Abstract group 1944.3794: $S_3\times C_3^3:C_{12}$

• Label: 1944.3794
• Order: \( 2^{3} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3^{2} \)
• Perm deg.: $19$
• Trans deg.: $72$
• Rank: $3$

Group information

Description:	$S_3\times C_3^3:C_{12}$
Order:	\(1944\)\(\medspace = 2^{3} \cdot 3^{5} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^2.Q_8^2.S_3^3.C_2^2$, of order \(497664\)\(\medspace = 2^{11} \cdot 3^{5} \)
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	4	6	12
Elements	1	7	242	72	722	900	1944
Conjugacy classes	1	3	89	4	177	50	324
Divisions	1	3	49	2	97	13	165
Autjugacy classes	1	2	7	2	10	4	26

Dimension	1	2	4	8
Irr. complex chars.	72	180	72	0	324
Irr. rational chars.	4	36	89	36	165

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	$4368$

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:	$\langle a, b, c, d, e \mid a^{3}=b^{12}=c^{3}=d^{6}=e^{3}=[a,b]=[a,c]=[a,d]=[a,e]=[b,e]=[c,d]=[c,e]=1, c^{b}=c^{2}, d^{b}=d^{5}, e^{d}=e^{2} \rangle$
Permutation group:	Degree $19$ $\langle(8,9)(13,14)(15,16), (11,12)(13,15,14,16)(18,19), (1,2,3), (4,5,6), (13,14)(15,16), (10,11,12)(17,18,19), (10,11,12), (7,8,9)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 3 & 16 \\ 14 & 15 \end{array}\right), \left(\begin{array}{rr} 7 & 15 \\ 3 & 16 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 6 & 7 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/18\Z)$
Direct product:	$C_3$ ${}^2$ $\, \times\, $ $S_3$ $\, \times\, $ $(C_3^2:C_4)$
Semidirect product:	$(S_3\times C_3^4)$ $\,\rtimes\,$ $C_4$	$C_3^4$ $\,\rtimes\,$ $(C_4\times S_3)$	$C_3^3$ $\,\rtimes\,$ $(C_6.D_6)$	$C_3^3$ $\,\rtimes\,$ $(C_6.D_6)$	all 26
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_6\times C_3^3)$ . $S_3$	$(D_6\times C_3^4)$ . $C_2$	$(D_6\times C_3^3)$ . $C_6$	$(C_3^3\times C_6)$ . $D_6$ (2)	all 25

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{18}\Z)$.

Homology

Abelianization:	$C_{6} \times C_{12} \simeq C_{2} \times C_{4} \times C_{3}^{2}$
Schur multiplier:	$C_{3} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 10120 subgroups in 2768 conjugacy classes, 276 normal (28 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3\times C_6$	$G/Z \simeq$ $C_3:S_3^2$
Commutator:	$G' \simeq$ $C_3^3$	$G/G' \simeq$ $C_6\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_3^3:C_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^4\times C_6$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $S_3\times C_3^3:C_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
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: $S_3\times C_3^3:C_{12}$

Order 972: $D_6\times C_3^4$, $C_3^4:C_{12}$, $C_3^4:C_{12}$

Order 648: $S_3\times C_3^2:C_{12}$ x 4, $S_3\times C_3^2:C_{12}$ x 4, $C_6\times C_3^2:C_{12}$

Order 486: $S_3\times C_3^4$ x 2, $C_3^4\times C_6$

Order 324: $D_6\times C_3^3$ x 24, $C_3^3:C_{12}$ x 18, $C_3^3:C_{12}$ x 4, $C_3^3:C_{12}$ x 4, $C_3^2\times C_6^2$

Order 243: $C_3^5$

Order 216: $C_6.S_3^2$ x 16, $C_6^2.C_6$ x 4, $C_6.C_6^2$ x 4, $C_6.C_6^2$, $C_6.S_3^2$

