Abstract group 864.4723: $C_6^3:C_2^2$

• Label: 864.4723
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{4} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{5} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{14} \cdot 3^{3} \cdot 5 \cdot 7 \)
• Perm deg.: $17$
• Trans deg.: $288$
• Rank: $5$

Group information

Description:	$C_6^3:C_2^2$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$C_2\times C_2^4.A_8\times \AGL(2,3)$, of order \(278691840\)\(\medspace = 2^{15} \cdot 3^{5} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_3$ x 3
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	159	26	678	864
Conjugacy classes	1	31	14	242	288
Divisions	1	31	9	151	192
Autjugacy classes	1	2	3	4	10

Dimension	1	2	4
Irr. complex chars.	96	192	0	288
Irr. rational chars.	32	96	64	192

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$288$
Rank:	$5$
Inequivalent generating 5-tuples:	$487630$

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:	$\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{6}=d^{6}=e^{6}=[a,b]=[a,c]=[a,d]=[a,e]=[b,c]=[c,d]=[c,e]=[d,e]=1, d^{b}=d^{5}, e^{b}=e^{5} \rangle$
Permutation group:	Degree $17$ $\langle(1,2)(3,4)(5,6)(13,14)(16,17), (1,3)(2,4)(5,6)(10,11), (1,4)(2,3)(10,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 9 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 24 & 25 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 35 & 0 \\ 0 & 35 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 16 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Direct product:	$C_2$ ${}^4$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_3:S_3)$
Semidirect product:	$C_6^3$ $\,\rtimes\,$ $C_2^2$	$C_3^3$ $\,\rtimes\,$ $C_2^5$	$(C_2\times C_6^2)$ $\,\rtimes\,$ $D_6$	$C_6^2$ $\,\rtimes\,$ $(C_2\times D_6)$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{36}\Z)$.

Homology

Abelianization:	$C_{2}^{4} \times C_{6} \simeq C_{2}^{5} \times C_{3}$
Schur multiplier:	$C_{2}^{9} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 15384 subgroups in 5024 conjugacy classes, 1418 normal (10 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

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 30, $C_2\times C_6^3$

Order 288: $C_6^2:C_2^3$ x 4, $C_6^2:C_2^3$

Order 216: $C_6^2:C_6$ x 140, $C_6^3$ x 15

Order 144: $C_6^2:C_2^2$ x 120, $C_6^2:C_2^2$ x 30, $C_2^2\times C_6^2$ x 9

Order 108: $C_3^2:D_6$ x 120, $C_3\times C_6^2$ x 35

Order 96: $C_2^3\times D_6$ x 4, $C_2^4\times C_6$

Order 72: $C_6\times D_6$ x 560, $C_6:D_6$ x 140, $C_2\times C_6^2$ x 135

Order 54: $C_3^2:C_6$ x 16, $C_3^2\times C_6$ x 15

Order 48: $C_2^2\times D_6$ x 120, $C_2^3\times C_6$ x 39

Order 36: $C_6\times S_3$ x 480, $C_6^2$ x 315, $C_6:S_3$ x 120

Order 32: $C_2^5$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 560, $C_2^2\times C_6$ x 275

Order 18: $C_3\times C_6$ x 135, $C_3\times S_3$ x 64, $C_3:S_3$ x 16

Order 16: $C_2^4$ x 31

Order 12: $D_6$ x 480, $C_2\times C_6$ x 435

Order 9: $C_3^2$ x 9

Order 8: $C_2^3$ x 155

Order 6: $C_6$ x 151, $S_3$ x 64

Order 4: $C_2^2$ x 155

Order 3: $C_3$ x 9

Order 2: $C_2$ x 31

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 432: $C_2\times C_6^3$, $C_6^3:C_2$

