Abstract group 1944.3966: $C_3^5:C_2^3$

• Label: 1944.3966
• Order: \( 2^{3} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 3^{2} \)
• Perm deg.: $17$
• Trans deg.: $72$
• Rank: $3$

Group information

Description:	$C_3^5:C_2^3$
Order:	\(1944\)\(\medspace = 2^{3} \cdot 3^{5} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$C_3^4.Q_8^2.C_3^3.C_2^5.C_2$, of order \(8957952\)\(\medspace = 2^{12} \cdot 3^{7} \)
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
Elements	1	271	242	1430	1944
Conjugacy classes	1	7	65	83	156
Divisions	1	7	49	67	124
Autjugacy classes	1	3	5	7	16

Dimension	1	2	4	8
Irr. complex chars.	8	36	112	0	156
Irr. rational chars.	8	36	48	32	124

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	$84$

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:	${\langle a, b, c, d, e, f \mid a^{2}=b^{3}=c^{6}=d^{3}=e^{3}=f^{6}=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $17$ $\langle(1,2)(3,6)(4,5)(8,9)(11,12)(13,14)(16,17), (1,2)(3,5)(4,6)(8,9)(11,12)(13,14) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(C_3^3:S_3^2)$
Semidirect product:	$C_3^5$ $\,\rtimes\,$ $C_2^3$	$(C_3^3:D_6)$ $\,\rtimes\,$ $S_3$	$(C_3^3:D_6)$ $\,\rtimes\,$ $S_3$	$(C_3^3:S_3)$ $\,\rtimes\,$ $D_6$	all 26
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 17.

Homology

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

Subgroups

There are 36344 subgroups in 3536 conjugacy classes, 191 normal (11 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^3:S_3^2$
Commutator:	$G' \simeq$ $C_3^5$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^5:C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^4\times C_6$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3^5:C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5$

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: $C_3^5:C_2^3$

Order 972: $C_3^3:S_3^2$ x 4, $C_3^3:C_6^2$ x 2, $C_3^4:D_6$

Order 648: $C_3^4:C_2^3$ x 8, $C_3^4:C_2^3$

Order 486: $C_3^4:C_6$ x 4, $C_3^4:C_6$ x 2, $C_3^4\times C_6$

Order 324: $C_3^3:D_6$ x 32, $C_3^2:S_3^2$ x 32, $C_3^2:C_6^2$ x 18, $C_3^2:S_3^2$ x 4, $C_3^3:D_6$

Order 243: $C_3^5$

Order 216: $C_6:S_3^2$ x 16, $C_6:S_3^2$ x 10

Order 162: $C_3^3:C_6$ x 64, $C_3^3\times C_6$ x 49, $C_3^2\wr C_2$ x 36, $C_3^3:S_3$ x 2

Order 108: $C_3^2:D_6$ x 146, $C_3:S_3^2$ x 64, $C_3:S_3^2$ x 40, $C_3^2:D_6$ x 26, $C_3^2\times D_6$ x 18

Order 81: $C_3^4$ x 49

Order 72: $S_3\times D_6$ x 24, $C_6:D_6$ x 2

Order 54: $C_3^2\times C_6$ x 378, $C_3^2:C_6$ x 292, $C_3^2:S_3$ x 52, $S_3\times C_3^2$ x 36

Order 36: $C_6:S_3$ x 100, $C_6\times S_3$ x 96, $S_3^2$ x 96, $C_6^2$ x 2

Order 27: $C_3^3$ x 378

Order 24: $C_2\times D_6$ x 9

Order 18: $C_3\times C_6$ x 382, $C_3\times S_3$ x 192, $C_3:S_3$ x 184

Order 12: $D_6$ x 78, $C_2\times C_6$ x 9

Order 9: $C_3^2$ x 378

Order 8: $C_2^3$

Order 6: $S_3$ x 84, $C_6$ x 67

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 49

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1944: $C_3^5:C_2^3$

Order 972: $C_3^4:D_6$, $C_3^3:C_6^2$, $C_3^3:S_3^2$

Order 648: $C_3^4:C_2^3$, $C_3^4:C_2^3$

Order 486: $C_3^4\times C_6$, $C_3^4:C_6$, $C_3^4:C_6$

Order 324: $C_3^3:D_6$ x 3, $C_3^2:C_6^2$ x 3, $C_3^3:D_6$, $C_3^2:S_3^2$, $C_3^2:S_3^2$

Order 243: $C_3^5$

Order 216: $C_6:S_3^2$ x 2, $C_6:S_3^2$

Order 162: $C_3^3\times C_6$ x 5, $C_3^3:C_6$ x 3, $C_3^2\wr C_2$ x 3, $C_3^3:S_3$

Order 108: $C_3^2:D_6$ x 7, $C_3^2:D_6$ x 3, $C_3^2\times D_6$ x 3, $C_3:S_3^2$ x 2, $C_3:S_3^2$

