Abstract group 864.4639: $C_6^3:C_2^2$

• Label: 864.4639
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \cdot 3^{2} \)
• Perm deg.: $17$
• Trans deg.: $144$
• Rank: $4$

Group information

Description:	$C_6^3:C_2^2$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$(C_2^4\times C_3:S_3).C_2^6.C_3^2.C_2^3$, of order \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \)
Composition factors:	$C_2$ x 5, $C_3$ 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	87	26	72	534	144	864
Conjugacy classes	1	15	14	4	210	8	252
Divisions	1	15	9	4	119	4	152
Autjugacy classes	1	4	3	1	10	1	20

Dimension	1	2	4
Irr. complex chars.	48	204	0	252
Irr. rational chars.	16	52	84	152

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$144$
Rank:	$4$
Inequivalent generating quadruples:	$109200$

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:	$\langle a, b, c, d \mid a^{6}=b^{4}=c^{6}=d^{6}=[a,c]=[a,d]=[c,d]=1, b^{a}=b^{3}, c^{b}=c^{5}, d^{b}=d^{5} \rangle$
Permutation group:	Degree $17$ $\langle(3,4)(11,12)(13,14)(16,17), (8,9), (1,2)(3,4)(13,14), (1,3)(2,4)(8,9), (5,6,7), (1,2)(3,4), (10,11,12), (15,16,17)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 24 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 35 & 0 \\ 0 & 35 \end{array}\right), \left(\begin{array}{rr} 7 & 34 \\ 6 & 29 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 21 \\ 6 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_6^2:C_2)$
Semidirect product:	$C_6^3$ $\,\rtimes\,$ $C_2^2$	$(C_3\times C_6^2)$ $\,\rtimes\,$ $D_4$	$(C_2\times C_6^2)$ $\,\rtimes\,$ $D_6$	$C_6^2$ $\,\rtimes\,$ $(C_3:D_4)$	all 41
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^3$ . $C_2^2$	$(C_2\times C_6^2)$ . $D_6$	$C_6^2$ . $(C_2\times D_6)$	$C_2$ . $(C_6^3:C_2)$	all 19

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

Homology

Abelianization:	$C_{2}^{3} \times C_{6} \simeq C_{2}^{4} \times C_{3}$
Schur multiplier:	$C_{2}^{5} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 5880 subgroups in 1952 conjugacy classes, 426 normal (22 characteristic).

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

Special subgroups

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 864: $C_6^3:C_2^2$

Order 432: $C_6^3:C_2$ x 12, $C_2\times C_6^3$, $C_6^3:C_2$, $C_6^2:C_{12}$

Order 288: $C_6^2:D_4$ x 4, $C_6^2:D_4$

Order 216: $C_6^2:C_6$ x 16, $C_6^3$ x 11, $C_6^2:C_6$ x 10, $C_6^2.C_6$ x 6

Order 144: $C_6^2:C_2^2$ x 48, $C_6^2:C_2^2$ x 12, $C_2^2\times C_6^2$ x 9, $C_6^2:C_2^2$ x 4, $C_6^2:C_4$ x 4, $C_6^2:C_2^2$, $C_6^2:C_4$

Order 108: $C_3\times C_6^2$ x 23, $C_3^2:D_6$ x 16, $C_3^2:C_{12}$ x 4

Order 96: $C_2^3:D_6$ x 4, $C_{12}:C_2^3$

Order 72: $C_2\times C_6^2$ x 115, $C_6\wr C_2$ x 64, $C_6\times D_6$ x 40, $C_6:C_{12}$ x 24, $C_6^2:C_2$ x 16, $C_6:D_6$ x 10, $C_6.D_6$ x 6

Order 54: $C_3^2\times C_6$ x 11, $C_3^2:C_6$ x 4

Order 48: $C_6:D_4$ x 48, $C_6\times D_4$ x 12, $C_2^3\times C_6$ x 10, $C_2^2\times D_6$ x 4, $C_6.C_2^3$ x 4, $C_2^2\times C_{12}$

Order 36: $C_6^2$ x 255, $C_6\times S_3$ x 64, $C_3:C_{12}$ x 16, $C_6:S_3$ x 16, $C_3^2:C_4$ x 4

Order 32: $C_2^2\times D_4$

Order 27: $C_3^3$

Order 24: $C_2^2\times C_6$ x 125, $C_3:D_4$ x 64, $C_2\times D_6$ x 40, $C_6:C_4$ x 24, $C_3\times D_4$ x 16, $C_2\times C_{12}$ x 6

