Abstract group 384.1698: $\OD_{32}.D_6$

• Label: 384.1698
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $35$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$\OD_{32}.D_6$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_4\times C_{12}:C_2^4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \)
Composition factors:	$C_2$ x 7, $C_3$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	3	4	6	8	12	16	24	48
Elements	1	19	2	44	14	64	16	128	32	64	384
Conjugacy classes	1	4	1	9	3	14	4	28	8	12	84
Divisions	1	4	1	6	3	6	3	5	3	2	34
Autjugacy classes	1	4	1	5	3	6	3	5	3	2	33

Dimension	1	2	4	8	16	32
Irr. complex chars.	32	40	12	0	0	0	84
Irr. rational chars.	8	10	9	4	2	1	34

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$96$
Rank:	$3$
Inequivalent generating triples:	$21504$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	32
Arbitrary	4	6	18

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=c^{48}=[a,b]=1, b^{4}=c^{36}, c^{a}=b^{2}c^{19}, c^{b}=c^{17} \rangle$
Permutation group:	Degree $35$ $\langle(1,2,6,10,5,11,20,28,7,12,22,29,21,9,25,27)(3,13,15,18,16,24,30,19,17,26,31,32,14,8,23,4) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 13 & 17 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 0 & 9 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 33 & 0 \\ 0 & 33 \end{array}\right), \left(\begin{array}{rr} 33 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 17 \\ 17 & 18 \end{array}\right), \left(\begin{array}{rr} 12 & 17 \\ 17 & 12 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/34\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_8.D_{12})$ $\,\rtimes\,$ $C_2$	$(C_{24}.D_4)$ $\,\rtimes\,$ $C_2$	$C_3$ $\,\rtimes\,$ $(C_{16}.D_4)$	$(C_{48}:C_4)$ $\,\rtimes\,$ $C_2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$\OD_{32}$ . $D_6$	$(Q_8:S_3)$ . $C_8$	$Q_8$ . $(S_3\times C_8)$	$D_4$ . $(S_3\times C_8)$	all 51

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

Homology

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

Subgroups

There are 334 subgroups in 146 conjugacy classes, 63 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_8$	$G/Z \simeq$ $S_3\times D_4$
Commutator:	$G' \simeq$ $C_{12}$	$G/G' \simeq$ $C_2^2\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_8$	$G/\Phi \simeq$ $C_2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $\OD_{32}:C_6$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $\OD_{32}.D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_8\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{16}.D_4$
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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 384: $\OD_{32}.D_6$

Order 192: $C_{24}.D_4$, $C_{24}.D_4$, $C_{24}.C_2^3$, $C_8.D_{12}$, $C_{48}:C_4$, $\OD_{32}:S_3$, $\OD_{32}:C_6$

Order 128: $C_{16}.D_4$

Order 96: $C_3:\OD_{32}$ x 2, $C_6:C_{16}$ x 2, $C_3\times \OD_{32}$, $C_2\times C_{48}$, $C_{48}:C_2$, $C_{12}.D_4$, $S_3\times C_{16}$, $D_{12}:C_4$, $C_{12}.D_4$, $C_{24}:C_4$, $\OD_{16}:C_6$, $C_{12}.D_4$, $C_8.D_6$

Order 64: $\OD_{32}:C_2$ x 2, $\OD_{32}:C_2$ x 2, $C_8.C_8$, $C_4\times C_{16}$, $D_8:C_4$

Order 48: $C_3:C_{16}$ x 3, $C_3\times \OD_{16}$ x 2, $C_2\times C_{24}$ x 2, $C_{48}$ x 2, $C_6:C_8$, $C_{24}:C_2$, $D_4:C_6$, $S_3\times C_8$, $D_{12}:C_2$, $C_3:Q_{16}$, $Q_8:S_3$, $C_3:\SD_{16}$, $C_3:D_8$, $C_{12}:C_4$, $C_3:\OD_{16}$

Order 32: $\OD_{32}$ x 4, $C_2\times C_{16}$ x 4, $\OD_{16}:C_2$ x 2, $C_4\wr C_2$ x 2, $D_8:C_2$, $C_4\times C_8$, $C_8.C_4$

Order 24: $C_{24}$ x 3, $C_3:C_8$ x 3, $C_2\times C_{12}$ x 2, $C_3\times D_4$ x 2, $C_3:D_4$, $C_6:C_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $C_3\times Q_8$

Order 16: $C_{16}$ x 5, $\OD_{16}$ x 4, $C_2\times C_8$ x 4, $\SD_{16}$ x 2, $D_4:C_2$ x 2, $Q_{16}$, $D_8$, $C_4^2$

Order 12: $C_{12}$ x 3, $C_3:C_4$ x 3, $C_2\times C_6$ x 2, $D_6$

Order 8: $C_8$ x 6, $D_4$ x 4, $C_2\times C_4$ x 4, $Q_8$ x 2

Order 6: $C_6$ x 3, $S_3$

Order 4: $C_4$ x 6, $C_2^2$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $\OD_{32}.D_6$

