x 2, $C_9:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^2:D_{18}$ x 2, $\He_3.D_6$, $\He_3.D_6$, $(C_3\times C_{18}):S_3$, $C_3^2:S_3^2$, $C_9:S_3^2$, $C_3^2:S_3^2$, $C_3^2:D_{18}$, $C_3^2.S_3^2$, $C_3^3:D_6$, $C_9\times S_3^2$

Order 243: $C_3^2.C_3^3$

Order 216: $S_3^3$ x 2, $S_3\times D_{18}$ x 2, $C_6.S_3^2$

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

Order 108: $S_3\times D_9$ x 11, $C_3:S_3^2$ x 6, $C_3\times S_3^2$ x 6, $C_3^2:D_6$ x 6, $C_3:S_3^2$ x 2, $C_3:D_{18}$ x 2, $C_3^2:D_6$ x 2, $S_3\times C_{18}$ x 2, $C_3\times D_{18}$ x 2, $C_3^2:D_6$, $C_{18}:C_6$

Order 81: $C_3.\He_3$ x 4, $C_9:C_3^2$, $C_3\times \He_3$, $C_3^2\times C_9$

Order 72: $S_3\times D_6$ x 5, $C_2\times D_{18}$

Order 54: $C_3^2:C_6$ x 8, $C_3:D_9$ x 7, $S_3\times C_9$ x 7, $C_3\times D_9$ x 7, $C_3^2:C_6$ x 6, $S_3\times C_3^2$ x 6, $C_3^2:S_3$ x 4, $C_9:C_6$ x 4, $C_3\times C_{18}$ x 2, $C_3^2:S_3$ x 2, $C_9:C_6$, $C_2\times \He_3$

Order 36: $S_3^2$ x 24, $C_6\times S_3$ x 10, $D_{18}$ x 8, $C_6:S_3$ x 5, $C_2\times C_{18}$

Order 27: $C_3\times C_9$ x 5, $C_9:C_3$ x 3, $\He_3$ x 3, $C_3^3$ x 2

Order 24: $C_2\times D_6$ x 4

Order 18: $C_3\times S_3$ x 28, $C_3:S_3$ x 14, $D_9$ x 9, $C_3\times C_6$ x 5, $C_{18}$ x 5

Order 12: $D_6$ x 27, $C_2\times C_6$ x 4

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

Order 8: $C_2^3$

Order 6: $S_3$ x 22, $C_6$ x 15

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 972: $C_3^3.S_3^2$ ($C_2$), $\He_3.S_3^2$ ($C_2$), $C_3^3.S_3^2$ ($C_2$), $C_3^3.S_3^2$ ($C_2$), $\He_3.S_3^2$ ($C_2$), $(C_3\times C_9):S_3^2$ ($C_2$), $C_3^3.S_3^2$ ($C_2$)

Order 486: $(C_3^2\times C_9):S_3$ ($C_2^2$), $(C_3^2\times C_9):C_6$ ($C_2^2$), $(C_3^2\times C_9):S_3$ ($C_2^2$), $(C_3^2\times C_9):S_3$ ($C_2^2$), $(C_3^2\times C_9):C_6$ ($C_2^2$), $(C_3^2\times C_9):C_6$ ($C_2^2$), $(C_3^2\times C_9):S_3$ ($C_2^2$)

Order 324: $C_3^2.S_3^2$ ($S_3$), $C_3^2:S_3^2$ ($S_3$), $C_9:S_3^2$ ($S_3$)

Order 243: $C_3^2.C_3^3$ ($C_2^3$)

Order 162: $C_3^3:S_3$ ($D_6$), $C_3^2:D_9$ ($D_6$), $C_3^3:S_3$ ($D_6$), $C_3^2:D_9$ ($D_6$), $S_3\times \He_3$ ($D_6$), $C_3^2:C_{18}$ ($D_6$), $\He_3.S_3$ ($D_6$), $\He_3.S_3$ ($D_6$), $\He_3.C_6$ ($D_6$)

Order 81: $C_3.\He_3$ ($C_2\times D_6$), $C_3\times \He_3$ ($C_2\times D_6$), $C_3^2\times C_9$ ($C_2\times D_6$)

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

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

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

Order 9: $C_3^2$ ($S_3^3$), $C_3^2$ ($C_6.S_3^2$)

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

Order 3: $C_3$ ($C_3.S_3^3$), $C_3$ ($(C_3\times C_{18}):D_6$)

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: $C_3^3.S_3^2$ ($C_2$), $\He_3.S_3^2$ ($C_2$), $C_3^3.S_3^2$ ($C_2$), $C_3^3.S_3^2$ ($C_2$), $\He_3.S_3^2$ ($C_2$), $(C_3\times C_9):S_3^2$ ($C_2$), $C_3^3.S_3^2$ ($C_2$)

Order 486: $(C_3^2\times C_9):S_3$ ($C_2^2$), $(C_3^2\times C_9):C_6$ ($C_2^2$), $(C_3^2\times C_9):S_3$ ($C_2^2$), $(C_3^2\times C_9):S_3$ ($C_2^2$), $(C_3^2\times C_9):C_6$ ($C_2^2$), $(C_3^2\times C_9):C_6$ ($C_2^2$), $(C_3^2\times C_9):S_3$ ($C_2^2$)

Order 324: $C_3^2.S_3^2$ ($S_3$), $C_3^2:S_3^2$ ($S_3$), $C_9:S_3^2$ ($S_3$)

Order 243: $C_3^2.C_3^3$ ($C_2^3$)

Order 162: $C_3^3:S_3$ ($D_6$), $C_3^2:D_9$ ($D_6$), $C_3^3:S_3$ ($D_6$), $C_3^2:D_9$ ($D_6$), $S_3\times \He_3$ ($D_6$), $C_3^2:C_{18}$ ($D_6$), $\He_3.S_3$ ($D_6$), $\He_3.S_3$ ($D_6$), $\He_3.C_6$ ($D_6$)

Order 81: $C_3.\He_3$ ($C_2\times D_6$), $C_3\times \He_3$ ($C_2\times D_6$), $C_3^2\times C_9$ ($C_2\times D_6$)

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

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

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

Order 9: $C_3^2$ ($S_3^3$), $C_3^2$ ($C_6.S_3^2$)

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

Order 3: $C_3$ ($C_3.S_3^3$), $C_3$ ($(C_3\times C_{18}):D_6$)

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

Series

Derived series

$C_3^2.S_3^3$

$\rhd$

$C_3^2.C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_3^2.S_3^3$

$\rhd$

$\He_3.S_3^2$

$\rhd$

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

$\rhd$

$\He_3.S_3$

$\rhd$

$C_3.\He_3$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^2.S_3^3$

$\rhd$

$C_3^2.C_3^3$

Upper central series

$C_1$

Supergroups

Character theory

Complex character table

See the $51 \times 51$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $39 \times 39$ rational character table.