Abstract group 136.6: $D_{68}$

• Label: 136.6
• Order: \( 2^{3} \cdot 17 \)
• Exponent: \( 2^{2} \cdot 17 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $21$
• Trans deg.: $68$
• Rank: $2$

Group information

Description:	$D_{68}$
Order:	\(136\)\(\medspace = 2^{3} \cdot 17 \)
Exponent:	\(68\)\(\medspace = 2^{2} \cdot 17 \)
Automorphism group:	$D_4\times F_{17}$, of order \(2176\)\(\medspace = 2^{7} \cdot 17 \)
Composition factors:	$C_2$ x 3, $C_{17}$
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	17	34	68
Elements	1	69	2	16	16	32	136
Conjugacy classes	1	3	1	8	8	16	37
Divisions	1	3	1	1	1	1	8
Autjugacy classes	1	2	1	1	1	1	7

Dimension	1	2	16	32
Irr. complex chars.	4	33	0	0	37
Irr. rational chars.	4	1	2	1	8

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$68$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	2	32
Arbitrary	2	2	18

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{68}=1, b^{a}=b^{67} \rangle$
Permutation group:	Degree $21$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21), (18,20,21,19), (18,21)(19,20), (1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 66 \end{array}\right), \left(\begin{array}{rr} 9 & 19 \\ 43 & 9 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{67})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{34}$ $\,\rtimes\,$ $C_2$ (2)	$C_{17}$ $\,\rtimes\,$ $D_4$	$C_4$ $\,\rtimes\,$ $D_{17}$	$C_{68}$ $\,\rtimes\,$ $C_2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $D_{34}$	$C_{34}$ . $C_2^2$			more information

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

Homology

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

Subgroups

There are 132 subgroups in 16 conjugacy classes, 9 normal (7 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{34}$
Commutator:	$G' \simeq$ $C_{34}$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_{34}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{68}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{68}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{34}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
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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Classes of subgroups up to conjugation

Order 136: $D_{68}$

Order 68: $D_{34}$ x 2, $C_{68}$

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

Order 17: $C_{17}$

Order 8: $D_4$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 136: $D_{68}$

Order 68: $D_{34}$, $C_{68}$

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

Order 17: $C_{17}$

Order 8: $D_4$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 136: $D_{68}$ ($C_1$)

Order 68: $D_{34}$ ($C_2$) x 2, $C_{68}$ ($C_2$)

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

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

Order 4: $C_4$ ($D_{17}$)

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 136: $D_{68}$ ($C_1$)

Order 68: $D_{34}$ ($C_2$), $C_{68}$ ($C_2$)

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

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

Order 4: $C_4$ ($D_{17}$)

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

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

Series

Derived series	$D_{68}$	$\rhd$	$C_{34}$	$\rhd$	$C_1$
Chief series	$D_{68}$	$\rhd$	$D_{34}$	$\rhd$	$C_{34}$	$\rhd$	$C_{17}$	$\rhd$	$C_1$
Lower central series	$D_{68}$	$\rhd$	$C_{34}$	$\rhd$	$C_{17}$
Upper central series	$C_1$	$\lhd$	$C_2$	$\lhd$	$C_4$

Supergroups

This group is a maximal subgroup of 27 larger groups in the database.

This group is a maximal quotient of 23 larger groups in the database.

Character theory

Complex character table

See the $37 \times 37$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	4A	17A	34A	68A
	Size	1	1	34	34	2	16	16	32
	2 P	1A	1A	1A	1A	2A	17A	17A	34A
	17 P	1A	2A	2B	2C	4A	1A	2A	4A
136.6.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
136.6.1b		$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
136.6.1c		$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$
136.6.1d		$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$
136.6.2a		$2$	$-2$	$0$	$0$	$0$	$2$	$-2$	$0$
136.6.2b		$16$	$16$	$0$	$0$	$16$	$-1$	$-1$	$-1$
136.6.2c		$16$	$16$	$0$	$0$	$-16$	$-1$	$-1$	$1$
136.6.2d		$32$	$-32$	$0$	$0$	$0$	$-2$	$2$	$0$