Abstract group 1728.46942: $C_3:(S_3\times \GL(2,\mathbb{Z}/4))$

• Label: 1728.46942
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $14$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3:(S_3\times \GL(2,\mathbb{Z}/4))$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$(C_3^2\times A_4).C_2^6.C_2$, of order \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	231	80	216	768	432	1728
Conjugacy classes	1	13	8	6	38	6	72
Divisions	1	13	7	6	31	6	64
Autjugacy classes	1	10	5	3	19	3	41

Dimension	1	2	3	4	6	8	12
Irr. complex chars.	8	18	8	22	10	0	6	72
Irr. rational chars.	8	14	8	12	10	6	6	64

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$80640$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	7	7	7

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{4}=c^{6}=d^{6}=e^{6}=[a,c]=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(2,4)(7,8)(9,10)(11,14)(12,13), (5,6)(7,8)(11,12)(13,14), (5,7)(6,8), (5,6) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 4 & 45 \\ 45 & 14 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 36 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 30 \\ 30 & 1 \end{array}\right), \left(\begin{array}{rr} 31 & 0 \\ 30 & 31 \end{array}\right), \left(\begin{array}{rr} 23 & 42 \\ 51 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 27 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 45 & 31 \end{array}\right), \left(\begin{array}{rr} 1 & 30 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/60\Z)$
Transitive group:	36T2415				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6:S_4)$ $\,\rtimes\,$ $D_6$ (2)	$(C_6.S_4)$ $\,\rtimes\,$ $D_6$ (2)	$(C_6:D_6)$ $\,\rtimes\,$ $S_4$	$A_4$ $\,\rtimes\,$ $(D_6:D_6)$	all 35
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(D_6\times S_4)$ (2)	$C_2^3$ . $(C_6:S_3^2)$	$(C_6^2:C_2^2)$ . $D_6$	$(C_6^2:C_6)$ . $C_2^3$	all 12

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{60}\Z)$.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 18124 subgroups in 1346 conjugacy classes, 68 normal (30 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2\times A_4:S_3^2$
Commutator:	$G' \simeq$ $C_6^2:C_6$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times A_4:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_6^2$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_3:(S_3\times \GL(2,\mathbb{Z}/4))$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

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 1728: $C_3:(S_3\times \GL(2,\mathbb{Z}/4))$

Order 864: $C_6^2:S_4$ x 2, $C_3^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_6:D_6\times A_4$, $C_2\times A_4:S_3^2$, $C_6.(S_3\times S_4)$

Order 576: $S_3\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:S_3^2$

Order 432: $C_6:S_3\times A_4$ x 4, $C_6^2:D_6$ x 2, $A_4:S_3^2$ x 2, $C_3\times C_6.S_4$ x 2, $A_4\times C_6^2$, $C_6^2:D_6$

Order 288: $C_6^2:D_4$ x 6, $C_3\times \GL(2,\mathbb{Z}/4)$ x 4, $D_6^2:C_2$ x 3, $C_2^3.S_3^2$ x 3, $C_2\times A_4\times D_6$ x 3, $C_3:\GL(2,\mathbb{Z}/4)$ x 2, $D_6:S_4$ x 2, $D_6:S_4$ x 2, $D_6.S_4$ x 2, $C_6^2:D_4$ x 2, $D_6\times S_4$ x 2, $C_6^2:C_2^3$

Order 216: $C_6^2:C_6$ x 3, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_6^2:C_6$ x 2, $C_3^2:D_{12}$ x 2, $C_6^2:C_6$, $C_6:S_3^2$, $C_6.S_3^2$

Order 192: $C_2^4:D_6$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$

Order 144: $A_4\times D_6$ x 14, $D_6:D_6$ x 12, $C_6^2:C_2^2$ x 10, $C_6^2:C_4$ x 6, $C_6.D_{12}$ x 6, $C_2^2:C_6^2$ x 6, $C_6^2:C_2^2$ x 6, $C_6:D_{12}$ x 6, $C_6\times S_4$ x 4, $S_3\times S_4$ x 4, $A_4:C_{12}$ x 4, $C_2^2.S_3^2$ x 3, $C_6:S_4$ x 2, $D_6:D_6$ x 2, $C_6.S_4$ x 2, $C_2^2\times C_6^2$, $D_6^2$

Order 108: $C_3^2:D_6$ x 6, $C_3:S_3^2$ x 2, $C_3^2:C_{12}$ x 2, $C_3\times C_6^2$, $C_3^2\times A_4$

Order 96: $C_2^2:D_{12}$ x 6, $D_6.D_4$ x 6, $D_4\times D_6$ x 6, $\GL(2,\mathbb{Z}/4)$ x 6, $D_6:D_4$ x 6, $C_2^3\times D_6$ x 3, $C_2^4:C_6$ x 2, $C_2^4:S_3$ x 2, $C_2^3\times A_4$, $C_2^2\times S_4$, $C_2^2.S_4$

