Abstract group 1728.22769: $C_2^3.S_3^3$

• Label: 1728.22769
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $21$
• Trans deg.: $192$
• Rank: $3$

Group information

Description:	$C_2^3.S_3^3$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^3.C_2^6.C_2^6.C_2$, of order \(221184\)\(\medspace = 2^{13} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	7	26	504	182	1008	1728
Conjugacy classes	1	7	7	32	49	60	156
Divisions	1	7	7	18	49	34	116
Autjugacy classes	1	5	5	5	29	7	52

Dimension	1	2	4	8
Irr. complex chars.	32	56	60	8	156
Irr. rational chars.	8	36	52	20	116

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$192$
Rank:	$3$
Inequivalent generating triples:	$5376$

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:	$\langle a, b, c, d \mid a^{4}=b^{12}=c^{12}=d^{3}=[a,d]=[b,d]=1, b^{a}=b^{5}c^{6}, c^{a}=c^{7}, c^{b}=c^{11}, d^{c}=d^{2} \rangle$
Permutation group:	Degree $21$ $\langle(1,2,4,3)(6,7)(8,9)(10,11)(15,16)(17,18), (2,3)(8,9)(10,11)(15,17,16,18) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6^2.D_6)$ $\,\rtimes\,$ $C_4$	$(C_6^2.D_6)$ $\,\rtimes\,$ $C_4$	$(C_3^2:C_{12})$ $\,\rtimes\,$ $C_4^2$	$C_3^2$ $\,\rtimes\,$ $(C_{12}:C_4^2)$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^3$ . $S_3^3$	$C_6^3$ . $C_2^3$	$(C_6^2.S_3)$ . $Q_8$	$C_6^2$ . $(S_3\times Q_8)$ (2)	all 82

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

Homology

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

Subgroups

There are 6516 subgroups in 1154 conjugacy classes, 256 normal (42 characteristic).

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

Special subgroups

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1728: $C_2^3.S_3^3$

Order 864: $C_6^3.C_2^2$ x 3, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$, $C_6^2.D_{12}$

Order 576: $C_6.(S_3\times C_4^2)$ x 2, $(C_3\times C_{12}):C_4^2$

Order 432: $C_6^2.D_6$ x 8, $C_6^2.D_6$ x 8, $C_6^2:C_{12}$ x 3, $C_6^2:C_{12}$ x 3, $C_6^3.C_2$

Order 288: $C_2^3.S_3^2$ x 9, $C_6^2.D_4$ x 4, $C_6^2.C_2^3$ x 3, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2

Order 216: $C_6.C_6^2$ x 14, $C_6^2.C_6$ x 12, $C_6^2.S_3$ x 6, $C_6^3$

Order 192: $C_{12}:C_4^2$ x 2, $C_6.(C_4\times D_4)$

Order 144: $C_3^2:C_4^2$ x 28, $C_6^2:C_4$ x 18, $C_6^2:C_4$ x 10, $C_3^2:C_4^2$ x 8, $C_{12}:C_{12}$ x 8, $C_6^2.C_2^2$ x 8, $C_6.D_{12}$ x 8, $C_2\times C_6\times C_{12}$ x 3

Order 108: $C_3^2:C_{12}$ x 8, $C_3\times C_6^2$ x 7, $C_3^2:C_{12}$ x 6, $C_3^3:C_4$ x 4

Order 96: $C_6:C_4^2$ x 12, $C_6.C_4^2$ x 5, $C_2\times C_4:C_{12}$ x 2, $C_2^2.D_{12}$ x 2, $C_2^3.D_6$ x 2, $C_6.C_4^2$

Order 72: $C_6:C_{12}$ x 88, $C_6.D_6$ x 54, $C_6\times C_{12}$ x 14, $C_2\times C_6^2$ x 7

