Abstract group 1536.408528972: $C_4^4:C_6$

• Label: 1536.408528972
• Order: \( 2^{9} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{5} \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \cdot 3 \)
• Perm deg.: $18$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$C_4^4:C_6$
Order:	\(1536\)\(\medspace = 2^{9} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_4^2:C_3.D_4\times C_2^6.S_4$, of order \(589824\)\(\medspace = 2^{16} \cdot 3^{2} \)
Composition factors:	$C_2$ x 9, $C_3$
Derived length:	$2$

This group is nonabelian, monomial (hence solvable), metabelian, and an A-group.

Group statistics

Order	1	2	3	4	6	12
Elements	1	31	32	480	224	768	1536
Conjugacy classes	1	15	2	176	14	48	256
Divisions	1	15	1	88	7	12	124
Autjugacy classes	1	5	1	6	2	1	16

Dimension	1	2	3	4	6
Irr. complex chars.	96	0	160	0	0	256
Irr. rational chars.	8	20	8	12	76	124

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$96$
Rank:	$3$
Inequivalent generating triples:	$1820$

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:	${\langle a, b, c, d, e \mid a^{12}=b^{4}=c^{4}=d^{4}=e^{2}=[b,c]=[b,d]=[b,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(1,2,6)(3,9,10)(4,8,11)(5,7,12), (1,3)(4,5)(6,10)(11,12)(13,14,16,15)(17,18) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 3 & 12 \\ 4 & 3 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 4 & 5 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 7 & 12 \\ 4 & 3 \end{array}\right), \left(\begin{array}{rr} 3 & 5 \\ 7 & 12 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_4$ ${}^2$ $\, \times\, $ $(C_4^2:C_3)$
Semidirect product:	$C_4^4$ $\,\rtimes\,$ $C_6$	$(C_2\times C_4^4)$ $\,\rtimes\,$ $C_3$	$(C_2\times C_4^3)$ $\,\rtimes\,$ $C_{12}$	$C_4^3$ $\,\rtimes\,$ $(C_2\times C_{12})$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^5.A_4)$ . $C_2^2$	$(C_2^4.A_4)$ . $C_2^3$	$(C_2^3\times C_4^2)$ . $A_4$	$C_2^5$ . $(C_2^2\times A_4)$	all 30

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{16}\Z)$.

Homology

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

Subgroups

There are 14640 subgroups in 4772 conjugacy classes, 216 normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4^2$	$G/Z \simeq$ $C_4^2:C_3$
Commutator:	$G' \simeq$ $C_4^2$	$G/G' \simeq$ $C_2\times C_4\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2^4$	$G/\Phi \simeq$ $C_2^3\times A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_4^4$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_4^4:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_2^2\times A_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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 1536: $C_4^4:C_6$

Order 768: $C_4^3:C_{12}$ x 4, $C_4\wr C_3\times C_2^2$ x 3

Order 512: $C_2\times C_4^4$

Order 384: $C_4^3:C_6$ x 18, $C_2\times A_4\times C_4^2$, $C_2^5.A_4$

Order 256: $C_4^4$ x 8, $C_2^2\times C_4^3$ x 7

Order 192: $C_4\wr C_3$ x 12, $C_2^4.A_4$ x 7, $A_4\times C_4^2$ x 4, $C_2^4:C_{12}$ x 3

Order 128: $C_2\times C_4^3$ x 132, $C_2^3\times C_4^2$ x 13

Order 96: $C_2^3:C_{12}$ x 18, $C_4^2:C_6$ x 7, $C_2^3\times A_4$, $C_2\times C_4\times C_{12}$

Order 64: $C_2^2\times C_4^2$ x 334, $C_4^3$ x 328, $C_2^4\times C_4$ x 7

Order 48: $C_4\times A_4$ x 12, $C_2^2\times A_4$ x 7, $C_4\times C_{12}$ x 4, $C_2^2\times C_{12}$ x 3, $C_4^2:C_3$