Order 72: $C_6:D_6$ x 33, $C_6\wr C_2$ x 16, $C_6\times A_4$ x 14, $C_3:D_{12}$ x 14, $S_3\times A_4$ x 10, $S_3\times D_6$ x 8, $C_6:C_{12}$ x 6, $C_6.D_6$ x 6, $C_2\times C_6^2$ x 5, $C_6\times D_6$ x 5, $C_3\times S_4$ x 4, $C_3:S_4$ x 2, $C_6^2:C_2$ x 2, $D_6:S_3$ x 2, $C_6.D_6$ x 2

Order 64: $C_2^3:D_4$

Order 54: $C_3^2:C_6$ x 5, $C_3^2\times C_6$ x 2

Order 48: $C_2^2\times D_6$ x 36, $S_3\times D_4$ x 24, $C_6:D_4$ x 13, $D_6:C_4$ x 12, $C_2^2\times A_4$ x 8, $C_2\times S_4$ x 6, $C_6\times D_4$ x 6, $C_2\times D_{12}$ x 6, $C_4\times D_6$ x 6, $C_2^2:C_{12}$ x 6, $C_6.D_4$ x 6, $A_4:C_4$ x 4, $C_2^3\times C_6$ x 3

Order 36: $C_6:S_3$ x 30, $C_6\times S_3$ x 24, $C_6^2$ x 16, $C_3:C_{12}$ x 10, $S_3^2$ x 8, $C_3\times A_4$ x 6, $C_3^2:C_4$ x 2

Order 32: $C_2^2\wr C_2$ x 8, $C_2^2\times D_4$ x 3, $C_2^3:C_4$ x 3, $C_2^5$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 131, $C_3:D_4$ x 30, $C_2^2\times C_6$ x 20, $D_{12}$ x 12, $C_4\times S_3$ x 12, $C_2\times A_4$ x 12, $C_3\times D_4$ x 12, $C_6:C_4$ x 7, $C_2\times C_{12}$ x 6, $S_4$ x 4

Order 18: $C_3\times C_6$ x 19, $C_3\times S_3$ x 16, $C_3:S_3$ x 8

Order 16: $C_2\times D_4$ x 24, $C_2^2:C_4$ x 12, $C_2^4$ x 12, $C_2^2\times C_4$ x 3

Order 12: $D_6$ x 120, $C_2\times C_6$ x 47, $C_3:C_4$ x 10, $C_{12}$ x 6, $A_4$ x 4

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 46, $D_4$ x 24, $C_2\times C_4$ x 12

Order 6: $C_6$ x 31, $S_3$ x 26

Order 4: $C_2^2$ x 48, $C_4$ x 6

Order 3: $C_3$ x 7

Order 2: $C_2$ x 13

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $C_3:(S_3\times \GL(2,\mathbb{Z}/4))$

Order 864: $C_6:D_6\times A_4$, $C_2\times A_4:S_3^2$, $C_6^2:S_4$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6.(S_3\times S_4)$

Order 576: $S_3\times \GL(2,\mathbb{Z}/4)$, $C_2^4:S_3^2$

Order 432: $C_6:S_3\times A_4$ x 3, $A_4\times C_6^2$, $C_6^2:D_6$, $A_4:S_3^2$, $C_6^2:D_6$, $C_3\times C_6.S_4$

Order 288: $D_6^2:C_2$ x 3, $C_6^2:D_4$ x 3, $C_2^3.S_3^2$ x 3, $C_3\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2\times A_4\times D_6$ x 2, $C_3:\GL(2,\mathbb{Z}/4)$, $D_6:S_4$, $D_6:S_4$, $D_6.S_4$, $C_6^2:D_4$, $C_6^2:C_2^3$, $D_6\times S_4$

Order 216: $C_6^2:C_6$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$, $C_6:S_3^2$, $C_3^2:S_4$, $C_6^2:C_6$, $C_3^2:D_{12}$, $C_6.S_3^2$

Order 192: $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^4:D_6$

Order 144: $C_6^2:C_2^2$ x 8, $D_6:D_6$ x 7, $A_4\times D_6$ x 6, $C_2^2:C_6^2$ x 4, $C_6^2:C_4$ x 3, $C_6.D_{12}$ x 3, $C_6^2:C_2^2$ x 3, $C_6:D_{12}$ x 3, $C_2^2.S_3^2$ x 3, $C_6\times S_4$ x 2, $A_4:C_{12}$ x 2, $C_2^2\times C_6^2$, $D_6^2$, $C_6:S_4$, $S_3\times S_4$, $D_6:D_6$, $C_6.S_4$

Order 108: $C_3^2:D_6$ x 4, $C_3\times C_6^2$, $C_3^2\times A_4$, $C_3:S_3^2$, $C_3^2:C_{12}$

