Abstract group 576.2103: $C_6^2.\OD_{16}$

• Label: 576.2103
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $38$
• Trans deg.: $96$
• Rank: $2$

Group information

Description:	$C_6^2.\OD_{16}$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$(C_2^2\times F_9).C_2^4$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	12	16
Elements	1	7	8	8	56	144	64	288	576
Conjugacy classes	1	3	2	4	8	8	10	12	48
Divisions	1	3	2	3	8	3	8	2	30
Autjugacy classes	1	3	1	3	3	3	3	2	19

Dimension	1	2	4	8	16
Irr. complex chars.	16	12	16	4	0	48
Irr. rational chars.	4	4	19	0	3	30

Minimal presentations

Permutation degree:	$38$
Transitive degree:	$96$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	16
Arbitrary	6	8	16

Constructions

Presentation:	$\langle a, b, c \mid b^{6}=c^{12}=1, a^{8}=c^{6}, b^{a}=bc^{5}, c^{a}=b^{2}c^{11}, c^{b}=c^{7} \rangle$
Permutation group:	Degree $38$ $\langle(1,2,4,9,6,10,17,25,7,11,18,26,5,19,16,29)(3,12,13,27,14,8,21,22,15,23,28,31,20,24,30,32) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6^2.C_8)$ $\,\rtimes\,$ $C_2$	$(C_3^2:\OD_{32})$ $\,\rtimes\,$ $C_2$	$C_3^2$ $\,\rtimes\,$ $(\OD_{32}:C_2)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $\OD_{16}$	$(C_{12}.D_6)$ . $C_4$	$(Q_8\times C_3^2)$ . $C_8$	$Q_8$ . $(C_3^2:C_8)$	all 20

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

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 450 subgroups in 118 conjugacy classes, 26 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $C_6^2:C_4$
Commutator:	$G' \simeq$ $C_3\times C_{12}$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2\times C_3^2:C_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_4.C_6^2$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $C_6^2.\OD_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2:C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{32}:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

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 576: $C_6^2.\OD_{16}$

Order 288: $C_{12}.(C_2\times D_6)$, $C_3^2:\OD_{32}$, $C_6^2.C_8$

Order 144: $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_3^2:C_{16}$ x 2, $C_4.C_6^2$

Order 96: $C_{12}.C_2^3$ x 2

Order 72: $C_3^2:C_8$ x 3, $D_4\times C_3^2$ x 2, $C_6\times C_{12}$ x 2, $Q_8\times C_3^2$

Order 64: $\OD_{32}:C_2$

Order 48: $C_6:C_8$ x 6, $C_3:\OD_{16}$ x 6, $D_4:C_6$ x 2

Order 36: $C_3\times C_{12}$ x 3, $C_6^2$ x 2

Order 32: $\OD_{16}:C_2$, $\OD_{32}$, $C_2\times C_{16}$

Order 24: $C_3:C_8$ x 8, $C_2\times C_{12}$ x 6, $C_3\times D_4$ x 6, $C_3\times Q_8$ x 2

Order 18: $C_3\times C_6$ x 3

Order 16: $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_{16}$ x 2, $D_4:C_2$

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 6

Order 9: $C_3^2$

Order 8: $C_8$ x 3, $D_4$ x 2, $C_2\times C_4$ x 2, $Q_8$

Order 6: $C_6$ x 8

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $C_6^2.\OD_{16}$

Order 288: $C_{12}.(C_2\times D_6)$, $C_3^2:\OD_{32}$, $C_6^2.C_8$

Order 144: $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_3^2:C_{16}$ x 2, $C_4.C_6^2$

Order 96: $C_{12}.C_2^3$

Order 72: $C_3^2:C_8$ x 3, $D_4\times C_3^2$ x 2, $C_6\times C_{12}$ x 2, $Q_8\times C_3^2$

Order 64: $\OD_{32}:C_2$

Order 48: $C_6:C_8$ x 2, $C_3:\OD_{16}$ x 2, $D_4:C_6$

Order 36: $C_3\times C_{12}$ x 3, $C_6^2$ x 2

Order 32: $\OD_{16}:C_2$, $\OD_{32}$, $C_2\times C_{16}$

Order 24: $C_3:C_8$ x 3, $C_2\times C_{12}$ x 2, $C_3\times D_4$ x 2, $C_3\times Q_8$

