Abstract group 616.26: $C_{77}:D_4$

• Label: 616.26
• Order: \( 2^{3} \cdot 7 \cdot 11 \)
• Exponent: \( 2^{2} \cdot 7 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \cdot 5 \)
• Perm deg.: $22$
• Trans deg.: $308$
• Rank: $2$

Group information

Description:	$C_{77}:D_4$
Order:	\(616\)\(\medspace = 2^{3} \cdot 7 \cdot 11 \)
Exponent:	\(308\)\(\medspace = 2^{2} \cdot 7 \cdot 11 \)
Automorphism group:	$C_2\times C_{11}:(C_{10}\times C_2\times F_7)$, of order \(18480\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Composition factors:	$C_2$ x 3, $C_7$, $C_{11}$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	4	7	11	14	22	77	154
Elements	1	157	154	6	10	18	30	60	180	616
Conjugacy classes	1	3	1	3	5	9	15	30	90	157
Divisions	1	3	1	1	1	2	2	1	2	14
Autjugacy classes	1	3	1	1	1	2	2	1	2	14

Dimension	1	2	6	10	12	20	60	120
Irr. complex chars.	4	153	0	0	0	0	0	0	157
Irr. rational chars.	4	1	2	2	1	1	2	1	14

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$308$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	120
Arbitrary	2	4	18

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{154}=[a,b]=[b,c]=1, c^{a}=bc^{153} \rangle$
Permutation group:	Degree $22$ $\langle(2,3)(4,5)(6,7)(9,11)(13,14)(15,16)(17,18)(19,20)(21,22), (8,9)(10,11), (8,10)(9,11), (1,2,4,6,7,5,3), (12,13,15,17,19,21,22,20,18,16,14)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 462 \end{array}\right), \left(\begin{array}{rr} 27 & 0 \\ 0 & 343 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{463})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{77}$ $\,\rtimes\,$ $D_4$	$D_{154}$ $\,\rtimes\,$ $C_2$	$C_2^2$ $\,\rtimes\,$ $D_{77}$	$(C_2\times C_{22})$ $\,\rtimes\,$ $D_7$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{22}$ . $D_{14}$	$C_{14}$ . $D_{22}$	$C_2$ . $D_{154}$	$C_{154}$ . $C_2^2$	more information

Elements of the group are displayed as matrices in $\GL_{2}(\F_{463})$.

Homology

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

Subgroups

There are 500 subgroups in 32 conjugacy classes, 15 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{154}$
Commutator:	$G' \simeq$ $C_{154}$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_{154}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{154}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{77}:D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{154}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

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 616: $C_{77}:D_4$

Order 308: $C_2\times C_{154}$, $D_{154}$, $C_{77}:C_4$

Order 154: $C_{154}$ x 2, $D_{77}$

Order 88: $C_{11}:D_4$

Order 77: $C_{77}$

Order 56: $C_7:D_4$

Order 44: $C_2\times C_{22}$, $D_{22}$, $C_{11}:C_4$

Order 28: $C_2\times C_{14}$, $D_{14}$, $C_7:C_4$

Order 22: $C_{22}$ x 2, $D_{11}$

Order 14: $C_{14}$ x 2, $D_7$

Order 11: $C_{11}$

Order 8: $D_4$

Order 7: $C_7$

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 616: $C_{77}:D_4$

Order 308: $C_2\times C_{154}$, $D_{154}$, $C_{77}:C_4$

Order 154: $C_{154}$ x 2, $D_{77}$

Order 88: $C_{11}:D_4$

Order 77: $C_{77}$

Order 56: $C_7:D_4$

Order 44: $C_2\times C_{22}$, $D_{22}$, $C_{11}:C_4$

Order 28: $C_2\times C_{14}$, $D_{14}$, $C_7:C_4$

Order 22: $C_{22}$ x 2, $D_{11}$

Order 14: $C_{14}$ x 2, $D_7$

Order 11: $C_{11}$

Order 8: $D_4$

Order 7: $C_7$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 616: $C_{77}:D_4$ ($C_1$)

Order 308: $C_2\times C_{154}$ ($C_2$), $D_{154}$ ($C_2$), $C_{77}:C_4$ ($C_2$)

Order 154: $C_{154}$ ($C_2^2$)

