Abstract group 1944.2642: $C_3^3.(S_3\times A_4)$

• Label: 1944.2642
• Order: \( 2^{3} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{4} \)
• Perm deg.: $34$
• Trans deg.: $162$
• Rank: $3$

Group information

Description:	$C_3^3.(S_3\times A_4)$
Order:	\(1944\)\(\medspace = 2^{3} \cdot 3^{5} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^4.C_6^2.C_3^3.C_2^3$, of order \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \)
Composition factors:	$C_2$ x 3, $C_3$ x 5
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	327	674	726	54	162	1944
Conjugacy classes	1	3	17	15	9	27	72
Divisions	1	3	12	14	3	9	42
Autjugacy classes	1	3	7	5	1	1	18

Dimension	1	2	3	4	6	18
Irr. complex chars.	6	12	2	0	52	0	72
Irr. rational chars.	2	6	2	4	16	12	42

Minimal presentations

Permutation degree:	$34$
Transitive degree:	$162$
Rank:	$3$
Inequivalent generating triples:	$5460$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{6}=b^{6}=c^{6}=d^{9}=[b,c]=[b,d]=[c,d]=1, b^{a}=b^{2}c^{3}, c^{a}=b^{3}c^{5}d^{6}, d^{a}=c^{4}d^{2} \rangle$
Permutation group:	Degree $34$ $\langle(3,9)(4,11)(6,8)(7,17)(10,13)(12,22)(14,20)(15,24)(16,19)(18,25)(21,23) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^2:D_9)$ $\,\rtimes\,$ $A_4$	$(C_6^2:D_9)$ $\,\rtimes\,$ $C_3$	$(C_6^2.C_3^2)$ $\,\rtimes\,$ $S_3$	$(C_6^2.C_3^3)$ $\,\rtimes\,$ $C_2$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^3:A_4)$ . $S_3$	$C_3^3$ . $(S_3\times A_4)$	$C_6^2$ . $(C_3^2:C_6)$ (2)	$C_6^2$ . $(C_3^2:C_6)$	all 10

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

Homology

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

Subgroups

There are 10000 subgroups in 404 conjugacy classes, 37 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^3.(S_3\times A_4)$
Commutator:	$G' \simeq$ $C_3\times C_6\times C_{18}$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_6^2:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6\times C_{18}$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $C_3^3.(S_3\times A_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_3^2:C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3:C_3^2$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1944: $C_3^3.(S_3\times A_4)$

Order 972: $C_6^2.C_3^3$

Order 648: $(C_6\times C_{18}):C_6$ x 3, $C_6^2:D_9$, $(C_3\times C_6^2):C_6$

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

Order 324: $C_6^2.C_3^2$ x 6, $C_3^2:D_{18}$ x 2, $C_3^3:A_4$ x 2, $C_3\times C_6\times C_{18}$

Order 243: $\He_3:C_3^2$

Order 216: $C_6:D_{18}$ x 6, $C_6^2:C_6$ x 3, $C_6^2:S_3$, $C_6^2:C_6$

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

Order 108: $C_3:D_{18}$ x 24, $C_3^2:A_4$ x 9, $C_6\times C_{18}$ x 6, $C_3^2:D_6$ x 2, $C_3^2\times A_4$ x 2, $C_3\times C_6^2$

Order 81: $\He_3:C_3$ x 6, $C_3\times \He_3$ x 2, $C_3^2\times C_9$

Order 72: $C_6:D_6$ x 7, $S_3\times A_4$ x 4, $C_2\times D_{18}$ x 3

Order 54: $C_3:D_9$ x 18, $C_3\times C_{18}$ x 12, $C_3^2:C_6$ x 3, $C_3^2:S_3$ x 2, $C_3^2\times C_6$, $C_3^2:C_6$

Order 36: $C_6:S_3$ x 26, $D_{18}$ x 18, $C_3\times A_4$ x 11, $C_6^2$ x 7, $C_2\times C_{18}$ x 3

Order 27: $\He_3$ x 9, $C_3\times C_9$ x 6, $C_3^3$ x 3

Order 24: $C_2\times D_6$ x 7, $C_2\times A_4$

Order 18: $C_3:S_3$ x 20, $C_3\times C_6$ x 13, $D_9$ x 12, $C_{18}$ x 9, $C_3\times S_3$ x 4

Order 12: $D_6$ x 26, $C_2\times C_6$ x 7, $A_4$ x 5

Order 9: $C_3^2$ x 18, $C_9$ x 3

Order 8: $C_2^3$

Order 6: $S_3$ x 20, $C_6$ x 14

Order 4: $C_2^2$ x 3

Order 3: $C_3$ x 12

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1944: $C_3^3.(S_3\times A_4)$