Order 81: $C_3^4$ x 5

Order 72: $S_3\times D_6$ x 2, $C_6:D_6$

Order 54: $C_3^2\times C_6$ x 12, $C_3^2:C_6$ x 7, $C_3^2:S_3$ x 3, $S_3\times C_3^2$ x 3

Order 36: $C_6:S_3$ x 8, $C_6\times S_3$ x 5, $S_3^2$ x 2, $C_6^2$

Order 27: $C_3^3$ x 12

Order 24: $C_2\times D_6$ x 2

Order 18: $C_3\times C_6$ x 13, $C_3:S_3$ x 7, $C_3\times S_3$ x 5

Order 12: $D_6$ x 7, $C_2\times C_6$ x 2

Order 9: $C_3^2$ x 12

Order 8: $C_2^3$

Order 6: $C_6$ x 7, $S_3$ x 5

Order 4: $C_2^2$ x 3

Order 3: $C_3$ x 5

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1944: $C_3^5:C_2^3$ ($C_1$)

Order 972: $C_3^3:S_3^2$ ($C_2$) x 4, $C_3^3:C_6^2$ ($C_2$) x 2, $C_3^4:D_6$ ($C_2$)

Order 486: $C_3^4:C_6$ ($C_2^2$) x 4, $C_3^4:C_6$ ($C_2^2$) x 2, $C_3^4\times C_6$ ($C_2^2$)

Order 324: $C_3^3:D_6$ ($S_3$) x 8, $C_3^3:D_6$ ($S_3$)

Order 243: $C_3^5$ ($C_2^3$)

Order 162: $C_3^3:C_6$ ($D_6$) x 16, $C_3^3\times C_6$ ($D_6$) x 9, $C_3^3:S_3$ ($D_6$) x 2

Order 108: $C_3^2:D_6$ ($C_3:S_3$) x 2

Order 81: $C_3^4$ ($C_2\times D_6$) x 9

Order 54: $C_3^2\times C_6$ ($S_3^2$) x 24, $C_3^2:S_3$ ($C_6:S_3$) x 4, $C_3^2\times C_6$ ($C_6:S_3$) x 2

Order 27: $C_3^3$ ($S_3\times D_6$) x 24, $C_3^3$ ($C_6:D_6$) x 2

Order 18: $C_3\times C_6$ ($C_3:S_3^2$) x 16, $C_3\times C_6$ ($C_3:S_3^2$) x 10

Order 9: $C_3^2$ ($C_6:S_3^2$) x 16, $C_3^2$ ($C_6:S_3^2$) x 10

Order 6: $C_6$ ($C_3^2:S_3^2$) x 8, $C_6$ ($C_3^2:S_3^2$)

Order 3: $C_3$ ($C_3^4:C_2^3$) x 8, $C_3$ ($C_3^4:C_2^3$)

Order 2: $C_2$ ($C_3^3:S_3^2$)

Order 1: $C_1$ ($C_3^5:C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1944: $C_3^5:C_2^3$ ($C_1$)

Order 972: $C_3^4:D_6$ ($C_2$), $C_3^3:C_6^2$ ($C_2$), $C_3^3:S_3^2$ ($C_2$)

Order 486: $C_3^4\times C_6$ ($C_2^2$), $C_3^4:C_6$ ($C_2^2$), $C_3^4:C_6$ ($C_2^2$)

Order 324: $C_3^3:D_6$ ($S_3$), $C_3^3:D_6$ ($S_3$)

Order 243: $C_3^5$ ($C_2^3$)

Order 162: $C_3^3\times C_6$ ($D_6$) x 2, $C_3^3:S_3$ ($D_6$), $C_3^3:C_6$ ($D_6$)

Order 108: $C_3^2:D_6$ ($C_3:S_3$)

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

Order 54: $C_3^2\times C_6$ ($S_3^2$) x 2, $C_3^2\times C_6$ ($C_6:S_3$), $C_3^2:S_3$ ($C_6:S_3$)

Order 27: $C_3^3$ ($S_3\times D_6$) x 2, $C_3^3$ ($C_6:D_6$)

Order 18: $C_3\times C_6$ ($C_3:S_3^2$) x 2, $C_3\times C_6$ ($C_3:S_3^2$)

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

Order 6: $C_6$ ($C_3^2:S_3^2$), $C_6$ ($C_3^2:S_3^2$)

Order 3: $C_3$ ($C_3^4:C_2^3$), $C_3$ ($C_3^4:C_2^3$)

Order 2: $C_2$ ($C_3^3:S_3^2$)

Order 1: $C_1$ ($C_3^5:C_2^3$)

Series

Derived series

$C_3^5:C_2^3$

$\rhd$

$C_3^5$

$\rhd$

$C_1$

Chief series

$C_3^5:C_2^3$

$\rhd$

$C_3^3:C_6^2$

$\rhd$

$C_3^4\times C_6$

$\rhd$

$C_3^5$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^5:C_2^3$

$\rhd$

$C_3^5$

Upper central series

$C_1$

$\lhd$

$C_2$

Character theory

Complex character table

See the $156 \times 156$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $124 \times 124$ rational character table.