Abstract group 544.193: $\OD_{16}:C_{34}$

• Label: 544.193
• Order: \( 2^{5} \cdot 17 \)
• Exponent: \( 2^{3} \cdot 17 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 17 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 17 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $33$
• Trans deg.: $272$
• Rank: $3$

Group information

Description:	$\OD_{16}:C_{34}$
Order:	\(544\)\(\medspace = 2^{5} \cdot 17 \)
Exponent:	\(136\)\(\medspace = 2^{3} \cdot 17 \)
Automorphism group:	$C_2^2\times C_{16}\times S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_{17}$
Nilpotency class:	$2$
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	8	16	16	112	128	256	544
Conjugacy classes	1	4	5	10	16	64	80	160	340
Divisions	1	4	4	4	1	4	4	4	26
Autjugacy classes	1	2	2	2	1	2	2	2	14

Dimension	1	2	8	16	32	128
Irr. complex chars.	272	68	0	0	0	0	340
Irr. rational chars.	8	4	1	8	4	1	26

Minimal presentations

Permutation degree:	$33$
Transitive degree:	$272$
Rank:	$3$
Inequivalent generating triples:	$34384$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	128
Arbitrary	2	4	24

Constructions

Presentation:	$\langle a, b, c \mid a^{34}=b^{2}=c^{8}=[a,c]=[b,c]=1, b^{a}=bc^{4} \rangle$
Permutation group:	Degree $33$ $\langle(1,12,4,10,2,11,3,9)(5,16,8,14,6,15,7,13), (1,7)(2,8)(3,6)(4,5)(9,15)(10,16) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 100 & 0 \\ 0 & 37 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{137})$
Direct product:	$C_{17}$ $\, \times\, $ $(\OD_{16}:C_2)$
Semidirect product:	$\OD_{16}$ $\,\rtimes\,$ $C_{34}$ (3)	$(C_2\times C_8)$ $\,\rtimes\,$ $C_{34}$ (3)	$(C_2\times C_{136})$ $\,\rtimes\,$ $C_2$ (3)	$(\OD_{16}\times C_{17})$ $\,\rtimes\,$ $C_2$ (3)	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$Q_8$ . $C_{68}$	$D_4$ . $C_{68}$ (3)	$C_{68}$ . $C_2^3$	$C_{136}$ . $C_2^2$ (4)	all 18

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:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 68 subgroups in 62 conjugacy classes, 56 normal (14 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 544: $\OD_{16}:C_{34}$

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

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

Order 68: $C_{68}$ x 4, $C_2\times C_{34}$ x 3

Order 34: $C_{34}$ x 4

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

Order 17: $C_{17}$

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

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

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

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

Order 34: $C_{34}$ x 2

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

Order 17: $C_{17}$

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

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

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 544: $\OD_{16}:C_{34}$ ($C_1$)

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

Order 136: $C_{136}$ ($C_2^2$) x 4, $C_2\times C_{68}$ ($C_2^2$) x 3, $D_4\times C_{17}$ ($C_4$) x 3, $Q_8\times C_{17}$ ($C_4$)

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

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

Order 32: $\OD_{16}:C_2$ ($C_{17}$)

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

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

Order 8: $C_8$ ($C_2\times C_{34}$) x 4, $D_4$ ($C_{68}$) x 3, $C_2\times C_4$ ($C_2\times C_{34}$) x 3, $Q_8$ ($C_{68}$)

Order 4: $C_2^2$ ($C_2\times C_{68}$) x 3, $C_4$ ($C_2\times C_{68}$) x 3, $C_4$ ($C_2^2\times C_{34}$)

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

Order 1: $C_1$ ($\OD_{16}:C_{34}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 544: $\OD_{16}:C_{34}$ ($C_1$)

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

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

Order 68: $C_2\times C_{34}$ ($C_2\times C_4$), $C_{68}$ ($C_2^3$), $C_{68}$ ($C_2\times C_4$)

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

Order 32: $\OD_{16}:C_2$ ($C_{17}$)

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

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

Order 8: $C_8$ ($C_2\times C_{34}$) x 2, $Q_8$ ($C_{68}$), $D_4$ ($C_{68}$), $C_2\times C_4$ ($C_2\times C_{34}$)

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

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

Order 1: $C_1$ ($\OD_{16}:C_{34}$)

Series

Derived series

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

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

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

$\rhd$

$C_2\times C_{136}$

$\rhd$

$C_2\times C_{68}$

$\rhd$

$C_{68}$

$\rhd$

$C_{34}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{136}$

$\lhd$

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

Supergroups

This group is a maximal subgroup of 30 larger groups in the database.

This group is a maximal quotient of 15 larger groups in the database.

Character theory

Complex character table

See the $340 \times 340$ 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 $26 \times 26$ rational character table.