Order 972: $C_6^2.C_3^3$

Order 648: $C_6^2:D_9$, $(C_3\times C_6^2):C_6$, $(C_6\times C_{18}):C_6$

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

Order 324: $C_6^2.C_3^2$ x 2, $C_3^2:D_{18}$ x 2, $C_3^3:A_4$ x 2, $C_3\times C_6\times C_{18}$

Order 243: $\He_3:C_3^2$

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

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

Order 108: $C_3:D_{18}$ x 6, $C_3^2:A_4$ x 5, $C_6\times C_{18}$ x 3, $C_3^2:D_6$ x 2, $C_3^2\times A_4$ x 2, $C_3\times C_6^2$

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

Order 72: $C_6:D_6$ x 4, $S_3\times A_4$ x 2, $C_2\times D_{18}$

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

Order 36: $C_6:S_3$ x 8, $C_3\times A_4$ x 5, $C_6^2$ x 4, $D_{18}$ x 2, $C_2\times C_{18}$

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

Order 24: $C_2\times D_6$ x 4, $C_2\times A_4$

Order 18: $C_3:S_3$ x 8, $C_3\times C_6$ x 4, $C_3\times S_3$ x 2, $D_9$ x 2, $C_{18}$

Order 12: $D_6$ x 8, $C_2\times C_6$ x 4, $A_4$ x 3

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

Order 8: $C_2^3$

Order 6: $S_3$ x 8, $C_6$ 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: $C_3^3.(S_3\times A_4)$ ($C_1$)

Order 972: $C_6^2.C_3^3$ ($C_2$)

Order 648: $C_6^2:D_9$ ($C_3$)

Order 324: $C_6^2.C_3^2$ ($S_3$) x 3, $C_3\times C_6\times C_{18}$ ($C_6$), $C_3^3:A_4$ ($S_3$)

Order 162: $C_3^2:D_9$ ($A_4$)

Order 108: $C_6\times C_{18}$ ($C_3\times S_3$) x 3, $C_3\times C_6^2$ ($C_3\times S_3$), $C_3^2:A_4$ ($C_3:S_3$)

Order 81: $C_3^2\times C_9$ ($C_2\times A_4$)

Order 36: $C_6^2$ ($C_3^2:C_6$) x 3, $C_6^2$ ($C_3^2:C_6$)

Order 27: $C_3\times C_9$ ($S_3\times A_4$) x 3, $C_3^3$ ($S_3\times A_4$)

Order 12: $C_2\times C_6$ ($\He_3.S_3$) x 3, $C_2\times C_6$ ($C_3^3:C_6$)

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

Order 4: $C_2^2$ ($(C_3^2\times C_9):C_6$)

Order 3: $C_3$ ($(C_6\times C_{18}):C_6$) x 3, $C_3$ ($(C_3\times C_6^2):C_6$)

Order 1: $C_1$ ($C_3^3.(S_3\times A_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1944: $C_3^3.(S_3\times A_4)$ ($C_1$)

Order 972: $C_6^2.C_3^3$ ($C_2$)

Order 648: $C_6^2:D_9$ ($C_3$)

Order 324: $C_6^2.C_3^2$ ($S_3$), $C_3\times C_6\times C_{18}$ ($C_6$), $C_3^3:A_4$ ($S_3$)

Order 162: $C_3^2:D_9$ ($A_4$)

Order 108: $C_3\times C_6^2$ ($C_3\times S_3$), $C_6\times C_{18}$ ($C_3\times S_3$), $C_3^2:A_4$ ($C_3:S_3$)

Order 81: $C_3^2\times C_9$ ($C_2\times A_4$)

Order 36: $C_6^2$ ($C_3^2:C_6$) x 2, $C_6^2$ ($C_3^2:C_6$)

Order 27: $C_3^3$ ($S_3\times A_4$), $C_3\times C_9$ ($S_3\times A_4$)

Order 12: $C_2\times C_6$ ($C_3^3:C_6$), $C_2\times C_6$ ($\He_3.S_3$)

Order 9: $C_3^2$ ($C_6^2:C_6$) x 2, $C_3^2$ ($C_6^2:C_6$)

Order 4: $C_2^2$ ($(C_3^2\times C_9):C_6$)

Order 3: $C_3$ ($(C_3\times C_6^2):C_6$), $C_3$ ($(C_6\times C_{18}):C_6$)

Order 1: $C_1$ ($C_3^3.(S_3\times A_4)$)

Series

Derived series

$C_3^3.(S_3\times A_4)$

$\rhd$

$C_3\times C_6\times C_{18}$

$\rhd$

$C_1$

Chief series

$C_3^3.(S_3\times A_4)$

$\rhd$

$C_6^2.C_3^3$

$\rhd$

$C_3\times C_6\times C_{18}$

$\rhd$

$C_3^2\times C_9$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^3.(S_3\times A_4)$

$\rhd$

$C_3\times C_6\times C_{18}$

Upper central series

$C_1$

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $42 \times 42$ rational character table.