Abstract group 1088.31: $\OD_{32}:C_{34}$

• Label: 1088.31
• Order: \( 2^{6} \cdot 17 \)
• Exponent: \( 2^{4} \cdot 17 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 17 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 17 \)
• $\card{\Aut(G)}$: \( 2^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $49$
• Trans deg.: $544$
• Rank: $2$

Group information

Description:	$\OD_{32}:C_{34}$
Order:	\(1088\)\(\medspace = 2^{6} \cdot 17 \)
Exponent:	\(272\)\(\medspace = 2^{4} \cdot 17 \)
Automorphism group:	$(C_2^2\times C_8).C_2^5$, of order \(1024\)\(\medspace = 2^{10} \)
Composition factors:	$C_2$ x 6, $C_{17}$
Nilpotency class:	$3$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	16	17	34	68	136	272
Elements	1	7	8	16	32	16	112	128	256	512	1088
Conjugacy classes	1	3	4	8	12	16	48	64	128	192	476
Divisions	1	3	3	3	2	1	3	3	3	2	24
Autjugacy classes	1	3	3	3	2	1	3	3	3	2	24

Dimension	1	2	4	16	32	64	256
Irr. complex chars.	272	204	0	0	0	0	0	476
Irr. rational chars.	4	4	3	5	4	3	1	24

Minimal presentations

Permutation degree:	$49$
Transitive degree:	$544$
Rank:	$2$
Inequivalent generating pairs:	$432$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	256
Arbitrary	2	4	32

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{16}=c^{34}=[b,c]=1, b^{a}=b^{13}c^{17}, c^{a}=b^{8}c \rangle$
Permutation group:	Degree $49$ $\langle(1,24,8,20,4,22,6,18,2,23,7,19,3,21,5,17)(9,31,15,28,12,29,13,26,10,32,16,27,11,30,14,25) \!\cdots\! \rangle$
Direct product:	$C_{17}$ $\, \times\, $ $(\OD_{32}:C_2)$
Semidirect product:	$\OD_{32}$ $\,\rtimes\,$ $C_{34}$	$(C_2\times C_{272})$ $\,\rtimes\,$ $C_2$	$(C_2\times C_{16})$ $\,\rtimes\,$ $C_{34}$	$(C_{17}\times \OD_{32})$ $\,\rtimes\,$ $C_2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$Q_8$ . $C_{136}$	$C_{136}$ . $D_4$ (2)	$D_4$ . $C_{136}$	$\OD_{16}$ . $C_{68}$	all 24

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

Homology

Abelianization:	$C_{2} \times C_{136} \simeq C_{2} \times C_{8} \times C_{17}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 82 subgroups in 56 conjugacy classes, 34 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{136}$	$G/Z \simeq$ $D_4$
Commutator:	$G' \simeq$ $C_4$	$G/G' \simeq$ $C_2\times C_{136}$
Frattini:	$\Phi \simeq$ $C_2\times C_8$	$G/\Phi \simeq$ $C_2\times C_{34}$
Fitting:	$\operatorname{Fit} \simeq$ $\OD_{32}:C_{34}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $\OD_{32}:C_{34}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{34}$	$G/\operatorname{soc} \simeq$ $C_2^2:C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{32}:C_2$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1088: $\OD_{32}:C_{34}$

Order 544: $C_{17}\times \OD_{32}$, $C_2\times C_{272}$, $\OD_{16}:C_{34}$

Order 272: $\OD_{16}\times C_{17}$ x 2, $C_2\times C_{136}$ x 2, $C_{272}$ x 2, $D_4:C_{34}$

Order 136: $C_{136}$ x 3, $C_2\times C_{68}$ x 2, $D_4\times C_{17}$ x 2, $Q_8\times C_{17}$

Order 68: $C_{68}$ x 3, $C_2\times C_{34}$ x 2

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

Order 34: $C_{34}$ x 3

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

Order 17: $C_{17}$

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

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

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1088: $\OD_{32}:C_{34}$

Order 544: $C_{17}\times \OD_{32}$, $C_2\times C_{272}$, $\OD_{16}:C_{34}$

Order 272: $\OD_{16}\times C_{17}$ x 2, $C_2\times C_{136}$ x 2, $C_{272}$ x 2, $D_4:C_{34}$

Order 136: $C_{136}$ x 3, $C_2\times C_{68}$ x 2, $D_4\times C_{17}$ x 2, $Q_8\times C_{17}$

Order 68: $C_{68}$ x 3, $C_2\times C_{34}$ x 2

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

Order 34: $C_{34}$ x 3

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

Order 17: $C_{17}$

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

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

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1088: $\OD_{32}:C_{34}$ ($C_1$)