Order 64: $C_4:C_4^2$

Order 54: $C_3^2\times C_6$ x 7

Order 48: $C_{12}:C_4$ x 36, $C_6.C_2^3$ x 19, $C_2^2\times C_{12}$ x 12, $C_4:C_{12}$ x 8, $C_{12}:C_4$ x 8, $C_6.D_4$ x 8

Order 36: $C_3:C_{12}$ x 52, $C_6^2$ x 49, $C_3^2:C_4$ x 34, $C_3\times C_{12}$ x 8

Order 32: $C_2\times C_4^2$ x 3, $C_2^2.D_4$ x 2, $C_2.C_4^2$ x 2

Order 27: $C_3^3$

Order 24: $C_6:C_4$ x 98, $C_2\times C_{12}$ x 58, $C_2^2\times C_6$ x 7

Order 18: $C_3\times C_6$ x 49

Order 16: $C_4:C_4$ x 8, $C_4^2$ x 8, $C_2^2\times C_4$ x 7

Order 12: $C_3:C_4$ x 60, $C_2\times C_6$ x 49, $C_{12}$ x 34

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 32, $C_2^3$

Order 6: $C_6$ x 49

Order 4: $C_4$ x 18, $C_2^2$ x 7

Order 3: $C_3$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $C_2^3.S_3^3$

Order 864: $C_6^3.C_2^2$ x 2, $C_6^2.D_{12}$, $C_6^2.D_{12}$, $C_6^2.D_{12}$

Order 576: $(C_3\times C_{12}):C_4^2$, $C_6.(S_3\times C_4^2)$

Order 432: $C_6^2:C_{12}$ x 2, $C_6^2:C_{12}$ x 2, $C_6^2.D_6$ x 2, $C_6^3.C_2$, $C_6^2.D_6$

Order 288: $C_2^3.S_3^2$ x 5, $C_6^2.D_4$ x 3, $C_6^2.C_2^3$ x 2, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$

Order 216: $C_6^2.C_6$ x 5, $C_6.C_6^2$ x 5, $C_6^2.S_3$ x 3, $C_6^3$

Order 192: $C_{12}:C_4^2$, $C_6.(C_4\times D_4)$

Order 144: $C_6^2:C_4$ x 10, $C_6^2:C_4$ x 7, $C_3^2:C_4^2$ x 5, $C_3^2:C_4^2$ x 2, $C_2\times C_6\times C_{12}$ x 2, $C_{12}:C_{12}$, $C_6^2.C_2^2$, $C_6.D_{12}$

Order 108: $C_3\times C_6^2$ x 5, $C_3^2:C_{12}$ x 2, $C_3^2:C_{12}$ x 2, $C_3^3:C_4$

Order 96: $C_6:C_4^2$ x 7, $C_6.C_4^2$ x 3, $C_6.C_4^2$, $C_2\times C_4:C_{12}$, $C_2^2.D_{12}$, $C_2^3.D_6$

Order 72: $C_6:C_{12}$ x 28, $C_6.D_6$ x 20, $C_2\times C_6^2$ x 5, $C_6\times C_{12}$ x 5

Order 64: $C_4:C_4^2$

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

Order 48: $C_6.C_2^3$ x 12, $C_2^2\times C_{12}$ x 7, $C_{12}:C_4$ x 7, $C_4:C_{12}$, $C_{12}:C_4$, $C_6.D_4$

Order 36: $C_6^2$ x 29, $C_3:C_{12}$ x 10, $C_3^2:C_4$ x 7, $C_3\times C_{12}$ x 2

