Abstract group 1088.390: $C_{17}:Q_{64}$

• Label: 1088.390
• Order: \( 2^{6} \cdot 17 \)
• Exponent: \( 2^{5} \cdot 17 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $81$
• Trans deg.: $1088$
• Rank: $2$

Group information

Description:	$C_{17}:Q_{64}$
Order:	\(1088\)\(\medspace = 2^{6} \cdot 17 \)
Exponent:	\(544\)\(\medspace = 2^{5} \cdot 17 \)
Automorphism group:	$C_{17}:(C_4.C_4^3.C_2.C_2^4)$, of order \(139264\)\(\medspace = 2^{13} \cdot 17 \)
Composition factors:	$C_2$ x 6, $C_{17}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	16	17	32	34	68	136	272	544
Elements	1	1	546	4	8	16	16	16	32	64	128	256	1088
Conjugacy classes	1	1	3	2	4	8	8	8	16	32	64	128	275
Divisions	1	1	3	1	1	1	1	1	1	1	1	1	14
Autjugacy classes	1	1	2	1	1	1	1	1	1	1	1	1	13

Dimension	1	2	4	8	16	32	64	128	256
Irr. complex chars.	4	271	0	0	0	0	0	0	0	275
Irr. rational chars.	4	1	1	1	3	1	1	1	1	14

Minimal presentations

Permutation degree:	$81$
Transitive degree:	$1088$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	512
Arbitrary	2	4	48

Constructions

Presentation:	$\langle a, b \mid b^{544}=1, a^{2}=b^{272}, b^{a}=b^{543} \rangle$
Permutation group:	Degree $81$ $\langle(1,2,7,12)(3,13,17,44)(4,18,23,42)(5,24,27,58)(6,28,26,11)(8,30,33,14)(9,34,37,21) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{17}$ $\,\rtimes\,$ $Q_{64}$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{68}$ . $D_8$	$C_8$ . $D_{68}$	$C_{136}$ . $D_4$	$C_{34}$ . $D_{16}$	all 11

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 570 subgroups in 30 conjugacy classes, 15 normal (13 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{272}$
Commutator:	$G' \simeq$ $C_{272}$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $D_{34}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{544}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{17}:Q_{64}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{34}$	$G/\operatorname{soc} \simeq$ $D_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{64}$
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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Subgroup information Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

Order 1088: $C_{17}:Q_{64}$

Order 544: $C_{17}:Q_{32}$ x 2, $C_{544}$

Order 272: $C_{17}:Q_{16}$ x 2, $C_{272}$

Order 136: $C_{17}:Q_8$ x 2, $C_{136}$

Order 68: $C_{17}:C_4$ x 2, $C_{68}$

Order 64: $Q_{64}$

Order 34: $C_{34}$

Order 32: $Q_{32}$ x 2, $C_{32}$

Order 17: $C_{17}$

Order 16: $Q_{16}$ x 2, $C_{16}$

Order 8: $Q_8$ x 2, $C_8$

Order 4: $C_4$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1088: $C_{17}:Q_{64}$

Order 544: $C_{17}:Q_{32}$, $C_{544}$

Order 272: $C_{17}:Q_{16}$, $C_{272}$

Order 136: $C_{17}:Q_8$, $C_{136}$

Order 68: $C_{68}$, $C_{17}:C_4$

Order 64: $Q_{64}$

Order 34: $C_{34}$

Order 32: $Q_{32}$, $C_{32}$

Order 17: $C_{17}$

Order 16: $Q_{16}$, $C_{16}$

Order 8: $Q_8$, $C_8$

Order 4: $C_4$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1088: $C_{17}:Q_{64}$ ($C_1$)

Order 544: $C_{17}:Q_{32}$ ($C_2$) x 2, $C_{544}$ ($C_2$)

Order 272: $C_{272}$ ($C_2^2$)

Order 136: $C_{136}$ ($D_4$)

Order 68: $C_{68}$ ($D_8$)

Order 34: $C_{34}$ ($D_{16}$)

Order 32: $C_{32}$ ($D_{17}$)

Order 17: $C_{17}$ ($Q_{64}$)