Order 162: $C_3^3\times C_6$ x 51, $S_3\times C_3^3$ x 48

Order 108: $C_3^2\times D_6$ x 74, $C_3^2:C_{12}$ x 45, $C_3^2:C_{12}$ x 45, $C_3\times C_6^2$ x 24, $C_3^2\times C_{12}$, $C_3^3:C_4$

Order 81: $C_3^4$ x 49

Order 72: $C_6:C_{12}$ x 16, $S_3\times C_{12}$ x 4, $C_6.D_6$ x 4, $C_6\times C_{12}$, $C_6.D_6$

Order 54: $C_3^2\times C_6$ x 426, $S_3\times C_3^2$ x 148

Order 36: $C_6^2$ x 74, $C_3:C_{12}$ x 72, $C_6\times S_3$ x 24, $C_3\times C_{12}$ x 10, $C_3^2:C_4$ x 10

Order 27: $C_3^3$ x 378

Order 24: $C_2\times C_{12}$ x 4, $C_6:C_4$ x 4, $C_4\times S_3$

Order 18: $C_3\times C_6$ x 526, $C_3\times S_3$ x 48

Order 12: $C_2\times C_6$ x 24, $C_{12}$ x 13, $C_3:C_4$ x 13, $D_6$

Order 9: $C_3^2$ x 378

Order 8: $C_2\times C_4$

Order 6: $C_6$ x 97, $S_3$ x 2

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

Order 3: $C_3$ x 49

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1944: $S_3\times C_3^3:C_{12}$

Order 972: $D_6\times C_3^4$, $C_3^4:C_{12}$, $C_3^4:C_{12}$

Order 648: $C_6\times C_3^2:C_{12}$, $S_3\times C_3^2:C_{12}$, $S_3\times C_3^2:C_{12}$

Order 486: $C_3^4\times C_6$, $S_3\times C_3^4$

Order 324: $C_3^3:C_{12}$ x 6, $D_6\times C_3^3$ x 3, $C_3^2\times C_6^2$, $C_3^3:C_{12}$, $C_3^3:C_{12}$

Order 243: $C_3^5$

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

Order 162: $C_3^3\times C_6$ x 8, $S_3\times C_3^3$ x 3

Order 108: $C_3^2\times D_6$ x 6, $C_3^2:C_{12}$ x 6, $C_3^2:C_{12}$ x 6, $C_3\times C_6^2$ x 3, $C_3^2\times C_{12}$, $C_3^3:C_4$

Order 81: $C_3^4$ x 7

Order 72: $C_6\times C_{12}$, $C_6.D_6$, $C_6:C_{12}$, $S_3\times C_{12}$, $C_6.D_6$

Order 54: $C_3^2\times C_6$ x 22, $S_3\times C_3^2$ x 6

Order 36: $C_3:C_{12}$ x 6, $C_6^2$ x 6, $C_3\times C_{12}$ x 4, $C_3^2:C_4$ x 4, $C_6\times S_3$ x 3

Order 27: $C_3^3$ x 19

Order 24: $C_2\times C_{12}$, $C_6:C_4$, $C_4\times S_3$

Order 18: $C_3\times C_6$ x 25, $C_3\times S_3$ x 3

Order 12: $C_{12}$ x 4, $C_3:C_4$ x 4, $C_2\times C_6$ x 3, $D_6$

Order 9: $C_3^2$ x 19

Order 8: $C_2\times C_4$

Order 6: $C_6$ x 10, $S_3$

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

Order 3: $C_3$ x 7

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1944: $S_3\times C_3^3:C_{12}$ ($C_1$)

Order 972: $D_6\times C_3^4$ ($C_2$), $C_3^4:C_{12}$ ($C_2$), $C_3^4:C_{12}$ ($C_2$)

Order 648: $S_3\times C_3^2:C_{12}$ ($C_3$) x 4

Order 486: $S_3\times C_3^4$ ($C_4$) x 2, $C_3^4\times C_6$ ($C_2^2$)

