Abstract group 9248.e: $C_{136}.D_{34}$

• Label: 9248.e
• Order: \( 2^{5} \cdot 17^{2} \)
• Exponent: \( 2^{3} \cdot 17 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 17 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 17 \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \)
• Perm deg.: $50$
• Trans deg.: $272$
• Rank: $3$

Group information

Description:	$C_{136}.D_{34}$
Order:	\(9248\)\(\medspace = 2^{5} \cdot 17^{2} \)
Exponent:	\(136\)\(\medspace = 2^{3} \cdot 17 \)
Automorphism group:	$C_{17}:((C_2\times C_8^2).C_2^6)$, of order \(139264\)\(\medspace = 2^{13} \cdot 17 \)
Composition factors:	$C_2$ x 5, $C_{17}$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	4	8	17	34	68	136
Elements	1	71	72	144	288	1952	2240	4480	9248
Conjugacy classes	1	4	5	10	152	472	624	1248	2516
Divisions	1	4	4	4	10	30	22	22	97
Autjugacy classes	1	3	3	3	3	7	7	7	34

Minimal presentations

Permutation degree:	$50$
Transitive degree:	$272$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{136}=c^{34}=[a,b]=[b,c]=1, c^{a}=b^{52}c^{33} \rangle$
Permutation group:	Degree $50$ $\langle(1,2,3,5,7,9,11,13,15,17,16,14,12,10,8,6,4)(18,19,22,27)(20,23,28,32)(21,26,29,24) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 61 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{137})$
Direct product:	not computed
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not computed
Possibly split product:	$D_{68}$ . $C_{68}$	$C_{136}$ . $D_{34}$ (2)	$(C_{17}:Q_8)$ . $C_{68}$	$(D_{68}:C_{34})$ . $C_2$	all 53

Elements of the group are displayed as matrices in $\GL_{2}(\F_{137})$.

Homology

Abelianization:	$C_{2}^{2} \times C_{68} \simeq C_{2}^{2} \times C_{4} \times C_{17}$
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 1048 subgroups in 212 conjugacy classes, 74 normal (46 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{136}$	$G/Z \simeq$ $D_{34}$
Commutator:	$G' \simeq$ $C_{34}$	$G/G' \simeq$ $C_2^2\times C_{68}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_{34}\times D_{34}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{34}\times C_{136}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{136}.D_{34}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{17}\times C_{34}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{16}:C_2$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

Order 9248: $C_{136}.D_{34}$

Order 4624: $C_{136}:C_{34}$ x 2, $D_{17}\times C_{136}$ x 2, $C_{68}.C_{68}$, $D_{68}:C_{34}$, $C_{34}\times C_{136}$

Order 2312: $D_{17}\times C_{68}$ x 2, $C_{17}:C_{136}$ x 2, $C_{17}\times C_{136}$ x 2, $C_{34}\wr C_2$ x 2, $C_{17}\times D_{68}$, $C_{34}\times C_{68}$, $C_{17}^2:Q_8$

Order 1156: $C_{17}\times C_{68}$ x 2, $C_{17}:C_{68}$ x 2, $C_{17}\times D_{34}$ x 2, $C_{34}^2$

Order 578: $C_{17}\times C_{34}$ x 2, $C_{17}\times D_{17}$ x 2

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

Order 289: $C_{17}^2$

Order 272: $C_2\times C_{136}$ x 12, $\OD_{16}\times C_{17}$ x 3, $C_{136}:C_2$ x 2, $C_8\times D_{17}$ x 2, $D_4:C_{34}$, $D_{68}:C_2$, $C_{17}:\OD_{16}$

Order 136: $C_{136}$ x 22, $C_2\times C_{68}$ x 12, $D_4\times C_{17}$ x 3, $C_{17}:D_4$ x 2, $C_4\times D_{17}$ x 2, $C_{17}:C_8$ x 2, $D_{68}$, $C_{17}:Q_8$, $Q_8\times C_{17}$