Order 18: $C_3\times C_6$ x 115, $C_3\times S_3$ x 16, $C_3:S_3$ x 4

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

Order 12: $C_2\times C_6$ x 271, $D_6$ x 64, $C_3:C_4$ x 16, $C_{12}$ x 4

Order 9: $C_3^2$ x 9

Order 8: $C_2^3$ x 21, $D_4$ x 16, $C_2\times C_4$ x 6

Order 6: $C_6$ x 119, $S_3$ x 16

Order 4: $C_2^2$ x 39, $C_4$ x 4

Order 3: $C_3$ x 9

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 864: $C_6^3:C_2^2$

Order 432: $C_2\times C_6^3$, $C_6^3:C_2$, $C_6^3:C_2$, $C_6^2:C_{12}$

Order 288: $C_6^2:D_4$, $C_6^2:D_4$

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

Order 144: $C_2^2\times C_6^2$ x 3, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_6^2:C_4$, $C_6^2:C_2^2$, $C_6^2:C_4$

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

Order 96: $C_{12}:C_2^3$, $C_2^3:D_6$

Order 72: $C_2\times C_6^2$ x 9, $C_6:D_6$ x 2, $C_6\times D_6$ x 2, $C_6^2:C_2$, $C_6.D_6$, $C_6\wr C_2$, $C_6:C_{12}$

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

Order 48: $C_2^3\times C_6$ x 4, $C_2^2\times D_6$, $C_6\times D_4$, $C_2^2\times C_{12}$, $C_6:D_4$, $C_6.C_2^3$

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

Order 32: $C_2^2\times D_4$

Order 27: $C_3^3$

Order 24: $C_2^2\times C_6$ x 11, $C_2\times D_6$ x 2, $C_2\times C_{12}$, $C_3:D_4$, $C_6:C_4$, $C_3\times D_4$

Order 18: $C_3\times C_6$ x 9, $C_3:S_3$, $C_3\times S_3$

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

Order 12: $C_2\times C_6$ x 14, $D_6$ x 2, $C_{12}$, $C_3:C_4$

Order 9: $C_3^2$ x 3

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

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

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 864: $C_6^3:C_2^2$ ($C_1$)

Order 432: $C_6^3:C_2$ ($C_2$) x 12, $C_2\times C_6^3$ ($C_2$), $C_6^3:C_2$ ($C_2$), $C_6^2:C_{12}$ ($C_2$)

Order 288: $C_6^2:D_4$ ($C_3$)

Order 216: $C_6^2:C_6$ ($C_2^2$) x 16, $C_6^3$ ($C_2^2$) x 7, $C_6^2:C_6$ ($C_2^2$) x 6, $C_6^2.C_6$ ($C_2^2$) x 6

Order 144: $C_6^2:C_2^2$ ($C_6$) x 12, $C_2^2\times C_6^2$ ($S_3$) x 4, $C_2^2\times C_6^2$ ($C_6$), $C_6^2:C_2^2$ ($C_6$), $C_6^2:C_4$ ($C_6$)

Order 108: $C_3\times C_6^2$ ($C_2^3$) x 7, $C_3\times C_6^2$ ($D_4$) x 4, $C_3^2:D_6$ ($C_2^3$) x 4, $C_3^2:C_{12}$ ($C_2^3$) x 4

Order 72: $C_2\times C_6^2$ ($D_6$) x 28, $C_6^2:C_2$ ($C_2\times C_6$) x 16, $C_2\times C_6^2$ ($C_2\times C_6$) x 7, $C_6:D_6$ ($C_2\times C_6$) x 6, $C_6.D_6$ ($C_2\times C_6$) x 6

Order 54: $C_3^2\times C_6$ ($C_2\times D_4$) x 6, $C_3^2\times C_6$ ($C_2^4$)

Order 48: $C_2^3\times C_6$ ($C_3\times S_3$) x 4, $C_2^3\times C_6$ ($C_3:S_3$)

Order 36: $C_6^2$ ($C_2\times D_6$) x 28, $C_6^2$ ($C_3:D_4$) x 16, $C_6^2$ ($C_2^2\times C_6$) x 7, $C_3^2:C_4$ ($C_2^2\times C_6$) x 4, $C_6^2$ ($C_3\times D_4$) x 4, $C_6:S_3$ ($C_2^2\times C_6$) x 4

Order 27: $C_3^3$ ($C_2^2\times D_4$)

Order 24: $C_2^2\times C_6$ ($C_6\times S_3$) x 28, $C_2^2\times C_6$ ($C_6:S_3$) x 7

Order 18: $C_3\times C_6$ ($C_6:D_4$) x 24, $C_3\times C_6$ ($C_6\times D_4$) x 6, $C_3\times C_6$ ($C_2^2\times D_6$) x 4, $C_3\times C_6$ ($C_2^3\times C_6$)

Order 16: $C_2^4$ ($C_3^2:C_6$)