Order 77: $C_{77}$ ($D_4$)

Order 44: $C_2\times C_{22}$ ($D_7$)

Order 28: $C_2\times C_{14}$ ($D_{11}$)

Order 22: $C_{22}$ ($D_{14}$)

Order 14: $C_{14}$ ($D_{22}$)

Order 11: $C_{11}$ ($C_7:D_4$)

Order 7: $C_7$ ($C_{11}:D_4$)

Order 4: $C_2^2$ ($D_{77}$)

Order 2: $C_2$ ($D_{154}$)

Order 1: $C_1$ ($C_{77}:D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 616: $C_{77}:D_4$ ($C_1$)

Order 308: $C_2\times C_{154}$ ($C_2$), $D_{154}$ ($C_2$), $C_{77}:C_4$ ($C_2$)

Order 154: $C_{154}$ ($C_2^2$)

Order 77: $C_{77}$ ($D_4$)

Order 44: $C_2\times C_{22}$ ($D_7$)

Order 28: $C_2\times C_{14}$ ($D_{11}$)

Order 22: $C_{22}$ ($D_{14}$)

Order 14: $C_{14}$ ($D_{22}$)

Order 11: $C_{11}$ ($C_7:D_4$)

Order 7: $C_7$ ($C_{11}:D_4$)

Order 4: $C_2^2$ ($D_{77}$)

Order 2: $C_2$ ($D_{154}$)

Order 1: $C_1$ ($C_{77}:D_4$)

Series

Derived series

$C_{77}:D_4$

$\rhd$

$C_{154}$

$\rhd$

$C_1$

Chief series

$C_{77}:D_4$

$\rhd$

$C_2\times C_{154}$

$\rhd$

$C_{154}$

$\rhd$

$C_{77}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{77}:D_4$

$\rhd$

$C_{154}$

$\rhd$

$C_{77}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 16 larger groups in the database.

This group is a maximal quotient of 14 larger groups in the database.

Character theory

Complex character table

See the $157 \times 157$ 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	7A	11A	14A	14B	22A	22B	77A	154A	154B
	Size	1	1	2	154	154	6	10	6	12	10	20	60	60	120
	2 P	1A	1A	1A	1A	2A	7A	11A	7A	7A	11A	11A	77A	77A	77A
	7 P	1A	2A	2B	2C	4A	7A	11A	14A	14B	22A	22B	77A	154A	154B
	11 P	1A	2A	2B	2C	4A	7A	11A	14A	14B	22A	22B	77A	154A	154B
616.26.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
616.26.1b		$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$
616.26.1c		$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$
616.26.1d		$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
616.26.2a		$2$	$-2$	$0$	$0$	$0$	$2$	$2$	$0$	$-2$	$0$	$0$	$2$	$-2$	$0$
616.26.2b		$6$	$6$	$6$	$0$	$0$	$-1$	$6$	$-1$	$-1$	$6$	$6$	$-1$	$-1$	$-1$
616.26.2c		$6$	$6$	$-6$	$0$	$0$	$-1$	$6$	$1$	$-1$	$-6$	$-6$	$-1$	$-1$	$1$
616.26.2d		$10$	$10$	$10$	$0$	$0$	$10$	$-1$	$10$	$10$	$-1$	$-1$	$-1$	$-1$	$-1$
616.26.2e		$10$	$10$	$-10$	$0$	$0$	$10$	$-1$	$-10$	$10$	$1$	$1$	$-1$	$-1$	$1$
616.26.2f		$12$	$-12$	$0$	$0$	$0$	$-2$	$12$	$0$	$2$	$0$	$0$	$-2$	$2$	$0$
616.26.2g		$20$	$-20$	$0$	$0$	$0$	$20$	$-2$	$0$	$-20$	$0$	$0$	$-2$	$2$	$0$
616.26.2h		$60$	$60$	$60$	$0$	$0$	$-10$	$-6$	$-10$	$-10$	$-6$	$-6$	$1$	$1$	$1$
616.26.2i		$60$	$60$	$-60$	$0$	$0$	$-10$	$-6$	$10$	$-10$	$6$	$6$	$1$	$1$	$-1$
616.26.2j		$120$	$-120$	$0$	$0$	$0$	$-20$	$-12$	$0$	$20$	$0$	$0$	$2$	$-2$	$0$