Abstract group 136.9: $C_2\times C_{68}$

• Label: 136.9
• Order: \( 2^{3} \cdot 17 \)
• Exponent: \( 2^{2} \cdot 17 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{7} \)
• Perm deg.: $23$
• Trans deg.: $136$
• Rank: $2$

Group information

Description:	$C_2\times C_{68}$
Order:	\(136\)\(\medspace = 2^{3} \cdot 17 \)
Exponent:	\(68\)\(\medspace = 2^{2} \cdot 17 \)
Automorphism group:	$D_4\times C_{16}$, of order \(128\)\(\medspace = 2^{7} \)
Composition factors:	$C_2$ x 3, $C_{17}$
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.

Group statistics

Order	1	2	4	17	34	68
Elements	1	3	4	16	48	64	136
Conjugacy classes	1	3	4	16	48	64	136
Divisions	1	3	2	1	3	2	12
Autjugacy classes	1	2	1	1	2	1	8

Dimension	1	2	16	32
Irr. complex chars.	136	0	0	0	136
Irr. rational chars.	4	2	4	2	12

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$136$
Rank:	$2$
Inequivalent generating pairs:	$54$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	2	3	18

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{2}=b^{68}=1 \rangle$
Permutation group:	Degree $23$ $\langle(3,6,4,5), (1,2), (7,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8), (3,4)(5,6)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 100 & 0 \\ 0 & 59 \end{array}\right), \left(\begin{array}{rr} 136 & 0 \\ 0 & 136 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{137})$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \times\, $ $C_{17}$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{34}$ . $C_2^2$	$C_2^2$ . $C_{34}$	$(C_2\times C_{34})$ . $C_2$	$C_2$ . $(C_2\times C_{34})$	more information

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

Homology

Primary decomposition:	$C_{2} \times C_{4} \times C_{17}$
Schur multiplier:	$C_{2}$
Commutator length:	$0$

Subgroups

There are 16 subgroups, all normal (8 characteristic).

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

Special subgroups

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

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

Order 34: $C_{34}$ x 3

Order 17: $C_{17}$

Order 8: $C_2\times C_4$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 136: $C_2\times C_{68}$

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

Order 34: $C_{34}$ x 2

Order 17: $C_{17}$

Order 8: $C_2\times C_4$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 136: $C_2\times C_{68}$ ($C_1$)

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

Order 34: $C_{34}$ ($C_4$) x 2, $C_{34}$ ($C_2^2$)

Order 17: $C_{17}$ ($C_2\times C_4$)

Order 8: $C_2\times C_4$ ($C_{17}$)

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

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

Order 1: $C_1$ ($C_2\times C_{68}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 136: $C_2\times C_{68}$ ($C_1$)

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

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

Order 17: $C_{17}$ ($C_2\times C_4$)

Order 8: $C_2\times C_4$ ($C_{17}$)

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

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

Order 1: $C_1$ ($C_2\times C_{68}$)

Series

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

Supergroups

This group is a maximal subgroup of 49 larger groups in the database.

This group is a maximal quotient of 36 larger groups in the database.

Character theory

Complex character table

See the $136 \times 136$ 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	2B	2C	4A	4B	17A	34A	34B	34C	68A	68B
	Size	1	1	1	1	2	2	16	16	16	16	32	32
	2 P	1A	1A	1A	1A	2C	2C	17A	17A	17A	17A	34C	34C
	17 P	1A	2A	2B	2C	4A	4B	1A	2A	2B	2C	4A	4B
136.9.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
136.9.1b		$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$
136.9.1c		$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$
136.9.1d		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$
136.9.1e		$2$	$-2$	$2$	$-2$	$0$	$0$	$2$	$-2$	$-2$	$-2$	$0$	$0$
136.9.1f		$2$	$2$	$-2$	$-2$	$0$	$0$	$2$	$2$	$2$	$2$	$0$	$0$
136.9.1g		$16$	$16$	$16$	$16$	$16$	$16$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
136.9.1h		$16$	$-16$	$-16$	$16$	$-16$	$16$	$-1$	$1$	$1$	$1$	$-1$	$1$
136.9.1i		$16$	$-16$	$-16$	$16$	$16$	$-16$	$-1$	$1$	$1$	$1$	$1$	$-1$
136.9.1j		$16$	$16$	$16$	$16$	$-16$	$-16$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
136.9.1k		$32$	$-32$	$32$	$-32$	$0$	$0$	$-2$	$2$	$2$	$2$	$0$	$0$
136.9.1l		$32$	$32$	$-32$	$-32$	$0$	$0$	$-2$	$-2$	$-2$	$-2$	$0$	$0$