Order 324: $D_6\times C_3^3$ ($C_6$) x 4, $D_6\times C_3^3$ ($S_3$) x 4, $C_3^3:C_{12}$ ($C_6$) x 4, $C_3^3:C_{12}$ ($C_6$) x 4, $C_3^3:C_{12}$ ($S_3$)

Order 243: $C_3^5$ ($C_2\times C_4$)

Order 216: $C_6.S_3^2$ ($C_3^2$)

Order 162: $S_3\times C_3^3$ ($C_{12}$) x 8, $S_3\times C_3^3$ ($C_3:C_4$) x 8, $C_3^3\times C_6$ ($D_6$) x 5, $C_3^3\times C_6$ ($C_2\times C_6$) x 4

Order 108: $C_3^2\times D_6$ ($C_3\times S_3$) x 16, $C_3^2:C_{12}$ ($C_3\times S_3$) x 4, $C_3^2\times D_6$ ($C_3\times C_6$), $C_3^2\times D_6$ ($C_3:S_3$), $C_3^3:C_4$ ($C_3\times C_6$), $C_3^2:C_{12}$ ($C_3\times C_6$)

Order 81: $C_3^4$ ($C_2\times C_{12}$) x 4, $C_3^4$ ($C_6:C_4$) x 4, $C_3^4$ ($C_4\times S_3$)

Order 54: $S_3\times C_3^2$ ($C_3:C_{12}$) x 32, $C_3^2\times C_6$ ($C_6\times S_3$) x 20, $C_3^2\times C_6$ ($S_3^2$) x 4, $S_3\times C_3^2$ ($C_3\times C_{12}$) x 2, $S_3\times C_3^2$ ($C_3^2:C_4$) x 2, $C_3^2\times C_6$ ($C_6^2$), $C_3^2\times C_6$ ($C_6:S_3$)

Order 36: $C_6\times S_3$ ($C_3^2:C_6$) x 4, $C_6\times S_3$ ($S_3\times C_3^2$) x 4, $C_3^2:C_4$ ($S_3\times C_3^2$)

Order 27: $C_3^3$ ($C_6:C_{12}$) x 16, $C_3^3$ ($S_3\times C_{12}$) x 4, $C_3^3$ ($C_6.D_6$) x 4, $C_3^3$ ($C_6\times C_{12}$), $C_3^3$ ($C_6.D_6$)

Order 18: $C_3\times C_6$ ($C_3\times S_3^2$) x 16, $C_3\times S_3$ ($C_3^2:C_{12}$) x 8, $C_3\times S_3$ ($C_3^2:C_{12}$) x 8, $C_3\times C_6$ ($C_3^2\times D_6$) x 5, $C_3\times C_6$ ($C_3^2:D_6$) x 4, $C_3\times C_6$ ($C_3:S_3^2$)

Order 12: $D_6$ ($C_3^2\wr C_2$)

Order 9: $C_3^2$ ($C_6.S_3^2$) x 16, $C_3^2$ ($C_6^2.C_6$) x 4, $C_3^2$ ($C_6.C_6^2$) x 4, $C_3^2$ ($C_6.C_6^2$), $C_3^2$ ($C_6.S_3^2$)

Order 6: $C_6$ ($C_3^2:S_3^2$) x 4, $C_6$ ($C_3^2\times S_3^2$) x 4, $S_3$ ($C_3^3:C_{12}$) x 2, $C_6$ ($C_3^2:C_6^2$)

Order 3: $C_3$ ($S_3\times C_3^2:C_{12}$) x 4, $C_3$ ($S_3\times C_3^2:C_{12}$) x 4, $C_3$ ($C_6\times C_3^2:C_{12}$)

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

Order 1: $C_1$ ($S_3\times C_3^3:C_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1944: $S_3\times C_3^3:C_{12}$ ($C_1$)

Order 972: $D_6\times C_3^4$ ($C_2$), $C_3^4:C_{12}$ ($C_2$), $C_3^4:C_{12}$ ($C_2$)