Order 192: $C_{24}.D_4$, $C_{24}.D_4$, $C_{24}.C_2^3$, $C_8.D_{12}$, $C_{48}:C_4$, $\OD_{32}:S_3$, $\OD_{32}:C_6$

Order 128: $C_{16}.D_4$

Order 96: $C_3:\OD_{32}$ x 2, $C_6:C_{16}$ x 2, $C_3\times \OD_{32}$, $C_2\times C_{48}$, $C_{48}:C_2$, $C_{12}.D_4$, $S_3\times C_{16}$, $D_{12}:C_4$, $C_{12}.D_4$, $C_{24}:C_4$, $\OD_{16}:C_6$, $C_{12}.D_4$, $C_8.D_6$

Order 64: $\OD_{32}:C_2$ x 2, $\OD_{32}:C_2$ x 2, $C_8.C_8$, $C_4\times C_{16}$, $D_8:C_4$

Order 48: $C_3:C_{16}$ x 3, $C_3\times \OD_{16}$ x 2, $C_2\times C_{24}$ x 2, $C_{48}$ x 2, $C_6:C_8$, $C_{24}:C_2$, $D_4:C_6$, $S_3\times C_8$, $D_{12}:C_2$, $C_3:Q_{16}$, $Q_8:S_3$, $C_3:\SD_{16}$, $C_3:D_8$, $C_{12}:C_4$, $C_3:\OD_{16}$

Order 32: $\OD_{32}$ x 4, $C_2\times C_{16}$ x 4, $\OD_{16}:C_2$ x 2, $C_4\wr C_2$ x 2, $D_8:C_2$, $C_4\times C_8$, $C_8.C_4$

Order 24: $C_{24}$ x 3, $C_3:C_8$ x 3, $C_2\times C_{12}$ x 2, $C_3\times D_4$ x 2, $C_3:D_4$, $C_6:C_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $C_3\times Q_8$

Order 16: $C_{16}$ x 5, $\OD_{16}$ x 4, $C_2\times C_8$ x 4, $\SD_{16}$ x 2, $D_4:C_2$ x 2, $Q_{16}$, $D_8$, $C_4^2$

Order 12: $C_{12}$ x 3, $C_2\times C_6$ x 2, $C_3:C_4$ x 2, $D_6$

Order 8: $C_8$ x 6, $D_4$ x 4, $C_2\times C_4$ x 4, $Q_8$ x 2

Order 6: $C_6$ x 3, $S_3$

Order 4: $C_4$ x 5, $C_2^2$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $\OD_{32}.D_6$ ($C_1$)

Order 192: $C_{24}.D_4$ ($C_2$), $C_{24}.D_4$ ($C_2$), $C_{24}.C_2^3$ ($C_2$), $C_8.D_{12}$ ($C_2$), $C_{48}:C_4$ ($C_2$), $\OD_{32}:S_3$ ($C_2$), $\OD_{32}:C_6$ ($C_2$)

Order 96: $C_3\times \OD_{32}$ ($C_2^2$), $C_2\times C_{48}$ ($C_2^2$), $C_{12}.D_4$ ($C_4$), $D_{12}:C_4$ ($C_4$), $C_{12}.D_4$ ($C_4$), $C_{24}:C_4$ ($C_2^2$), $C_3:\OD_{32}$ ($C_2^2$), $C_6:C_{16}$ ($C_2^2$), $\OD_{16}:C_6$ ($C_2^2$), $C_{12}.D_4$ ($C_4$), $C_8.D_6$ ($C_2^2$)

Order 64: $\OD_{32}:C_2$ ($S_3$)

Order 48: $C_3:C_{16}$ ($D_4$) x 2, $C_6:C_8$ ($C_2\times C_4$), $D_4:C_6$ ($C_2\times C_4$), $D_{12}:C_2$ ($C_2\times C_4$), $C_3\times \OD_{16}$ ($C_2\times C_4$), $C_2\times C_{24}$ ($C_2^3$), $C_3:Q_{16}$ ($C_8$), $Q_8:S_3$ ($C_8$), $C_3:\SD_{16}$ ($C_8$), $C_3:D_8$ ($C_8$), $C_{12}:C_4$ ($C_2\times C_4$), $C_3:\OD_{16}$ ($C_2\times C_4$)

Order 32: $\OD_{16}:C_2$ ($D_6$), $\OD_{32}$ ($D_6$), $C_2\times C_{16}$ ($D_6$)

Order 24: $C_3:C_8$ ($C_2\times C_8$) x 2, $C_2\times C_{12}$ ($C_2^2\times C_4$), $D_{12}$ ($C_2\times C_8$), $C_3:Q_8$ ($C_2\times C_8$), $C_{24}$ ($D_4:C_2$), $C_{24}$ ($C_2\times D_4$), $C_3\times Q_8$ ($C_2\times C_8$), $C_3\times D_4$ ($C_2\times C_8$)