Order 32: $C_2\times C_4^2$ x 2, $C_2.C_4^2$ x 2, $C_2^2.D_4$

Order 27: $C_3^3$

Order 24: $C_6:C_4$ x 34, $C_2\times C_{12}$ x 19, $C_2^2\times C_6$ x 5

Order 18: $C_3\times C_6$ x 29

Order 16: $C_2^2\times C_4$ x 5, $C_4^2$ x 2, $C_4:C_4$

Order 12: $C_2\times C_6$ x 29, $C_3:C_4$ x 12, $C_{12}$ x 7

Order 9: $C_3^2$ x 5

Order 8: $C_2\times C_4$ x 13, $C_2^3$

Order 6: $C_6$ x 29

Order 4: $C_2^2$ x 5, $C_4$ x 5

Order 3: $C_3$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $C_2^3.S_3^3$ ($C_1$)

Order 864: $C_6^3.C_2^2$ ($C_2$) x 3, $C_6^2.D_{12}$ ($C_2$) x 2, $C_6^2.D_{12}$ ($C_2$), $C_6^2.D_{12}$ ($C_2$)

Order 432: $C_6^2.D_6$ ($C_4$) x 8, $C_6^2.D_6$ ($C_4$) x 4, $C_6^2:C_{12}$ ($C_2^2$) x 3, $C_6^2:C_{12}$ ($C_2^2$) x 3, $C_6^3.C_2$ ($C_2^2$)

Order 288: $C_6^2.D_4$ ($S_3$) x 2, $C_6^2.D_4$ ($S_3$)

Order 216: $C_6.C_6^2$ ($C_2\times C_4$) x 10, $C_6^2.C_6$ ($C_2\times C_4$) x 6, $C_6^2.S_3$ ($Q_8$) x 2, $C_6^2.S_3$ ($D_4$) x 2, $C_6^2.S_3$ ($C_2\times C_4$) x 2, $C_6^3$ ($C_2^3$)

Order 144: $C_6.D_{12}$ ($C_3:C_4$) x 8, $C_6^2:C_4$ ($D_6$) x 6, $C_6^2:C_4$ ($D_6$) x 3

Order 108: $C_3^3:C_4$ ($C_4:C_4$) x 4, $C_3^2:C_{12}$ ($C_4^2$) x 4, $C_3\times C_6^2$ ($C_2^2\times C_4$) x 3, $C_3\times C_6^2$ ($D_4:C_2$) x 2, $C_3\times C_6^2$ ($C_2\times Q_8$), $C_3\times C_6^2$ ($C_2\times D_4$)

Order 72: $C_6:C_{12}$ ($C_4\times S_3$) x 12, $C_6:C_{12}$ ($C_6:C_4$) x 8, $C_6.D_6$ ($C_6:C_4$) x 4, $C_2\times C_6^2$ ($C_2\times D_6$) x 3, $C_6.D_6$ ($C_4\times S_3$) x 2

Order 54: $C_3^2\times C_6$ ($C_4\times Q_8$) x 2, $C_3^2\times C_6$ ($C_4\times D_4$) x 2, $C_3^2\times C_6$ ($C_4^2:C_2$), $C_3^2\times C_6$ ($C_2^2.D_4$), $C_3^2\times C_6$ ($C_2\times C_4^2$)

Order 48: $C_6.C_2^3$ ($S_3^2$) x 3

Order 36: $C_3:C_{12}$ ($C_{12}:C_4$) x 8, $C_6^2$ ($C_4\times D_6$) x 7, $C_6^2$ ($D_4:S_3$) x 4, $C_6^2$ ($S_3\times Q_8$) x 3, $C_6^2$ ($S_3\times D_4$) x 3, $C_6^2$ ($C_6.C_2^3$) x 2, $C_6^2$ ($D_{12}:C_2$) x 2

Order 27: $C_3^3$ ($C_4:C_4^2$)

Order 24: $C_6:C_4$ ($C_6.D_6$) x 8, $C_2^2\times C_6$ ($S_3\times D_6$) x 3, $C_6:C_4$ ($C_6.D_6$) x 2

