Abstract group 1944.2449: $C_3^3.(C_3\times S_4)$

• Label: 1944.2449
• Order: \( 2^{3} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3^{2} \)
• Perm deg.: $40$
• Trans deg.: $108$
• Rank: $3$

Group information

Description:	$C_3^3.(C_3\times S_4)$
Order:	\(1944\)\(\medspace = 2^{3} \cdot 3^{5} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3\times \He_3.A_4.C_6.C_2$, of order \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \)
Composition factors:	$C_2$ x 3, $C_3$ x 5
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	9	12
Elements	1	57	80	54	672	648	432	1944
Conjugacy classes	1	2	14	1	28	15	8	69
Divisions	1	2	8	1	15	6	4	37
Autjugacy classes	1	2	6	1	9	5	2	26

Dimension	1	2	3	4	6	12	18	36
Irr. complex chars.	6	12	30	0	18	0	3	0	69
Irr. rational chars.	2	6	2	4	15	4	2	2	37

Minimal presentations

Permutation degree:	$40$
Transitive degree:	$108$
Rank:	$3$
Inequivalent generating triples:	$294840$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	36	36
Arbitrary	9	12	24

Constructions

Presentation:	$\langle a, b, c, d \mid a^{6}=b^{9}=c^{6}=d^{6}=[a,c]=[c,d]=1, b^{a}=b^{8}, d^{a}=b^{3}c^{3}d^{5}, c^{b}=cd^{3}, d^{b}=cd^{4} \rangle$
Permutation group:	Degree $40$ $\langle(2,5)(3,6)(7,9)(11,13)(16,21)(17,24)(19,28)(20,31)(22,32)(23,33)(25,34) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^3.A_4)$ $\,\rtimes\,$ $S_3$ (2)	$(C_6^2:C_9)$ $\,\rtimes\,$ $S_3$	$(C_6^2:D_9)$ $\,\rtimes\,$ $C_3$	$(C_6^2:C_9)$ $\,\rtimes\,$ $C_6$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^3$ . $(C_3:S_4)$	$C_3^3$ . $(C_3\times S_4)$	$C_3$ . $(C_3^3:S_4)$	$(C_3\times \He_3)$ . $S_4$	all 15

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:	$C_{2}$
Commutator length:	$2$

Subgroups

There are 2620 subgroups in 276 conjugacy classes, 30 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $\He_3.S_4$
Commutator:	$G' \simeq$ $C_6^2:C_9$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_3^2:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2:C_6^2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_3^3.(C_3\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_3^2:C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
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: $C_3^3.(C_3\times S_4)$

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

Order 648: $C_3^3.S_4$ x 2, $C_6^2:D_9$, $C_3\times C_6^2:C_6$, $C_6^2:D_9$

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

Order 324: $C_6^2:C_9$ x 5, $C_3^3.A_4$ x 2, $C_3^3:C_{12}$, $C_3^2:C_6^2$, $C_3^2:C_6^2$, $C_6^2:C_9$

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

Order 216: $C_3^2.S_4$ x 6, $C_3^2.S_4$ x 6, $C_6^2:C_6$ x 3, $C_6^2:C_6$, $C_6^2:C_6$

Order 162: $D_9:C_3^2$ x 2, $C_6\times \He_3$, $C_3^3:C_6$, $C_3^2\times D_9$, $C_3^2:D_9$

Order 108: $C_3^2.A_4$ x 9, $C_3^2.A_4$ x 6, $C_3^2:C_{12}$ x 3, $C_3\times C_6^2$ x 3, $C_2^2\times \He_3$ x 3, $C_3^2:D_6$ x 3, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2:C_{12}$, $C_3^2:C_{12}$

Order 81: $C_3^2:C_9$ x 5, $C_9:C_3^2$ x 2, $C_3\times \He_3$, $C_3^2\times C_9$

Order 72: $C_6\wr C_2$ x 5, $C_2^2:D_9$ x 3, $D_4\times C_3^2$, $C_6^2:C_2$

Order 54: $C_9:C_6$ x 6, $C_3\times D_9$ x 6, $C_2\times \He_3$ x 6, $C_3^2:C_6$ x 3, $C_3^2\times C_6$ x 3, $C_3^2:C_6$, $S_3\times C_3^2$

Order 36: $C_6^2$ x 14, $C_2^2:C_9$ x 6, $C_3:C_{12}$ x 5, $C_6\times S_3$ x 5, $C_3\times C_{12}$, $C_3^2:C_4$, $C_6:S_3$

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

Order 24: $C_3\times D_4$ x 4, $C_3:D_4$ x 2

Order 18: $C_3\times C_6$ x 17, $C_3\times S_3$ x 5, $D_9$ x 3, $C_3:S_3$

Order 12: $C_2\times C_6$ x 12, $C_{12}$ x 4, $D_6$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 13, $C_9$ x 6

Order 8: $D_4$

Order 6: $C_6$ x 15, $S_3$ x 2

Order 4: $C_2^2$ x 2, $C_4$

Order 3: $C_3$ x 8

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1944: $C_3^3.(C_3\times S_4)$