Order 544: $C_{17}\times \OD_{32}$ ($C_2$), $C_2\times C_{272}$ ($C_2$), $\OD_{16}:C_{34}$ ($C_2$)

Order 272: $D_4:C_{34}$ ($C_4$), $\OD_{16}\times C_{17}$ ($C_4$), $C_2\times C_{136}$ ($C_2^2$)

Order 136: $C_{136}$ ($D_4$) x 2, $C_2\times C_{68}$ ($C_2\times C_4$), $Q_8\times C_{17}$ ($C_8$), $D_4\times C_{17}$ ($C_8$)

Order 68: $C_2\times C_{34}$ ($\OD_{16}$), $C_{68}$ ($C_2\times C_8$), $C_{68}$ ($C_2^2:C_4$)

Order 64: $\OD_{32}:C_2$ ($C_{17}$)

Order 34: $C_{34}$ ($C_2^2:C_8$)

Order 32: $\OD_{16}:C_2$ ($C_{34}$), $\OD_{32}$ ($C_{34}$), $C_2\times C_{16}$ ($C_{34}$)

Order 17: $C_{17}$ ($\OD_{32}:C_2$)

Order 16: $\OD_{16}$ ($C_{68}$), $C_2\times C_8$ ($C_2\times C_{34}$), $D_4:C_2$ ($C_{68}$)

Order 8: $C_8$ ($D_4\times C_{17}$) x 2, $Q_8$ ($C_{136}$), $D_4$ ($C_{136}$), $C_2\times C_4$ ($C_2\times C_{68}$)

Order 4: $C_2^2$ ($\OD_{16}\times C_{17}$), $C_4$ ($C_2\times C_{136}$), $C_4$ ($C_2^2:C_{68}$)

Order 2: $C_2$ ($C_2^2:C_{136}$)

Order 1: $C_1$ ($\OD_{32}:C_{34}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1088: $\OD_{32}:C_{34}$ ($C_1$)

Order 544: $C_{17}\times \OD_{32}$ ($C_2$), $C_2\times C_{272}$ ($C_2$), $\OD_{16}:C_{34}$ ($C_2$)

Order 272: $D_4:C_{34}$ ($C_4$), $\OD_{16}\times C_{17}$ ($C_4$), $C_2\times C_{136}$ ($C_2^2$)

Order 136: $C_{136}$ ($D_4$) x 2, $C_2\times C_{68}$ ($C_2\times C_4$), $Q_8\times C_{17}$ ($C_8$), $D_4\times C_{17}$ ($C_8$)

Order 68: $C_2\times C_{34}$ ($\OD_{16}$), $C_{68}$ ($C_2\times C_8$), $C_{68}$ ($C_2^2:C_4$)

Order 64: $\OD_{32}:C_2$ ($C_{17}$)

Order 34: $C_{34}$ ($C_2^2:C_8$)

Order 32: $\OD_{16}:C_2$ ($C_{34}$), $\OD_{32}$ ($C_{34}$), $C_2\times C_{16}$ ($C_{34}$)

Order 17: $C_{17}$ ($\OD_{32}:C_2$)

Order 16: $\OD_{16}$ ($C_{68}$), $C_2\times C_8$ ($C_2\times C_{34}$), $D_4:C_2$ ($C_{68}$)

Order 8: $C_8$ ($D_4\times C_{17}$) x 2, $Q_8$ ($C_{136}$), $D_4$ ($C_{136}$), $C_2\times C_4$ ($C_2\times C_{68}$)

Order 4: $C_2^2$ ($\OD_{16}\times C_{17}$), $C_4$ ($C_2\times C_{136}$), $C_4$ ($C_2^2:C_{68}$)

Order 2: $C_2$ ($C_2^2:C_{136}$)

Order 1: $C_1$ ($\OD_{32}:C_{34}$)

Series

Derived series

$\OD_{32}:C_{34}$

$\rhd$

$C_4$

$\rhd$

$C_1$

Chief series

$\OD_{32}:C_{34}$

$\rhd$

$\OD_{16}:C_{34}$

$\rhd$

$C_2\times C_{136}$

$\rhd$

$C_{136}$

$\rhd$

$C_{68}$

$\rhd$

$C_{34}$

$\rhd$

$C_{17}$

$\rhd$

$C_1$

Lower central series

$\OD_{32}:C_{34}$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{136}$

$\lhd$

$C_2\times C_{136}$

$\lhd$

$\OD_{32}:C_{34}$

Character theory

Complex character table

See the $476 \times 476$ 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 $24 \times 24$ rational character table.