Order 68: $C_{68}$ x 22, $C_2\times C_{34}$ x 12, $D_{34}$ x 2, $C_{17}:C_4$ x 2

Order 34: $C_{34}$ x 30, $D_{17}$ x 2

Order 32: $\OD_{16}:C_2$

Order 17: $C_{17}$ x 10

Order 16: $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $D_4:C_2$

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

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

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 9248: $C_{136}.D_{34}$

Order 4624: $C_{136}:C_{34}$

Order 289: $C_{17}^2$

Order 32: $\OD_{16}:C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 9248: $C_{136}.D_{34}$ ($C_1$)

Order 4624: $C_{136}:C_{34}$ ($C_2$) x 2, $D_{17}\times C_{136}$ ($C_2$) x 2, $C_{68}.C_{68}$ ($C_2$), $D_{68}:C_{34}$ ($C_2$), $C_{34}\times C_{136}$ ($C_2$)

Order 2312: $D_{17}\times C_{68}$ ($C_2^2$) x 2, $C_{17}:C_{136}$ ($C_2^2$) x 2, $C_{17}\times C_{136}$ ($C_2^2$) x 2, $C_{34}\wr C_2$ ($C_4$) x 2, $C_{17}\times D_{68}$ ($C_4$), $C_{34}\times C_{68}$ ($C_2^2$), $C_{17}^2:Q_8$ ($C_4$)

Order 1156: $C_{17}:C_{68}$ ($C_2\times C_4$) x 2, $C_{17}\times D_{34}$ ($C_2\times C_4$) x 2, $C_{17}\times C_{68}$ ($C_2^3$), $C_{17}\times C_{68}$ ($C_2\times C_4$), $C_{34}^2$ ($C_2\times C_4$)

Order 578: $C_{17}\times C_{34}$ ($C_2^2\times C_4$)

Order 544: $C_8.D_{34}$ ($C_{17}$)

Order 289: $C_{17}^2$ ($\OD_{16}:C_2$)

Order 272: $C_{136}:C_2$ ($C_{34}$) x 2, $C_8\times D_{17}$ ($C_{34}$) x 2, $D_{68}:C_2$ ($C_{34}$), $C_2\times C_{136}$ ($C_{34}$), $C_2\times C_{136}$ ($D_{17}$), $C_{17}:\OD_{16}$ ($C_{34}$)

Order 136: $C_{17}:D_4$ ($C_{68}$) x 2, $C_4\times D_{17}$ ($C_2\times C_{34}$) x 2, $C_{136}$ ($C_2\times C_{34}$) x 2, $C_{136}$ ($D_{34}$) x 2, $C_{17}:C_8$ ($C_2\times C_{34}$) x 2, $C_2\times C_{68}$ ($C_2\times C_{34}$), $C_2\times C_{68}$ ($D_{34}$), $D_{68}$ ($C_{68}$), $C_{17}:Q_8$ ($C_{68}$)

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

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

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

Order 16: $C_2\times C_8$ ($C_{17}\times D_{17}$)

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

Order 4: $C_2^2$ ($D_{17}\times C_{68}$), $C_4$ ($C_{34}\times D_{34}$), $C_4$ ($D_{17}\times C_{68}$)

Order 2: $C_2$ ($C_{68}\times D_{34}$)

Order 1: $C_1$ ($C_{136}.D_{34}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 9248: $C_{136}.D_{34}$ ($C_1$)

Order 4624: $C_{136}:C_{34}$ ($C_2$)

Order 289: $C_{17}^2$ ($\OD_{16}:C_2$)

Order 1: $C_1$ ($C_{136}.D_{34}$)

Series

Derived series

$C_{136}.D_{34}$

$\rhd$

$C_{34}$

$\rhd$

$C_1$

Chief series

$C_{136}.D_{34}$

$\rhd$

$C_{34}\times C_{136}$

$\rhd$

$C_{34}\times C_{68}$

$\rhd$

$C_{17}\times C_{68}$

$\rhd$

$C_{17}\times C_{34}$

$\rhd$

$C_{34}$

$\rhd$

$C_{17}$

$\rhd$

$C_1$

Lower central series

$C_{136}.D_{34}$

$\rhd$

$C_{34}$

$\rhd$

$C_{17}$

Upper central series

$C_1$

$\lhd$

$C_{136}$

$\lhd$

$C_2\times C_{136}$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

The $2516 \times 2516$ character table is not available for this group.

Rational character table

The $97 \times 97$ rational character table is not available for this group.