Order 16: $C_{16}$ ($D_{34}$)

Order 8: $C_8$ ($D_{68}$)

Order 4: $C_4$ ($D_{136}$)

Order 2: $C_2$ ($D_{272}$)

Order 1: $C_1$ ($C_{17}:Q_{64}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1088: $C_{17}:Q_{64}$ ($C_1$)

Order 544: $C_{17}:Q_{32}$ ($C_2$), $C_{544}$ ($C_2$)

Order 272: $C_{272}$ ($C_2^2$)

Order 136: $C_{136}$ ($D_4$)

Order 68: $C_{68}$ ($D_8$)

Order 34: $C_{34}$ ($D_{16}$)

Order 32: $C_{32}$ ($D_{17}$)

Order 17: $C_{17}$ ($Q_{64}$)

Order 16: $C_{16}$ ($D_{34}$)

Order 8: $C_8$ ($D_{68}$)

Order 4: $C_4$ ($D_{136}$)

Order 2: $C_2$ ($D_{272}$)

Order 1: $C_1$ ($C_{17}:Q_{64}$)

Series

Derived series

$C_{17}:Q_{64}$

$\rhd$

$C_{272}$

$\rhd$

$C_1$

Chief series

$C_{17}:Q_{64}$

$\rhd$

$C_{17}:Q_{32}$

$\rhd$

$C_{272}$

$\rhd$

$C_{136}$

$\rhd$

$C_{68}$

$\rhd$

$C_{34}$

$\rhd$

$C_{17}$

$\rhd$

$C_1$

Lower central series

$C_{17}:Q_{64}$

$\rhd$

$C_{272}$

$\rhd$

$C_{136}$

$\rhd$

$C_{68}$

$\rhd$

$C_{34}$

$\rhd$

$C_{17}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$C_8$

$\lhd$

$C_{16}$

$\lhd$

$C_{32}$

Character theory

Complex character table

See the $275 \times 275$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	4A	4B	4C	8A	16A	17A	32A	34A	68A	136A	272A	544A
	Size	1	1	2	272	272	4	8	16	16	16	32	64	128	256
	2 P	1A	1A	2A	2A	2A	4A	8A	17A	16A	17A	34A	68A	136A	272A
	17 P	1A	2A	4A	4B	4C	8A	16A	1A	32A	2A	4A	8A	16A	32A
	Schur
1088.390.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1088.390.1b		$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1088.390.1c		$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$
1088.390.1d		$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$
1088.390.2a		$2$	$2$	$2$	$0$	$0$	$2$	$-2$	$2$	$0$	$2$	$2$	$2$	$-2$	$0$
1088.390.2b		$4$	$4$	$4$	$0$	$0$	$-4$	$0$	$4$	$0$	$4$	$4$	$-4$	$0$	$0$
1088.390.2c		$8$	$8$	$-8$	$0$	$0$	$0$	$0$	$8$	$0$	$8$	$-8$	$0$	$0$	$0$
1088.390.2d		$16$	$16$	$16$	$0$	$0$	$16$	$16$	$-1$	$16$	$-1$	$-1$	$-1$	$-1$	$-1$
1088.390.2e		$16$	$16$	$16$	$0$	$0$	$16$	$16$	$-1$	$-16$	$-1$	$-1$	$-1$	$-1$	$1$
1088.390.2f	2	$16$	$-16$	$0$	$0$	$0$	$0$	$0$	$16$	$0$	$-16$	$0$	$0$	$0$	$0$
1088.390.2g		$32$	$32$	$32$	$0$	$0$	$32$	$-32$	$-2$	$0$	$-2$	$-2$	$-2$	$2$	$0$
1088.390.2h		$64$	$64$	$64$	$0$	$0$	$-64$	$0$	$-4$	$0$	$-4$	$-4$	$4$	$0$	$0$
1088.390.2i		$128$	$128$	$-128$	$0$	$0$	$0$	$0$	$-8$	$0$	$-8$	$8$	$0$	$0$	$0$
1088.390.2j	2	$256$	$-256$	$0$	$0$	$0$	$0$	$0$	$-16$	$0$	$16$	$0$	$0$	$0$	$0$