Abstract group 770.11: $D_{385}$

• Label: 770.11
• Order: \( 2 \cdot 5 \cdot 7 \cdot 11 \)
• Exponent: \( 2 \cdot 5 \cdot 7 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \cdot 5 \)
• Perm deg.: $23$
• Trans deg.: $385$
• Rank: $2$

Group information

Description:	$D_{385}$
Order:	\(770\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 11 \)
Exponent:	\(770\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 11 \)
Automorphism group:	$C_{385}.C_{30}.C_2^3$, of order \(92400\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Composition factors:	$C_2$, $C_5$, $C_7$, $C_{11}$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.

Group statistics

Order	1	2	5	7	11	35	55	77	385
Elements	1	385	4	6	10	24	40	60	240	770
Conjugacy classes	1	1	2	3	5	12	20	30	120	194
Divisions	1	1	1	1	1	1	1	1	1	9
Autjugacy classes	1	1	1	1	1	1	1	1	1	9

Dimension	1	2	4	6	10	24	40	60	240
Irr. complex chars.	2	192	0	0	0	0	0	0	0	194
Irr. rational chars.	2	0	1	1	1	1	1	1	1	9

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$385$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	2	240
Arbitrary	2	2	20

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{385}=1, b^{a}=b^{384} \rangle$
Permutation group:	Degree $23$ $\langle(2,3)(4,5)(6,7)(9,10)(11,12)(14,15)(16,17)(18,19)(20,21)(22,23), (8,9,11,12,10), (1,2,4,6,7,5,3), (13,14,16,18,20,22,23,21,19,17,15)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 768 \end{array}\right), \left(\begin{array}{rr} 371 & 87 \\ 707 & 371 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{769})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{77}$ $\,\rtimes\,$ $D_5$	$C_{55}$ $\,\rtimes\,$ $D_7$	$C_7$ $\,\rtimes\,$ $D_{55}$	$C_5$ $\,\rtimes\,$ $D_{77}$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 584 subgroups in 16 conjugacy classes, 9 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $D_{385}$
Commutator:	$G' \simeq$ $C_{385}$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_{385}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{385}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{385}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{385}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
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 770: $D_{385}$

Order 385: $C_{385}$

Order 154: $D_{77}$

Order 110: $D_{55}$

Order 77: $C_{77}$

Order 70: $D_{35}$

Order 55: $C_{55}$

Order 35: $C_{35}$

Order 22: $D_{11}$

Order 14: $D_7$

Order 11: $C_{11}$

Order 10: $D_5$

Order 7: $C_7$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 770: $D_{385}$

Order 385: $C_{385}$

Order 154: $D_{77}$

Order 110: $D_{55}$

Order 77: $C_{77}$

Order 70: $D_{35}$

Order 55: $C_{55}$

Order 35: $C_{35}$

Order 22: $D_{11}$

Order 14: $D_7$

Order 11: $C_{11}$

Order 10: $D_5$

Order 7: $C_7$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 770: $D_{385}$ ($C_1$)

Order 385: $C_{385}$ ($C_2$)

Order 77: $C_{77}$ ($D_5$)

Order 55: $C_{55}$ ($D_7$)

Order 35: $C_{35}$ ($D_{11}$)

Order 11: $C_{11}$ ($D_{35}$)

Order 7: $C_7$ ($D_{55}$)

Order 5: $C_5$ ($D_{77}$)

Order 1: $C_1$ ($D_{385}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 770: $D_{385}$ ($C_1$)

Order 385: $C_{385}$ ($C_2$)

Order 77: $C_{77}$ ($D_5$)

Order 55: $C_{55}$ ($D_7$)

Order 35: $C_{35}$ ($D_{11}$)

Order 11: $C_{11}$ ($D_{35}$)

Order 7: $C_7$ ($D_{55}$)

Order 5: $C_5$ ($D_{77}$)

Order 1: $C_1$ ($D_{385}$)

Series

Derived series	$D_{385}$	$\rhd$	$C_{385}$	$\rhd$	$C_1$
Chief series	$D_{385}$	$\rhd$	$C_{385}$	$\rhd$	$C_{77}$	$\rhd$	$C_{11}$	$\rhd$	$C_1$
Lower central series	$D_{385}$	$\rhd$	$C_{385}$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

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

Character theory

Complex character table

See the $194 \times 194$ 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	5A	7A	11A	35A	55A	77A	385A
	Size	1	385	4	6	10	24	40	60	240
	2 P	1A	1A	5A	7A	11A	35A	55A	77A	385A
	5 P	1A	2A	5A	7A	11A	35A	55A	77A	385A
	7 P	1A	2A	1A	7A	11A	7A	11A	77A	77A
	11 P	1A	2A	5A	1A	11A	5A	55A	11A	55A
770.11.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
770.11.1b		$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
770.11.2a		$4$	$0$	$-1$	$4$	$4$	$-1$	$-1$	$4$	$-1$
770.11.2b		$6$	$0$	$6$	$-1$	$6$	$-1$	$6$	$-1$	$-1$
770.11.2c		$10$	$0$	$10$	$10$	$-1$	$10$	$-1$	$-1$	$-1$
770.11.2d		$24$	$0$	$-6$	$-4$	$24$	$1$	$-6$	$-4$	$1$
770.11.2e		$40$	$0$	$-10$	$40$	$-4$	$-10$	$1$	$-4$	$1$
770.11.2f		$60$	$0$	$60$	$-10$	$-6$	$-10$	$-6$	$1$	$1$
770.11.2g		$240$	$0$	$-60$	$-40$	$-24$	$10$	$6$	$4$	$-1$