Order 32: $C_2\times C_4^2$ x 1312, $C_2^3\times C_4$ x 162, $C_2^5$

Order 24: $C_2\times C_{12}$ x 18, $C_2\times A_4$ x 7, $C_2^2\times C_6$

Order 16: $C_4^2$ x 750, $C_2^2\times C_4$ x 710, $C_2^4$ x 15

Order 12: $C_{12}$ x 12, $C_2\times C_6$ x 7, $A_4$

Order 8: $C_2\times C_4$ x 612, $C_2^3$ x 57

Order 6: $C_6$ x 7

Order 4: $C_4$ x 88, $C_2^2$ x 57

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1536: $C_4^4:C_6$

Order 768: $C_4\wr C_3\times C_2^2$, $C_4^3:C_{12}$

Order 512: $C_2\times C_4^4$

Order 384: $C_4^3:C_6$ x 2, $C_2\times A_4\times C_4^2$, $C_2^5.A_4$

Order 256: $C_2^2\times C_4^3$ x 3, $C_4^4$ x 2

Order 192: $C_2^4.A_4$ x 2, $A_4\times C_4^2$, $C_4\wr C_3$, $C_2^4:C_{12}$

Order 128: $C_2\times C_4^3$ x 12, $C_2^3\times C_4^2$ x 6

Order 96: $C_4^2:C_6$ x 2, $C_2^3:C_{12}$ x 2, $C_2^3\times A_4$, $C_2\times C_4\times C_{12}$

Order 64: $C_2^2\times C_4^2$ x 28, $C_4^3$ x 10, $C_2^4\times C_4$ x 3

Order 48: $C_2^2\times A_4$ x 2, $C_2^2\times C_{12}$, $C_4\times A_4$, $C_4^2:C_3$, $C_4\times C_{12}$

Order 32: $C_2\times C_4^2$ x 44, $C_2^3\times C_4$ x 19, $C_2^5$

Order 24: $C_2\times C_{12}$ x 2, $C_2\times A_4$ x 2, $C_2^2\times C_6$

Order 16: $C_2^2\times C_4$ x 39, $C_4^2$ x 20, $C_2^4$ x 5

Order 12: $C_2\times C_6$ x 2, $A_4$, $C_{12}$

Order 8: $C_2\times C_4$ x 29, $C_2^3$ x 12

Order 6: $C_6$ x 2

Order 4: $C_2^2$ x 12, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1536: $C_4^4:C_6$ ($C_1$)

Order 768: $C_4^3:C_{12}$ ($C_2$) x 4, $C_4\wr C_3\times C_2^2$ ($C_2$) x 3

Order 512: $C_2\times C_4^4$ ($C_3$)

Order 384: $C_4^3:C_6$ ($C_4$) x 12, $C_4^3:C_6$ ($C_2^2$) x 6, $C_2^5.A_4$ ($C_2^2$)

Order 256: $C_4^4$ ($C_6$) x 4, $C_2^2\times C_4^3$ ($C_6$) x 3

Order 192: $C_4\wr C_3$ ($C_2\times C_4$) x 12, $C_2^4.A_4$ ($C_2\times C_4$) x 6, $C_2^4.A_4$ ($C_2^3$)

Order 128: $C_2\times C_4^3$ ($C_{12}$) x 12, $C_2\times C_4^3$ ($C_2\times C_6$) x 6, $C_2^3\times C_4^2$ ($C_2\times C_6$), $C_2^3\times C_4^2$ ($A_4$)

Order 96: $C_4^2:C_6$ ($C_4^2$) x 4, $C_4^2:C_6$ ($C_2^2\times C_4$) x 3

Order 64: $C_4^3$ ($C_2\times C_{12}$) x 12, $C_2^2\times C_4^2$ ($C_2\times C_{12}$) x 6, $C_2^2\times C_4^2$ ($C_2\times A_4$) x 4, $C_2^4\times C_4$ ($C_2\times A_4$) x 3, $C_2^2\times C_4^2$ ($C_2^2\times C_6$)

Order 48: $C_4^2:C_3$ ($C_2\times C_4^2$)

Order 32: $C_2^3\times C_4$ ($C_4\times A_4$) x 12, $C_2^3\times C_4$ ($C_2^2\times A_4$) x 6, $C_2\times C_4^2$ ($C_4\times C_{12}$) x 4, $C_2\times C_4^2$ ($C_2^2\times C_{12}$) x 3, $C_2^5$ ($C_2^2\times A_4$), $C_2\times C_4^2$ ($C_4^2:C_3$)

Order 16: $C_2^2\times C_4$ ($C_2^3:C_{12}$) x 12, $C_2^4$ ($C_2^3:C_{12}$) x 6, $C_4^2$ ($C_4^2:C_6$) x 4, $C_2^2\times C_4$ ($C_4^2:C_6$) x 3, $C_4^2$ ($C_2\times C_4\times C_{12}$), $C_2^4$ ($C_2^3\times A_4$)