Order 18: $C_3\times C_6$ x 3

Order 16: $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_{16}$ x 2, $D_4:C_2$

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

Order 9: $C_3^2$

Order 8: $C_8$ x 3, $D_4$ x 2, $C_2\times C_4$ x 2, $Q_8$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $C_6^2.\OD_{16}$ ($C_1$)

Order 288: $C_{12}.(C_2\times D_6)$ ($C_2$), $C_3^2:\OD_{32}$ ($C_2$), $C_6^2.C_8$ ($C_2$)

Order 144: $C_{12}.D_6$ ($C_4$), $C_{12}.D_6$ ($C_2^2$), $C_4.C_6^2$ ($C_4$)

Order 72: $C_3^2:C_8$ ($D_4$) x 2, $Q_8\times C_3^2$ ($C_8$), $D_4\times C_3^2$ ($C_8$), $C_6\times C_{12}$ ($C_2\times C_4$)

Order 36: $C_3\times C_{12}$ ($C_2\times C_8$), $C_3\times C_{12}$ ($C_2^2:C_4$), $C_6^2$ ($\OD_{16}$)

Order 18: $C_3\times C_6$ ($C_2^2:C_8$)

Order 16: $D_4:C_2$ ($C_3^2:C_4$)

Order 9: $C_3^2$ ($\OD_{32}:C_2$)

Order 8: $Q_8$ ($C_3^2:C_8$), $D_4$ ($C_3^2:C_8$), $C_2\times C_4$ ($C_2\times C_3^2:C_4$)

Order 4: $C_2^2$ ($C_3^2:\OD_{16}$), $C_4$ ($C_6^2:C_4$), $C_4$ ($C_6^2.C_4$)

Order 2: $C_2$ ($C_6^2:C_8$)

Order 1: $C_1$ ($C_6^2.\OD_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $C_6^2.\OD_{16}$ ($C_1$)

Order 288: $C_{12}.(C_2\times D_6)$ ($C_2$), $C_3^2:\OD_{32}$ ($C_2$), $C_6^2.C_8$ ($C_2$)

Order 144: $C_{12}.D_6$ ($C_4$), $C_{12}.D_6$ ($C_2^2$), $C_4.C_6^2$ ($C_4$)

Order 72: $C_3^2:C_8$ ($D_4$) x 2, $Q_8\times C_3^2$ ($C_8$), $D_4\times C_3^2$ ($C_8$), $C_6\times C_{12}$ ($C_2\times C_4$)

Order 36: $C_3\times C_{12}$ ($C_2\times C_8$), $C_3\times C_{12}$ ($C_2^2:C_4$), $C_6^2$ ($\OD_{16}$)

Order 18: $C_3\times C_6$ ($C_2^2:C_8$)

Order 16: $D_4:C_2$ ($C_3^2:C_4$)

Order 9: $C_3^2$ ($\OD_{32}:C_2$)

Order 8: $Q_8$ ($C_3^2:C_8$), $D_4$ ($C_3^2:C_8$), $C_2\times C_4$ ($C_2\times C_3^2:C_4$)

Order 4: $C_2^2$ ($C_3^2:\OD_{16}$), $C_4$ ($C_6^2:C_4$), $C_4$ ($C_6^2.C_4$)

Order 2: $C_2$ ($C_6^2:C_8$)

Order 1: $C_1$ ($C_6^2.\OD_{16}$)

Series

Derived series

$C_6^2.\OD_{16}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_1$

Chief series

$C_6^2.\OD_{16}$

$\rhd$

$C_3^2:\OD_{32}$

$\rhd$

$C_{12}.D_6$

$\rhd$

$C_3^2:C_8$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_6^2.\OD_{16}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_4$

$\lhd$

$C_2\times C_4$

$\lhd$

$D_4:C_2$

Supergroups

This group is a maximal subgroup of 5 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 $48 \times 48$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.