Abstract group 1088.154: $C_{136}.D_4$

• Label: 1088.154
• Order: \( 2^{6} \cdot 17 \)
• Exponent: \( 2^{3} \cdot 17 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 17 \)
• $\card{Z(G)}$: \( 2 \cdot 17 \)
• $\card{\Aut(G)}$: \( 2^{12} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $49$
• Trans deg.: $544$
• Rank: $3$

Group information

Description:	$C_{136}.D_4$
Order:	\(1088\)\(\medspace = 2^{6} \cdot 17 \)
Exponent:	\(136\)\(\medspace = 2^{3} \cdot 17 \)
Automorphism group:	$C_{16}\times C_4.C_2^5.C_2$, of order \(4096\)\(\medspace = 2^{12} \)
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	17	34	68	136
Elements	1	7	24	32	16	112	384	512	1088
Conjugacy classes	1	3	5	7	16	48	80	112	272
Divisions	1	3	5	5	1	3	5	5	28
Autjugacy classes	1	3	4	4	1	3	4	4	24

Dimension	1	2	4	8	16	32	64	128
Irr. complex chars.	136	102	34	0	0	0	0	0	272
Irr. rational chars.	8	4	1	1	8	4	1	1	28

Minimal presentations

Permutation degree:	$49$
Transitive degree:	$544$
Rank:	$3$
Inequivalent generating triples:	$103152$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=c^{136}=[a,c]=1, b^{4}=c^{68}, b^{a}=b^{3}, c^{b}=c^{103} \rangle$
Permutation group:	Degree $49$ $\langle(1,18,2,17)(3,20,4,19)(5,22,6,21)(7,24,8,23)(9,26,10,25)(11,28,12,27)(13,30,14,29) \!\cdots\! \rangle$
Direct product:	$C_{17}$ $\, \times\, $ $(C_8.D_4)$
Semidirect product:	$(C_{68}.D_4)$ $\,\rtimes\,$ $C_2$ (2)	$(C_4.D_4)$ $\,\rtimes\,$ $C_{34}$ (2)	$(C_8.C_{68})$ $\,\rtimes\,$ $C_2$	$(C_8.C_4)$ $\,\rtimes\,$ $C_{34}$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{136}$ . $D_4$ (2)	$(Q_8\times C_{17})$ . $D_4$	$Q_8$ . $(D_4\times C_{17})$	$(D_4\times C_{17})$ . $D_4$	all 26

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

Homology

Abelianization:	$C_{2}^{2} \times C_{34} \simeq C_{2}^{3} \times C_{17}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 162 subgroups in 100 conjugacy classes, 50 normal (34 characteristic).

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

Special subgroups

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

Classes of subgroups up to conjugation

Order 1088: $C_{136}.D_4$

Order 544: $C_{68}.D_4$ x 2, $Q_{16}:C_{34}$ x 2, $C_8.C_{68}$, $Q_{16}\times C_{34}$, $\OD_{16}:C_{34}$

Order 272: $Q_{16}\times C_{17}$ x 4, $\OD_{16}\times C_{17}$ x 4, $Q_8\times C_{34}$ x 2, $\SD_{16}\times C_{17}$ x 2, $C_2\times C_{136}$ x 2, $D_4:C_{34}$

Order 136: $C_{136}$ x 5, $Q_8\times C_{17}$ x 5, $C_2\times C_{68}$ x 4, $D_4\times C_{17}$ x 2

Order 68: $C_{68}$ x 5, $C_2\times C_{34}$ x 2

Order 64: $C_8.D_4$

Order 34: $C_{34}$ x 3

Order 32: $C_4.D_4$ x 2, $Q_{16}:C_2$ x 2, $C_2\times Q_{16}$, $\OD_{16}:C_2$, $C_8.C_4$

Order 17: $C_{17}$

Order 16: $Q_{16}$ x 4, $\OD_{16}$ x 4, $\SD_{16}$ x 2, $C_2\times C_8$ x 2, $C_2\times Q_8$ x 2, $D_4:C_2$

Order 8: $Q_8$ x 5, $C_8$ x 5, $C_2\times C_4$ x 4, $D_4$ x 2

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1088: $C_{136}.D_4$

Order 544: $C_8.C_{68}$, $C_{68}.D_4$, $Q_{16}:C_{34}$, $Q_{16}\times C_{34}$, $\OD_{16}:C_{34}$

Order 272: $Q_{16}\times C_{17}$ x 3, $\OD_{16}\times C_{17}$ x 3, $C_2\times C_{136}$ x 2, $D_4:C_{34}$, $Q_8\times C_{34}$, $\SD_{16}\times C_{17}$

Order 136: $C_{136}$ x 4, $C_2\times C_{68}$ x 3, $Q_8\times C_{17}$ x 3, $D_4\times C_{17}$ x 2