Order 96: $C_2^2:D_{12}$ x 3, $D_6.D_4$ x 3, $D_4\times D_6$ x 3, $\GL(2,\mathbb{Z}/4)$ x 3, $D_6:D_4$ x 3, $C_2^3\times D_6$ x 2, $C_2^3\times A_4$, $C_2^2\times S_4$, $C_2^2.S_4$, $C_2^4:C_6$, $C_2^4:S_3$

Order 72: $C_6:D_6$ x 21, $C_6\times A_4$ x 8, $C_6\wr C_2$ x 7, $C_2\times C_6^2$ x 5, $S_3\times D_6$ x 5, $C_3:D_{12}$ x 5, $S_3\times A_4$ x 4, $C_6.D_6$ x 4, $C_6\times D_6$ x 3, $C_6:C_{12}$ x 3, $C_3\times S_4$ x 2, $C_3:S_4$, $C_6^2:C_2$, $D_6:S_3$, $C_6.D_6$

Order 64: $C_2^3:D_4$

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

Order 48: $C_2^2\times D_6$ x 17, $C_6:D_4$ x 7, $S_3\times D_4$ x 7, $C_2^2\times A_4$ x 6, $D_6:C_4$ x 6, $C_2\times S_4$ x 3, $C_6\times D_4$ x 3, $C_2\times D_{12}$ x 3, $C_4\times D_6$ x 3, $C_2^2:C_{12}$ x 3, $C_6.D_4$ x 3, $C_2^3\times C_6$ x 2, $A_4:C_4$ x 2

Order 36: $C_6:S_3$ x 18, $C_6^2$ x 13, $C_6\times S_3$ x 11, $C_3:C_{12}$ x 5, $C_3\times A_4$ x 4, $S_3^2$ x 3, $C_3^2:C_4$

Order 32: $C_2^2\wr C_2$ x 4, $C_2^2\times D_4$ x 3, $C_2^3:C_4$ x 3, $C_2^5$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 48, $C_3:D_4$ x 12, $C_2^2\times C_6$ x 12, $C_2\times A_4$ x 8, $C_3\times D_4$ x 5, $C_6:C_4$ x 4, $D_{12}$ x 4, $C_4\times S_3$ x 4, $C_2\times C_{12}$ x 3, $S_4$ x 2

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

Order 16: $C_2\times D_4$ x 13, $C_2^4$ x 10, $C_2^2:C_4$ x 6, $C_2^2\times C_4$ x 3

Order 12: $D_6$ x 42, $C_2\times C_6$ x 27, $C_3:C_4$ x 5, $A_4$ x 3, $C_{12}$ x 3

Order 9: $C_3^2$ x 5

Order 8: $C_2^3$ x 31, $D_4$ x 9, $C_2\times C_4$ x 7

Order 6: $C_6$ x 19, $S_3$ x 11

Order 4: $C_2^2$ x 31, $C_4$ x 3

Order 3: $C_3$ x 5

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $C_3:(S_3\times \GL(2,\mathbb{Z}/4))$ ($C_1$)

Order 864: $C_6^2:S_4$ ($C_2$) x 2, $C_3^2:\GL(2,\mathbb{Z}/4)$ ($C_2$) x 2, $C_6:D_6\times A_4$ ($C_2$), $C_2\times A_4:S_3^2$ ($C_2$), $C_6.(S_3\times S_4)$ ($C_2$)

Order 432: $C_6:S_3\times A_4$ ($C_2^2$) x 2, $C_6^2:D_6$ ($C_2^2$) x 2, $C_3\times C_6.S_4$ ($C_2^2$) x 2, $A_4\times C_6^2$ ($C_2^2$)

Order 288: $C_3:\GL(2,\mathbb{Z}/4)$ ($S_3$) x 2, $C_6^2:C_2^3$ ($S_3$)

Order 216: $C_6^2:C_6$ ($D_4$) x 2, $C_6^2:C_6$ ($C_2^3$)

Order 144: $C_6^2:C_2^2$ ($D_6$) x 2, $C_2^2:C_6^2$ ($D_6$) x 2, $C_6:S_4$ ($D_6$) x 2, $C_6.S_4$ ($D_6$) x 2, $C_2^2\times C_6^2$ ($D_6$)

Order 108: $C_3^2\times A_4$ ($C_2\times D_4$)

Order 72: $C_6:D_6$ ($C_3:D_4$) x 2, $C_6\times A_4$ ($C_2\times D_6$) x 2, $C_2\times C_6^2$ ($C_2\times D_6$), $C_6:D_6$ ($S_4$)

Order 48: $C_2^3\times C_6$ ($S_3^2$) x 2, $C_2^2\times A_4$ ($S_3^2$)