Order 8: $C_2\times C_4$ ($C_4\wr C_3$) x 12, $C_2\times C_4$ ($C_2^4.A_4$) x 6, $C_2^3$ ($A_4\times C_4^2$) x 4, $C_2^3$ ($C_2^4:C_{12}$) x 3, $C_2^3$ ($C_2^4.A_4$)

Order 4: $C_4$ ($C_4^3:C_6$) x 12, $C_2^2$ ($C_4^3:C_6$) x 6, $C_2^2$ ($C_2\times A_4\times C_4^2$), $C_2^2$ ($C_2^5.A_4$)

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

Order 1: $C_1$ ($C_4^4:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1536: $C_4^4:C_6$ ($C_1$)

Order 768: $C_4\wr C_3\times C_2^2$ ($C_2$), $C_4^3:C_{12}$ ($C_2$)

Order 512: $C_2\times C_4^4$ ($C_3$)

Order 384: $C_4^3:C_6$ ($C_2^2$), $C_4^3:C_6$ ($C_4$), $C_2^5.A_4$ ($C_2^2$)

Order 256: $C_4^4$ ($C_6$), $C_2^2\times C_4^3$ ($C_6$)

Order 192: $C_2^4.A_4$ ($C_2^3$), $C_2^4.A_4$ ($C_2\times C_4$), $C_4\wr C_3$ ($C_2\times C_4$)

Order 128: $C_2\times C_4^3$ ($C_2\times C_6$), $C_2\times C_4^3$ ($C_{12}$), $C_2^3\times C_4^2$ ($C_2\times C_6$), $C_2^3\times C_4^2$ ($A_4$)

Order 96: $C_4^2:C_6$ ($C_4^2$), $C_4^2:C_6$ ($C_2^2\times C_4$)

Order 64: $C_4^3$ ($C_2\times C_{12}$), $C_2^4\times C_4$ ($C_2\times A_4$), $C_2^2\times C_4^2$ ($C_2\times C_{12}$), $C_2^2\times C_4^2$ ($C_2^2\times C_6$), $C_2^2\times C_4^2$ ($C_2\times A_4$)

Order 48: $C_4^2:C_3$ ($C_2\times C_4^2$)

Order 32: $C_2^5$ ($C_2^2\times A_4$), $C_2^3\times C_4$ ($C_2^2\times A_4$), $C_2^3\times C_4$ ($C_4\times A_4$), $C_2\times C_4^2$ ($C_2^2\times C_{12}$), $C_2\times C_4^2$ ($C_4^2:C_3$), $C_2\times C_4^2$ ($C_4\times C_{12}$)

Order 16: $C_4^2$ ($C_4^2:C_6$), $C_4^2$ ($C_2\times C_4\times C_{12}$), $C_2^4$ ($C_2^3\times A_4$), $C_2^4$ ($C_2^3:C_{12}$), $C_2^2\times C_4$ ($C_4^2:C_6$), $C_2^2\times C_4$ ($C_2^3:C_{12}$)

Order 8: $C_2^3$ ($A_4\times C_4^2$), $C_2^3$ ($C_2^4.A_4$), $C_2^3$ ($C_2^4:C_{12}$), $C_2\times C_4$ ($C_2^4.A_4$), $C_2\times C_4$ ($C_4\wr C_3$)

Order 4: $C_2^2$ ($C_4^3:C_6$), $C_2^2$ ($C_2\times A_4\times C_4^2$), $C_2^2$ ($C_2^5.A_4$), $C_4$ ($C_4^3:C_6$)

Order 2: $C_2$ ($C_4\wr C_3\times C_2^2$), $C_2$ ($C_4^3:C_{12}$)

Order 1: $C_1$ ($C_4^4:C_6$)

Series

Derived series

$C_4^4:C_6$

$\rhd$

$C_4^2$

$\rhd$

$C_1$

Chief series

$C_4^4:C_6$

$\rhd$

$C_4^3:C_{12}$

$\rhd$

$C_4^3:C_6$

$\rhd$

$C_2^4.A_4$

$\rhd$

$C_2^2\times C_4^2$

$\rhd$

$C_2\times C_4^2$

$\rhd$

$C_4^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_4^4:C_6$

$\rhd$

$C_4^2$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4^2$

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 $256 \times 256$ 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 $124 \times 124$ rational character table.