Order 648: $S_3\times C_3^2:C_{12}$ ($C_3$)

Order 486: $C_3^4\times C_6$ ($C_2^2$), $S_3\times C_3^4$ ($C_4$)

Order 324: $D_6\times C_3^3$ ($C_6$), $D_6\times C_3^3$ ($S_3$), $C_3^3:C_{12}$ ($C_6$), $C_3^3:C_{12}$ ($C_6$), $C_3^3:C_{12}$ ($S_3$)

Order 243: $C_3^5$ ($C_2\times C_4$)

Order 216: $C_6.S_3^2$ ($C_3^2$)

Order 162: $C_3^3\times C_6$ ($D_6$) x 2, $C_3^3\times C_6$ ($C_2\times C_6$), $S_3\times C_3^3$ ($C_{12}$), $S_3\times C_3^3$ ($C_3:C_4$)

Order 108: $C_3^2\times D_6$ ($C_3\times C_6$), $C_3^2\times D_6$ ($C_3:S_3$), $C_3^2\times D_6$ ($C_3\times S_3$), $C_3^3:C_4$ ($C_3\times C_6$), $C_3^2:C_{12}$ ($C_3\times C_6$), $C_3^2:C_{12}$ ($C_3\times S_3$)

Order 81: $C_3^4$ ($C_2\times C_{12}$), $C_3^4$ ($C_6:C_4$), $C_3^4$ ($C_4\times S_3$)

Order 54: $C_3^2\times C_6$ ($C_6\times S_3$) x 2, $C_3^2\times C_6$ ($C_6^2$), $C_3^2\times C_6$ ($C_6:S_3$), $C_3^2\times C_6$ ($S_3^2$), $S_3\times C_3^2$ ($C_3\times C_{12}$), $S_3\times C_3^2$ ($C_3^2:C_4$), $S_3\times C_3^2$ ($C_3:C_{12}$)

Order 36: $C_3^2:C_4$ ($S_3\times C_3^2$), $C_6\times S_3$ ($C_3^2:C_6$), $C_6\times S_3$ ($S_3\times C_3^2$)

Order 27: $C_3^3$ ($C_6\times C_{12}$), $C_3^3$ ($C_6.D_6$), $C_3^3$ ($C_6:C_{12}$), $C_3^3$ ($S_3\times C_{12}$), $C_3^3$ ($C_6.D_6$)

Order 18: $C_3\times C_6$ ($C_3^2\times D_6$) x 2, $C_3\times C_6$ ($C_3^2:D_6$), $C_3\times C_6$ ($C_3:S_3^2$), $C_3\times C_6$ ($C_3\times S_3^2$), $C_3\times S_3$ ($C_3^2:C_{12}$), $C_3\times S_3$ ($C_3^2:C_{12}$)

Order 12: $D_6$ ($C_3^2\wr C_2$)

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

Order 6: $C_6$ ($C_3^2:C_6^2$), $C_6$ ($C_3^2:S_3^2$), $C_6$ ($C_3^2\times S_3^2$), $S_3$ ($C_3^3:C_{12}$)

Order 3: $C_3$ ($C_6\times C_3^2:C_{12}$), $C_3$ ($S_3\times C_3^2:C_{12}$), $C_3$ ($S_3\times C_3^2:C_{12}$)

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

Order 1: $C_1$ ($S_3\times C_3^3:C_{12}$)

Series

Derived series

$S_3\times C_3^3:C_{12}$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$S_3\times C_3^3:C_{12}$

$\rhd$

$S_3\times C_3^2:C_{12}$

$\rhd$

$D_6\times C_3^3$

$\rhd$

$S_3\times C_3^3$

$\rhd$

$S_3\times C_3^2$

$\rhd$

$C_3\times S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$S_3\times C_3^3:C_{12}$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_3\times C_6$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $324 \times 324$ 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 $165 \times 165$ rational character table (warning: may be slow to load).