Order 16: $\OD_{16}$ ($C_4\times S_3$), $C_2\times C_8$ ($C_2\times D_6$), $D_4:C_2$ ($C_4\times S_3$)

Order 12: $C_2\times C_6$ ($\OD_{16}:C_2$), $C_{12}$ ($C_2^2\times C_8$), $C_{12}$ ($C_4\times D_4$)

Order 8: $Q_8$ ($S_3\times C_8$), $D_4$ ($S_3\times C_8$), $C_2\times C_4$ ($C_4\times D_6$), $C_8$ ($D_4:S_3$), $C_8$ ($S_3\times D_4$)

Order 6: $C_6$ ($C_8\times D_4$)

Order 4: $C_2^2$ ($C_8.D_6$), $C_4$ ($C_2^3.D_6$), $C_4$ ($C_8\times D_6$)

Order 3: $C_3$ ($C_{16}.D_4$)

Order 2: $C_2$ ($C_3:(C_8\times D_4)$)

Order 1: $C_1$ ($\OD_{32}.D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $\OD_{32}.D_6$ ($C_1$)

Order 192: $C_{24}.D_4$ ($C_2$), $C_{24}.D_4$ ($C_2$), $C_{24}.C_2^3$ ($C_2$), $C_8.D_{12}$ ($C_2$), $C_{48}:C_4$ ($C_2$), $\OD_{32}:S_3$ ($C_2$), $\OD_{32}:C_6$ ($C_2$)

Order 96: $C_3\times \OD_{32}$ ($C_2^2$), $C_2\times C_{48}$ ($C_2^2$), $C_{12}.D_4$ ($C_4$), $D_{12}:C_4$ ($C_4$), $C_{12}.D_4$ ($C_4$), $C_{24}:C_4$ ($C_2^2$), $C_3:\OD_{32}$ ($C_2^2$), $C_6:C_{16}$ ($C_2^2$), $\OD_{16}:C_6$ ($C_2^2$), $C_{12}.D_4$ ($C_4$), $C_8.D_6$ ($C_2^2$)

Order 64: $\OD_{32}:C_2$ ($S_3$)

Order 48: $C_3:C_{16}$ ($D_4$) x 2, $C_6:C_8$ ($C_2\times C_4$), $D_4:C_6$ ($C_2\times C_4$), $D_{12}:C_2$ ($C_2\times C_4$), $C_3\times \OD_{16}$ ($C_2\times C_4$), $C_2\times C_{24}$ ($C_2^3$), $C_3:Q_{16}$ ($C_8$), $Q_8:S_3$ ($C_8$), $C_3:\SD_{16}$ ($C_8$), $C_3:D_8$ ($C_8$), $C_{12}:C_4$ ($C_2\times C_4$), $C_3:\OD_{16}$ ($C_2\times C_4$)

Order 32: $\OD_{16}:C_2$ ($D_6$), $\OD_{32}$ ($D_6$), $C_2\times C_{16}$ ($D_6$)

Order 24: $C_3:C_8$ ($C_2\times C_8$) x 2, $C_2\times C_{12}$ ($C_2^2\times C_4$), $D_{12}$ ($C_2\times C_8$), $C_3:Q_8$ ($C_2\times C_8$), $C_{24}$ ($D_4:C_2$), $C_{24}$ ($C_2\times D_4$), $C_3\times Q_8$ ($C_2\times C_8$), $C_3\times D_4$ ($C_2\times C_8$)

Order 16: $\OD_{16}$ ($C_4\times S_3$), $C_2\times C_8$ ($C_2\times D_6$), $D_4:C_2$ ($C_4\times S_3$)

Order 12: $C_2\times C_6$ ($\OD_{16}:C_2$), $C_{12}$ ($C_2^2\times C_8$), $C_{12}$ ($C_4\times D_4$)

Order 8: $Q_8$ ($S_3\times C_8$), $D_4$ ($S_3\times C_8$), $C_2\times C_4$ ($C_4\times D_6$), $C_8$ ($D_4:S_3$), $C_8$ ($S_3\times D_4$)

Order 6: $C_6$ ($C_8\times D_4$)

Order 4: $C_2^2$ ($C_8.D_6$), $C_4$ ($C_2^3.D_6$), $C_4$ ($C_8\times D_6$)

Order 3: $C_3$ ($C_{16}.D_4$)

Order 2: $C_2$ ($C_3:(C_8\times D_4)$)

Order 1: $C_1$ ($\OD_{32}.D_6$)

Series

Derived series

$\OD_{32}.D_6$

$\rhd$

$C_{12}$

$\rhd$

$C_1$

Chief series

$\OD_{32}.D_6$

$\rhd$

$C_{24}.C_2^3$

$\rhd$

$\OD_{16}:C_6$

$\rhd$

$C_2\times C_{24}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$\OD_{32}.D_6$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_8$

$\lhd$

$C_2\times C_8$

$\lhd$

$\OD_{32}:C_2$

Character theory

Complex character table

See the $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $34 \times 34$ rational character table.