Order 36: $C_6:S_3$ ($C_2\times S_4$) x 2, $C_3\times A_4$ ($S_3\times D_4$) x 2, $C_6^2$ ($C_2\times S_4$), $C_6^2$ ($C_6:D_4$)

Order 24: $C_2^2\times C_6$ ($S_3\times D_6$) x 2, $C_2\times A_4$ ($S_3\times D_6$)

Order 18: $C_3:S_3$ ($\GL(2,\mathbb{Z}/4)$) x 2, $C_3\times C_6$ ($C_2^2\times S_4$)

Order 16: $C_2^4$ ($C_3:S_3^2$)

Order 12: $C_2\times C_6$ ($S_3\times S_4$) x 2, $C_2\times C_6$ ($D_6:D_6$) x 2, $A_4$ ($D_6:D_6$)

Order 9: $C_3^2$ ($C_2\times \GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($C_6:S_3^2$)

Order 6: $C_6$ ($D_6\times S_4$) x 2

Order 4: $C_2^2$ ($A_4:S_3^2$), $C_2^2$ ($C_6^2:D_6$)

Order 3: $C_3$ ($S_3\times \GL(2,\mathbb{Z}/4)$) x 2

Order 2: $C_2$ ($C_2\times A_4:S_3^2$)

Order 1: $C_1$ ($C_3:(S_3\times \GL(2,\mathbb{Z}/4))$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $C_3:(S_3\times \GL(2,\mathbb{Z}/4))$ ($C_1$)

Order 864: $C_6:D_6\times A_4$ ($C_2$), $C_2\times A_4:S_3^2$ ($C_2$), $C_6^2:S_4$ ($C_2$), $C_3^2:\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_6.(S_3\times S_4)$ ($C_2$)

Order 432: $C_6:S_3\times A_4$ ($C_2^2$) x 2, $A_4\times C_6^2$ ($C_2^2$), $C_6^2:D_6$ ($C_2^2$), $C_3\times C_6.S_4$ ($C_2^2$)

Order 288: $C_3:\GL(2,\mathbb{Z}/4)$ ($S_3$), $C_6^2:C_2^3$ ($S_3$)

Order 216: $C_6^2:C_6$ ($C_2^3$), $C_6^2:C_6$ ($D_4$)

Order 144: $C_6^2:C_2^2$ ($D_6$) x 2, $C_2^2\times C_6^2$ ($D_6$), $C_2^2:C_6^2$ ($D_6$), $C_6:S_4$ ($D_6$), $C_6.S_4$ ($D_6$)

Order 108: $C_3^2\times A_4$ ($C_2\times D_4$)

Order 72: $C_2\times C_6^2$ ($C_2\times D_6$), $C_6:D_6$ ($C_3:D_4$), $C_6:D_6$ ($S_4$), $C_6\times A_4$ ($C_2\times D_6$)

Order 48: $C_2^3\times C_6$ ($S_3^2$), $C_2^2\times A_4$ ($S_3^2$)

Order 36: $C_6:S_3$ ($C_2\times S_4$) x 2, $C_6^2$ ($C_2\times S_4$), $C_6^2$ ($C_6:D_4$), $C_3\times A_4$ ($S_3\times D_4$)

Order 24: $C_2^2\times C_6$ ($S_3\times D_6$), $C_2\times A_4$ ($S_3\times D_6$)

Order 18: $C_3\times C_6$ ($C_2^2\times S_4$), $C_3:S_3$ ($\GL(2,\mathbb{Z}/4)$)

Order 16: $C_2^4$ ($C_3:S_3^2$)

Order 12: $C_2\times C_6$ ($S_3\times S_4$), $C_2\times C_6$ ($D_6:D_6$), $A_4$ ($D_6:D_6$)

Order 9: $C_3^2$ ($C_2\times \GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($C_6:S_3^2$)

Order 6: $C_6$ ($D_6\times S_4$)

Order 4: $C_2^2$ ($A_4:S_3^2$), $C_2^2$ ($C_6^2:D_6$)

Order 3: $C_3$ ($S_3\times \GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($C_2\times A_4:S_3^2$)

Order 1: $C_1$ ($C_3:(S_3\times \GL(2,\mathbb{Z}/4))$)

Series

Derived series

$C_3:(S_3\times \GL(2,\mathbb{Z}/4))$

$\rhd$

$C_6^2:C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_3:(S_3\times \GL(2,\mathbb{Z}/4))$

$\rhd$

$C_6^2:S_4$

$\rhd$

$A_4\times C_6^2$

$\rhd$

$C_6^2:C_6$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_3:(S_3\times \GL(2,\mathbb{Z}/4))$

$\rhd$

$C_6^2:C_6$

$\rhd$

$C_3^2\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 15 larger groups in the database.

This group is a maximal quotient of 5 larger groups in the database.

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 $64 \times 64$ rational character table.