Order 18: $C_3\times C_6$ ($C_3:(C_4\times Q_8)$) x 4, $C_3\times C_6$ ($D_6.D_4$) x 3, $C_3\times C_6$ ($(C_4\times S_3):C_4$) x 2, $C_3\times C_6$ ($C_2^3.D_6$) x 2, $C_3\times C_6$ ($C_3:C_4\times Q_8$) x 2, $C_3\times C_6$ ($C_2^3.D_6$) x 2, $C_3\times C_6$ ($C_6:C_4^2$) x 2, $C_3\times C_6$ ($D_{12}:C_4$) x 2, $C_3\times C_6$ ($C_2^3.D_6$), $C_3\times C_6$ ($S_3\times C_4^2$)

Order 12: $C_2\times C_6$ ($D_6.D_6$) x 4, $C_2\times C_6$ ($C_4\times S_3^2$) x 4, $C_2\times C_6$ ($D_{12}:S_3$) x 4, $C_3:C_4$ ($C_3^2:C_4^2$) x 4, $C_2\times C_6$ ($C_{12}.D_6$) x 3, $C_2\times C_6$ ($D_6:D_6$) x 2, $C_2\times C_6$ ($D_6.D_6$) x 2, $C_2\times C_6$ ($C_2^2.S_3^2$), $C_2\times C_6$ ($C_{12}:D_6$)

Order 9: $C_3^2$ ($C_{12}:C_4^2$) x 2, $C_3^2$ ($C_6.(C_4\times D_4)$)

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

Order 6: $C_6$ ($C_6^2.C_2^3$) x 4, $C_6$ ($C_2^3.S_3^2$) x 2, $C_6$ ($C_6^2.C_2^3$) x 2, $C_6$ ($C_6^2.C_2^3$) x 2, $C_6$ ($C_6^2.C_2^3$) x 2, $C_6$ ($C_6^2.C_2^3$) x 2, $C_6$ ($C_6^2.C_2^3$) x 2, $C_6$ ($C_6^2.C_2^3$) x 2, $C_6$ ($C_2^3.S_3^2$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2.C_2^3$)

Order 4: $C_2^2$ ($C_2.S_3^3$) x 2, $C_2^2$ ($C_2.S_3^3$) x 2, $C_2^2$ ($C_2.S_3^3$), $C_2^2$ ($D_6:S_3^2$), $C_2^2$ ($C_2.S_3^3$)

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

Order 2: $C_2$ ($C_2^2.S_3^3$) x 2, $C_2$ ($C_2^2.S_3^3$) x 2, $C_2$ ($C_2^2.S_3^3$), $C_2$ ($C_2^2.S_3^3$), $C_2$ ($C_2^2.S_3^3$)

Order 1: $C_1$ ($C_2^3.S_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $C_2^3.S_3^3$ ($C_1$)

Order 864: $C_6^3.C_2^2$ ($C_2$) x 2, $C_6^2.D_{12}$ ($C_2$), $C_6^2.D_{12}$ ($C_2$), $C_6^2.D_{12}$ ($C_2$)

Order 432: $C_6^2:C_{12}$ ($C_2^2$) x 2, $C_6^2:C_{12}$ ($C_2^2$) x 2, $C_6^3.C_2$ ($C_2^2$), $C_6^2.D_6$ ($C_4$), $C_6^2.D_6$ ($C_4$)

Order 288: $C_6^2.D_4$ ($S_3$), $C_6^2.D_4$ ($S_3$)

Order 216: $C_6.C_6^2$ ($C_2\times C_4$) x 3, $C_6^2.C_6$ ($C_2\times C_4$) x 2, $C_6^3$ ($C_2^3$), $C_6^2.S_3$ ($Q_8$), $C_6^2.S_3$ ($D_4$), $C_6^2.S_3$ ($C_2\times C_4$)

Order 144: $C_6^2:C_4$ ($D_6$) x 3, $C_6^2:C_4$ ($D_6$) x 2, $C_6.D_{12}$ ($C_3:C_4$)

