Abstract group 1088.334: $C_8.D_{68}$

• Label: 1088.334
• Order: \( 2^{6} \cdot 17 \)
• Exponent: \( 2^{4} \cdot 17 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $49$
• Trans deg.: $544$
• Rank: $2$

Group information

Description:	$C_8.D_{68}$
Order:	\(1088\)\(\medspace = 2^{6} \cdot 17 \)
Exponent:	\(272\)\(\medspace = 2^{4} \cdot 17 \)
Automorphism group:	$C_{17}:((C_2^2\times C_8).C_2^5)$, of order \(17408\)\(\medspace = 2^{10} \cdot 17 \)
Composition factors:	$C_2$ x 6, $C_{17}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	16	17	34	68	136	272
Elements	1	71	72	144	288	16	48	64	128	256	1088
Conjugacy classes	1	3	4	8	12	8	16	24	48	64	188
Divisions	1	3	3	3	2	1	2	2	2	1	20
Autjugacy classes	1	3	3	3	2	1	2	2	2	1	20

Dimension	1	2	4	16	32	64	128
Irr. complex chars.	16	140	32	0	0	0	0	188
Irr. rational chars.	4	4	3	3	3	2	1	20

Minimal presentations

Permutation degree:	$49$
Transitive degree:	$544$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	128
Arbitrary	4	6	32

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{16}=c^{34}=1, b^{a}=b^{5}c^{17}, c^{a}=b^{8}c^{33}, c^{b}=c^{33} \rangle$
Permutation group:	Degree $49$ $\langle(1,2,5,10,6,11,20,28,7,12,21,29,4,9,18,26)(3,13,15,30,14,24,31,27,16,25,19,32,17,8,22,23) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$\OD_{32}$ $\,\rtimes\,$ $D_{17}$	$(C_{34}:C_{16})$ $\,\rtimes\,$ $C_2$	$(C_{17}\times \OD_{32})$ $\,\rtimes\,$ $C_2$	$C_{17}$ $\,\rtimes\,$ $(\OD_{32}:C_2)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{68}$ . $C_8$	$C_8$ . $D_{68}$	$C_{136}$ . $D_4$ (2)	$(C_{17}:Q_8)$ . $C_8$	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:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 514 subgroups in 56 conjugacy classes, 27 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_8$	$G/Z \simeq$ $C_{17}:D_4$
Commutator:	$G' \simeq$ $C_{68}$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_8$	$G/\Phi \simeq$ $D_{34}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{17}\times \OD_{32}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_8.D_{68}$	$G/R \simeq$ $C_1$
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: $C_8.D_{68}$

Order 544: $C_{17}\times \OD_{32}$, $C_{34}:C_{16}$, $C_8.D_{34}$

Order 272: $C_{136}:C_2$, $C_8\times D_{17}$, $D_{68}:C_2$, $C_2\times C_{136}$, $C_{272}$, $C_{17}:\OD_{16}$, $C_{17}:C_{16}$

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

Order 68: $C_{68}$ x 2, $C_2\times C_{34}$, $D_{34}$, $C_{17}:C_4$

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

Order 34: $C_{34}$ x 2, $D_{17}$

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: $C_8.D_{68}$

Order 544: $C_{17}\times \OD_{32}$, $C_{34}:C_{16}$, $C_8.D_{34}$

Order 272: $C_{136}:C_2$, $C_8\times D_{17}$, $D_{68}:C_2$, $C_2\times C_{136}$, $C_{272}$, $C_{17}:\OD_{16}$, $C_{17}:C_{16}$

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

Order 68: $C_{68}$ x 2, $C_2\times C_{34}$, $D_{34}$, $C_{17}:C_4$

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

Order 34: $C_{34}$ x 2, $D_{17}$

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: $C_8.D_{68}$ ($C_1$)

Order 544: $C_{17}\times \OD_{32}$ ($C_2$), $C_{34}:C_{16}$ ($C_2$), $C_8.D_{34}$ ($C_2$)

Order 272: $D_{68}:C_2$ ($C_4$), $C_2\times C_{136}$ ($C_2^2$), $C_{17}:\OD_{16}$ ($C_4$)

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

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

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

Order 32: $\OD_{32}$ ($D_{17}$)

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

Order 16: $C_2\times C_8$ ($D_{34}$)

Order 8: $C_2\times C_4$ ($C_4\times D_{17}$), $C_8$ ($C_{17}:D_4$), $C_8$ ($D_{68}$)

Order 4: $C_2^2$ ($C_{136}:C_2$), $C_4$ ($C_8\times D_{17}$), $C_4$ ($D_{34}:C_4$)

Order 2: $C_2$ ($D_{34}:C_8$)

Order 1: $C_1$ ($C_8.D_{68}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1088: $C_8.D_{68}$ ($C_1$)

Order 544: $C_{17}\times \OD_{32}$ ($C_2$), $C_{34}:C_{16}$ ($C_2$), $C_8.D_{34}$ ($C_2$)

Order 272: $D_{68}:C_2$ ($C_4$), $C_2\times C_{136}$ ($C_2^2$), $C_{17}:\OD_{16}$ ($C_4$)

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

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

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

Order 32: $\OD_{32}$ ($D_{17}$)

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

Order 16: $C_2\times C_8$ ($D_{34}$)

Order 8: $C_2\times C_4$ ($C_4\times D_{17}$), $C_8$ ($C_{17}:D_4$), $C_8$ ($D_{68}$)

Order 4: $C_2^2$ ($C_{136}:C_2$), $C_4$ ($C_8\times D_{17}$), $C_4$ ($D_{34}:C_4$)

Order 2: $C_2$ ($D_{34}:C_8$)

Order 1: $C_1$ ($C_8.D_{68}$)

Series

Derived series

$C_8.D_{68}$

$\rhd$

$C_{68}$

$\rhd$

$C_1$

Chief series

$C_8.D_{68}$

$\rhd$

$C_8.D_{34}$

$\rhd$

$C_2\times C_{136}$

$\rhd$

$C_{136}$

$\rhd$

$C_{68}$

$\rhd$

$C_{34}$

$\rhd$

$C_{17}$

$\rhd$

$C_1$

Lower central series

$C_8.D_{68}$

$\rhd$

$C_{68}$

$\rhd$

$C_{34}$

$\rhd$

$C_{17}$

Upper central series

$C_1$

$\lhd$

$C_8$

$\lhd$

$C_2\times C_8$

$\lhd$

$\OD_{32}$

Character theory

Complex character table

See the $188 \times 188$ 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 $20 \times 20$ rational character table.