Order 68: $C_{68}$ x 4, $C_2\times C_{34}$ x 2

Order 64: $C_8.D_4$

Order 34: $C_{34}$ x 3

Order 32: $C_4.D_4$, $Q_{16}:C_2$, $C_2\times Q_{16}$, $\OD_{16}:C_2$, $C_8.C_4$

Order 17: $C_{17}$

Order 16: $Q_{16}$ x 3, $\OD_{16}$ x 3, $C_2\times C_8$ x 2, $\SD_{16}$, $D_4:C_2$, $C_2\times Q_8$

Order 8: $C_8$ x 4, $Q_8$ x 3, $C_2\times C_4$ x 3, $D_4$ x 2

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1088: $C_{136}.D_4$ ($C_1$)

Order 544: $C_{68}.D_4$ ($C_2$) x 2, $Q_{16}:C_{34}$ ($C_2$) x 2, $C_8.C_{68}$ ($C_2$), $Q_{16}\times C_{34}$ ($C_2$), $\OD_{16}:C_{34}$ ($C_2$)

Order 272: $\OD_{16}\times C_{17}$ ($C_2^2$) x 3, $Q_8\times C_{34}$ ($C_2^2$) x 2, $D_4:C_{34}$ ($C_2^2$), $C_2\times C_{136}$ ($C_2^2$)

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

Order 68: $C_{68}$ ($C_2\times D_4$) x 2, $C_2\times C_{34}$ ($D_4:C_2$)

Order 64: $C_8.D_4$ ($C_{17}$)

Order 34: $C_{34}$ ($C_4:D_4$)

Order 32: $C_4.D_4$ ($C_{34}$) x 2, $Q_{16}:C_2$ ($C_{34}$) x 2, $C_2\times Q_{16}$ ($C_{34}$), $\OD_{16}:C_2$ ($C_{34}$), $C_8.C_4$ ($C_{34}$)

Order 17: $C_{17}$ ($C_8.D_4$)

Order 16: $\OD_{16}$ ($C_2\times C_{34}$) x 3, $C_2\times Q_8$ ($C_2\times C_{34}$) x 2, $C_2\times C_8$ ($C_2\times C_{34}$), $D_4:C_2$ ($C_2\times C_{34}$)

Order 8: $C_8$ ($D_4\times C_{17}$) x 2, $Q_8$ ($D_4\times C_{17}$), $D_4$ ($D_4\times C_{17}$), $C_2\times C_4$ ($C_2^2\times C_{34}$)

Order 4: $C_4$ ($D_4\times C_{34}$) x 2, $C_2^2$ ($D_4:C_{34}$)

Order 2: $C_2$ ($C_{68}:D_4$)

Order 1: $C_1$ ($C_{136}.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1088: $C_{136}.D_4$ ($C_1$)

Order 544: $C_8.C_{68}$ ($C_2$), $C_{68}.D_4$ ($C_2$), $Q_{16}:C_{34}$ ($C_2$), $Q_{16}\times C_{34}$ ($C_2$), $\OD_{16}:C_{34}$ ($C_2$)

Order 272: $\OD_{16}\times C_{17}$ ($C_2^2$) x 2, $D_4:C_{34}$ ($C_2^2$), $Q_8\times C_{34}$ ($C_2^2$), $C_2\times C_{136}$ ($C_2^2$)

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

Order 68: $C_{68}$ ($C_2\times D_4$) x 2, $C_2\times C_{34}$ ($D_4:C_2$)

Order 64: $C_8.D_4$ ($C_{17}$)

Order 34: $C_{34}$ ($C_4:D_4$)

Order 32: $C_4.D_4$ ($C_{34}$), $Q_{16}:C_2$ ($C_{34}$), $C_2\times Q_{16}$ ($C_{34}$), $\OD_{16}:C_2$ ($C_{34}$), $C_8.C_4$ ($C_{34}$)

Order 17: $C_{17}$ ($C_8.D_4$)

Order 16: $\OD_{16}$ ($C_2\times C_{34}$) x 2, $C_2\times C_8$ ($C_2\times C_{34}$), $D_4:C_2$ ($C_2\times C_{34}$), $C_2\times Q_8$ ($C_2\times C_{34}$)

Order 8: $C_8$ ($D_4\times C_{17}$) x 2, $Q_8$ ($D_4\times C_{17}$), $D_4$ ($D_4\times C_{17}$), $C_2\times C_4$ ($C_2^2\times C_{34}$)

Order 4: $C_4$ ($D_4\times C_{34}$) x 2, $C_2^2$ ($D_4:C_{34}$)

Order 2: $C_2$ ($C_{68}:D_4$)

Order 1: $C_1$ ($C_{136}.D_4$)

Series

Derived series

$C_{136}.D_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_1$

Chief series

$C_{136}.D_4$

$\rhd$

$Q_{16}:C_{34}$

$\rhd$

$D_4:C_{34}$

$\rhd$

$C_2\times C_{68}$

$\rhd$

$C_{68}$

$\rhd$

$C_{34}$

$\rhd$

$C_{17}$

$\rhd$

$C_1$

Lower central series

$C_{136}.D_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{34}$

$\lhd$

$C_2\times C_{68}$

$\lhd$

$C_{136}.D_4$

Character theory

Complex character table

See the $272 \times 272$ 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 $28 \times 28$ rational character table.