Order 12: $C_2\times C_6$ ($C_6\times D_6$) x 28, $C_2\times C_6$ ($C_6\wr C_2$) x 16, $C_2\times C_6$ ($C_6:D_6$) x 7, $C_2\times C_6$ ($C_6^2:C_2$) x 4

Order 9: $C_3^2$ ($C_2^3:D_6$) x 4, $C_3^2$ ($C_{12}:C_2^3$)

Order 8: $C_2^3$ ($C_3^2:D_6$) x 7

Order 6: $C_6$ ($C_6^2:C_2^2$) x 24, $C_6$ ($C_6^2:C_2^2$) x 6, $C_6$ ($C_6^2:C_2^2$) x 4, $C_6$ ($C_6^2:C_2^2$)

Order 4: $C_2^2$ ($C_6^2:C_6$) x 7, $C_2^2$ ($C_6^2:C_6$) x 4

Order 3: $C_3$ ($C_6^2:D_4$) x 4, $C_3$ ($C_6^2:D_4$)

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

Order 1: $C_1$ ($C_6^3:C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 864: $C_6^3:C_2^2$ ($C_1$)

Order 432: $C_2\times C_6^3$ ($C_2$), $C_6^3:C_2$ ($C_2$), $C_6^3:C_2$ ($C_2$), $C_6^2:C_{12}$ ($C_2$)

Order 288: $C_6^2:D_4$ ($C_3$)

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

Order 144: $C_2^2\times C_6^2$ ($C_6$), $C_2^2\times C_6^2$ ($S_3$), $C_6^2:C_2^2$ ($C_6$), $C_6^2:C_2^2$ ($C_6$), $C_6^2:C_4$ ($C_6$)

Order 108: $C_3\times C_6^2$ ($C_2^3$) x 2, $C_3\times C_6^2$ ($D_4$), $C_3^2:D_6$ ($C_2^3$), $C_3^2:C_{12}$ ($C_2^3$)

Order 72: $C_2\times C_6^2$ ($C_2\times C_6$) x 2, $C_2\times C_6^2$ ($D_6$) x 2, $C_6:D_6$ ($C_2\times C_6$), $C_6^2:C_2$ ($C_2\times C_6$), $C_6.D_6$ ($C_2\times C_6$)

Order 54: $C_3^2\times C_6$ ($C_2^4$), $C_3^2\times C_6$ ($C_2\times D_4$)

Order 48: $C_2^3\times C_6$ ($C_3:S_3$), $C_2^3\times C_6$ ($C_3\times S_3$)

Order 36: $C_6^2$ ($C_2^2\times C_6$) x 2, $C_6^2$ ($C_2\times D_6$) x 2, $C_3^2:C_4$ ($C_2^2\times C_6$), $C_6^2$ ($C_3:D_4$), $C_6^2$ ($C_3\times D_4$), $C_6:S_3$ ($C_2^2\times C_6$)

Order 27: $C_3^3$ ($C_2^2\times D_4$)

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

Order 18: $C_3\times C_6$ ($C_2^3\times C_6$), $C_3\times C_6$ ($C_2^2\times D_6$), $C_3\times C_6$ ($C_6\times D_4$), $C_3\times C_6$ ($C_6:D_4$)

Order 16: $C_2^4$ ($C_3^2:C_6$)

Order 12: $C_2\times C_6$ ($C_6:D_6$) x 2, $C_2\times C_6$ ($C_6\times D_6$) x 2, $C_2\times C_6$ ($C_6^2:C_2$), $C_2\times C_6$ ($C_6\wr C_2$)

Order 9: $C_3^2$ ($C_{12}:C_2^3$), $C_3^2$ ($C_2^3:D_6$)

Order 8: $C_2^3$ ($C_3^2:D_6$) x 2

Order 6: $C_6$ ($C_6^2:C_2^2$), $C_6$ ($C_6^2:C_2^2$), $C_6$ ($C_6^2:C_2^2$), $C_6$ ($C_6^2:C_2^2$)

Order 4: $C_2^2$ ($C_6^2:C_6$) x 2, $C_2^2$ ($C_6^2:C_6$)

Order 3: $C_3$ ($C_6^2:D_4$), $C_3$ ($C_6^2:D_4$)

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

Order 1: $C_1$ ($C_6^3:C_2^2$)

Series

Derived series

$C_6^3:C_2^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_1$

Chief series

$C_6^3:C_2^2$

$\rhd$

$C_6^2:C_{12}$

$\rhd$

$C_6^2:C_4$

$\rhd$

$C_2\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^3:C_2^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_6$

$\lhd$

$C_2^3\times C_6$

Supergroups

This group is a maximal subgroup of 132 larger groups in the database.

This group is a maximal quotient of 49 larger groups in the database.

Character theory

Complex character table

See the $252 \times 252$ 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 $152 \times 152$ rational character table (warning: may be slow to load).