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

Order 648: $C_6^2:D_9$, $C_3^3.S_4$, $C_3\times C_6^2:C_6$, $C_6^2:D_9$

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

Order 324: $C_6^2:C_9$ x 4, $C_3^3:C_{12}$, $C_3^2:C_6^2$, $C_3^2:C_6^2$, $C_3^3.A_4$, $C_6^2:C_9$

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

Order 216: $C_3^2.S_4$ x 3, $C_3^2.S_4$, $C_6^2:C_6$, $C_6^2:C_6$, $C_6^2:C_6$

Order 162: $C_6\times \He_3$, $D_9:C_3^2$, $C_3^3:C_6$, $C_3^2\times D_9$, $C_3^2:D_9$

Order 108: $C_3^2.A_4$ x 5, $C_3\times C_6^2$ x 3, $C_3^2:C_{12}$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2:C_{12}$, $C_3^2:C_{12}$, $C_2^2\times \He_3$, $C_3^2:D_6$, $C_3^2.A_4$

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

Order 72: $C_6\wr C_2$ x 3, $C_2^2:D_9$ x 2, $D_4\times C_3^2$, $C_6^2:C_2$

Order 54: $C_3\times D_9$ x 3, $C_3^2\times C_6$ x 3, $C_2\times \He_3$ x 2, $C_9:C_6$, $C_3^2:C_6$, $C_3^2:C_6$, $S_3\times C_3^2$

Order 36: $C_6^2$ x 11, $C_2^2:C_9$ x 4, $C_3:C_{12}$ x 3, $C_6\times S_3$ x 3, $C_3\times C_{12}$, $C_3^2:C_4$, $C_6:S_3$

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

Order 24: $C_3:D_4$ x 2, $C_3\times D_4$ x 2

Order 18: $C_3\times C_6$ x 12, $C_3\times S_3$ x 3, $D_9$ x 2, $C_3:S_3$

Order 12: $C_2\times C_6$ x 8, $D_6$ x 2, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 10, $C_9$ x 4

Order 8: $D_4$

Order 6: $C_6$ x 9, $S_3$ x 2

Order 4: $C_2^2$ x 2, $C_4$

Order 3: $C_3$ x 6

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1944: $C_3^3.(C_3\times S_4)$ ($C_1$)

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

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

Order 324: $C_3^3.A_4$ ($S_3$) x 2, $C_6^2:C_9$ ($C_6$), $C_3^2:C_6^2$ ($S_3$), $C_6^2:C_9$ ($S_3$)

Order 108: $C_3^2.A_4$ ($C_3\times S_3$) x 3, $C_3\times C_6^2$ ($C_3:S_3$), $C_3\times C_6^2$ ($C_3\times S_3$)

Order 81: $C_3\times \He_3$ ($S_4$)

Order 36: $C_6^2$ ($C_3^2:S_3$) x 3, $C_6^2$ ($C_3^2:C_6$)

Order 27: $C_3^3$ ($C_3:S_4$), $C_3^3$ ($C_3\times S_4$)

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

Order 9: $C_3^2$ ($C_3^2:S_4$) x 3, $C_3^2$ ($C_3^2:S_4$)

Order 4: $C_2^2$ ($(C_3^2\times C_9):S_3$)

Order 3: $C_3$ ($C_3^3:S_4$), $C_3$ ($\He_3.S_4$)

Order 1: $C_1$ ($C_3^3.(C_3\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1944: $C_3^3.(C_3\times S_4)$ ($C_1$)

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

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

Order 324: $C_6^2:C_9$ ($C_6$), $C_3^2:C_6^2$ ($S_3$), $C_3^3.A_4$ ($S_3$), $C_6^2:C_9$ ($S_3$)

Order 108: $C_3^2.A_4$ ($C_3\times S_3$) x 2, $C_3\times C_6^2$ ($C_3:S_3$), $C_3\times C_6^2$ ($C_3\times S_3$)

Order 81: $C_3\times \He_3$ ($S_4$)

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

Order 27: $C_3^3$ ($C_3:S_4$), $C_3^3$ ($C_3\times S_4$)

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

Order 9: $C_3^2$ ($C_3^2:S_4$), $C_3^2$ ($C_3^2:S_4$)

Order 4: $C_2^2$ ($(C_3^2\times C_9):S_3$)

Order 3: $C_3$ ($C_3^3:S_4$), $C_3$ ($\He_3.S_4$)

Order 1: $C_1$ ($C_3^3.(C_3\times S_4)$)

Series

Derived series

$C_3^3.(C_3\times S_4)$

$\rhd$

$C_6^2:C_9$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$C_3^3.(C_3\times S_4)$

$\rhd$

$C_6^2.C_3^3$

$\rhd$

$C_6^2:C_9$

$\rhd$

$C_3^2.A_4$

$\rhd$

$C_6^2$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_3^3.(C_3\times S_4)$

$\rhd$

$C_6^2:C_9$

Upper central series

$C_1$

$\lhd$

$C_3$

Character theory

Complex character table

See the $69 \times 69$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $37 \times 37$ rational character table.