Order 288: $C_6^2:C_2^3$, $C_6^2:C_2^3$

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

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

Order 108: $C_3\times C_6^2$, $C_3^2:D_6$

Order 96: $C_2^4\times C_6$, $C_2^3\times D_6$

Order 72: $C_2\times C_6^2$ x 3, $C_6:D_6$, $C_6\times D_6$

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

Order 48: $C_2^3\times C_6$ x 4, $C_2^2\times D_6$

Order 36: $C_6^2$ x 3, $C_6:S_3$, $C_6\times S_3$

Order 32: $C_2^5$

Order 27: $C_3^3$

Order 24: $C_2^2\times C_6$ x 4, $C_2\times D_6$

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

Order 16: $C_2^4$ x 2

Order 12: $C_2\times C_6$ x 4, $D_6$

Order 9: $C_3^2$ x 3

Order 8: $C_2^3$ x 2

Order 6: $C_6$ x 4, $S_3$

Order 4: $C_2^2$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

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 30, $C_2\times C_6^3$ ($C_2$)

Order 288: $C_6^2:C_2^3$ ($C_3$)

Order 216: $C_6^2:C_6$ ($C_2^2$) x 140, $C_6^3$ ($C_2^2$) x 15

Order 144: $C_6^2:C_2^2$ ($C_6$) x 30, $C_2^2\times C_6^2$ ($S_3$) x 4, $C_2^2\times C_6^2$ ($C_6$)

Order 108: $C_3^2:D_6$ ($C_2^3$) x 120, $C_3\times C_6^2$ ($C_2^3$) x 35

Order 72: $C_6:D_6$ ($C_2\times C_6$) x 140, $C_2\times C_6^2$ ($D_6$) x 60, $C_2\times C_6^2$ ($C_2\times C_6$) x 15

Order 54: $C_3^2:C_6$ ($C_2^4$) x 16, $C_3^2\times C_6$ ($C_2^4$) x 15

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 140, $C_6:S_3$ ($C_2^2\times C_6$) x 120, $C_6^2$ ($C_2^2\times C_6$) x 35

Order 27: $C_3^3$ ($C_2^5$)

Order 24: $C_2^2\times C_6$ ($C_6\times S_3$) x 60, $C_2^2\times C_6$ ($C_6:S_3$) x 15

Order 18: $C_3\times C_6$ ($C_2^2\times D_6$) x 60, $C_3:S_3$ ($C_2^3\times C_6$) x 16, $C_3\times C_6$ ($C_2^3\times C_6$) x 15

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

Order 12: $C_2\times C_6$ ($C_6\times D_6$) x 140, $C_2\times C_6$ ($C_6:D_6$) x 35

Order 9: $C_3^2$ ($C_2^3\times D_6$) x 4, $C_3^2$ ($C_2^4\times C_6$)

Order 8: $C_2^3$ ($C_3^2:D_6$) x 15

Order 6: $C_6$ ($C_6^2:C_2^2$) x 60, $C_6$ ($C_6^2:C_2^2$) x 15

Order 4: $C_2^2$ ($C_6^2:C_6$) x 35

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

Order 2: $C_2$ ($C_6^3:C_2$) x 15

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: $C_2\times C_6^3$ ($C_2$), $C_6^3:C_2$ ($C_2$)

Order 288: $C_6^2:C_2^3$ ($C_3$)

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

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

Order 108: $C_3\times C_6^2$ ($C_2^3$), $C_3^2:D_6$ ($C_2^3$)

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

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

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

Order 27: $C_3^3$ ($C_2^5$)

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

Order 18: $C_3\times C_6$ ($C_2^3\times C_6$), $C_3\times C_6$ ($C_2^2\times D_6$), $C_3:S_3$ ($C_2^3\times C_6$)

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

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

Order 9: $C_3^2$ ($C_2^4\times C_6$), $C_3^2$ ($C_2^3\times D_6$)

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

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

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

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

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

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

Series

Derived series

$C_6^3:C_2^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_6^3:C_2^2$

$\rhd$

$C_6^3:C_2$

$\rhd$

$C_6^3$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^3:C_2^2$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2^3\times C_6$

Supergroups

This group is a maximal subgroup of 46 larger groups in the database.

This group is a maximal quotient of 21 larger groups in the database.

Character theory

Complex character table

See the $288 \times 288$ 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 $192 \times 192$ rational character table (warning: may be slow to load).