Order 108: $C_3\times C_6^2$ ($C_2^2\times C_4$) x 2, $C_3\times C_6^2$ ($D_4:C_2$), $C_3\times C_6^2$ ($C_2\times Q_8$), $C_3\times C_6^2$ ($C_2\times D_4$), $C_3^3:C_4$ ($C_4:C_4$), $C_3^2:C_{12}$ ($C_4^2$)

Order 72: $C_6:C_{12}$ ($C_4\times S_3$) x 3, $C_2\times C_6^2$ ($C_2\times D_6$) x 2, $C_6:C_{12}$ ($C_6:C_4$) x 2, $C_6.D_6$ ($C_6:C_4$), $C_6.D_6$ ($C_4\times S_3$)

Order 54: $C_3^2\times C_6$ ($C_4\times Q_8$), $C_3^2\times C_6$ ($C_4\times D_4$), $C_3^2\times C_6$ ($C_4^2:C_2$), $C_3^2\times C_6$ ($C_2^2.D_4$), $C_3^2\times C_6$ ($C_2\times C_4^2$)

Order 48: $C_6.C_2^3$ ($S_3^2$) x 2

Order 36: $C_6^2$ ($C_4\times D_6$) x 4, $C_6^2$ ($S_3\times Q_8$) x 2, $C_6^2$ ($D_4:S_3$) x 2, $C_6^2$ ($S_3\times D_4$) x 2, $C_3:C_{12}$ ($C_{12}:C_4$), $C_6^2$ ($C_6.C_2^3$), $C_6^2$ ($D_{12}:C_2$)

Order 27: $C_3^3$ ($C_4:C_4^2$)

Order 24: $C_6:C_4$ ($C_6.D_6$) x 2, $C_2^2\times C_6$ ($S_3\times D_6$) x 2, $C_6:C_4$ ($C_6.D_6$)

Order 18: $C_3\times C_6$ ($D_6.D_4$) x 2, $C_3\times C_6$ ($C_3:(C_4\times Q_8)$) x 2, $C_3\times C_6$ ($(C_4\times S_3):C_4$), $C_3\times C_6$ ($C_2^3.D_6$), $C_3\times C_6$ ($C_2^3.D_6$), $C_3\times C_6$ ($S_3\times C_4^2$), $C_3\times C_6$ ($C_3:C_4\times Q_8$), $C_3\times C_6$ ($C_2^3.D_6$), $C_3\times C_6$ ($C_6:C_4^2$), $C_3\times C_6$ ($D_{12}:C_4$)

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

Order 9: $C_3^2$ ($C_{12}:C_4^2$), $C_3^2$ ($C_6.(C_4\times D_4)$)

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

Order 6: $C_6$ ($C_6^2.C_2^3$) x 2, $C_6$ ($C_2^3.S_3^2$), $C_6$ ($C_2^3.S_3^2$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2.C_2^3$)

Order 4: $C_2^2$ ($C_2.S_3^3$), $C_2^2$ ($C_2.S_3^3$), $C_2^2$ ($D_6:S_3^2$), $C_2^2$ ($C_2.S_3^3$), $C_2^2$ ($C_2.S_3^3$)

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

Order 2: $C_2$ ($C_2^2.S_3^3$), $C_2$ ($C_2^2.S_3^3$), $C_2$ ($C_2^2.S_3^3$), $C_2$ ($C_2^2.S_3^3$), $C_2$ ($C_2^2.S_3^3$)

Order 1: $C_1$ ($C_2^3.S_3^3$)

Series

Derived series

$C_2^3.S_3^3$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_1$

Chief series

$C_2^3.S_3^3$

$\rhd$

$C_6^2.D_{12}$

$\rhd$

$C_6^2:C_{12}$

$\rhd$

$C_6.C_6^2$

$\rhd$

$C_3^2:C_{12}$

$\rhd$

$C_3:C_{12}$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2^3.S_3^3$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2^3$

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 $116 \times 116$ rational character table.