Abstract group 1610.7: $D_{805}$

• Label: 1610.7
• Order: \( 2 \cdot 5 \cdot 7 \cdot 23 \)
• Exponent: \( 2 \cdot 5 \cdot 7 \cdot 23 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 23 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \cdot 11 \)
• Perm deg.: $35$
• Trans deg.: $805$
• Rank: $2$

Group information

Description:	$D_{805}$
Order:	\(1610\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 23 \)
Exponent:	\(1610\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 23 \)
Automorphism group:	$C_{805}.C_{66}.C_2^3$, of order \(425040\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 23 \)
Composition factors:	$C_2$, $C_5$, $C_7$, $C_{23}$
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	23	35	115	161	805
Elements	1	805	4	6	22	24	88	132	528	1610
Conjugacy classes	1	1	2	3	11	12	44	66	264	404
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	22	24	88	132	528
Irr. complex chars.	2	402	0	0	0	0	0	0	0	404
Irr. rational chars.	2	0	1	1	1	1	1	1	1	9

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$805$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	2	528
Arbitrary	2	2	32

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{805}=1, b^{a}=b^{804} \rangle$
Permutation group:	Degree $35$ $\langle(2,3)(4,5)(6,7)(9,10)(11,12)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{161}$ $\,\rtimes\,$ $D_5$	$C_{115}$ $\,\rtimes\,$ $D_7$	$C_{35}$ $\,\rtimes\,$ $D_{23}$	$C_{23}$ $\,\rtimes\,$ $D_{35}$	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 1160 subgroups in 16 conjugacy classes, 9 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $D_{805}$
Commutator:	$G' \simeq$ $C_{805}$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_{805}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{805}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{805}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{805}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
23-Sylow subgroup:	$P_{ 23 } \simeq$ $C_{23}$

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 1610: $D_{805}$

Order 805: $C_{805}$

Order 322: $D_{161}$

Order 230: $D_{115}$

Order 161: $C_{161}$

Order 115: $C_{115}$

Order 70: $D_{35}$

Order 46: $D_{23}$

Order 35: $C_{35}$

Order 23: $C_{23}$

Order 14: $D_7$

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 1610: $D_{805}$

Order 805: $C_{805}$

Order 322: $D_{161}$

Order 230: $D_{115}$

Order 161: $C_{161}$

Order 115: $C_{115}$

Order 70: $D_{35}$

Order 46: $D_{23}$

Order 35: $C_{35}$

Order 23: $C_{23}$

Order 14: $D_7$

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 1610: $D_{805}$ ($C_1$)

Order 805: $C_{805}$ ($C_2$)

Order 161: $C_{161}$ ($D_5$)

Order 115: $C_{115}$ ($D_7$)

Order 35: $C_{35}$ ($D_{23}$)

Order 23: $C_{23}$ ($D_{35}$)

Order 7: $C_7$ ($D_{115}$)

Order 5: $C_5$ ($D_{161}$)

Order 1: $C_1$ ($D_{805}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1610: $D_{805}$ ($C_1$)

Order 805: $C_{805}$ ($C_2$)

Order 161: $C_{161}$ ($D_5$)

Order 115: $C_{115}$ ($D_7$)

Order 35: $C_{35}$ ($D_{23}$)

Order 23: $C_{23}$ ($D_{35}$)

Order 7: $C_7$ ($D_{115}$)

Order 5: $C_5$ ($D_{161}$)

Order 1: $C_1$ ($D_{805}$)

Series

Derived series	$D_{805}$	$\rhd$	$C_{805}$	$\rhd$	$C_1$
Chief series	$D_{805}$	$\rhd$	$C_{805}$	$\rhd$	$C_{161}$	$\rhd$	$C_{23}$	$\rhd$	$C_1$
Lower central series	$D_{805}$	$\rhd$	$C_{805}$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $404 \times 404$ 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	23A	35A	115A	161A	805A
	Size	1	805	4	6	22	24	88	132	528
	2 P	1A	1A	5A	7A	23A	35A	115A	161A	805A
	5 P	1A	2A	5A	7A	23A	35A	115A	161A	805A
	7 P	1A	2A	1A	7A	23A	7A	23A	161A	161A
	23 P	1A	2A	5A	1A	23A	5A	115A	23A	115A
1610.7.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1610.7.1b		$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1610.7.2a		$4$	$0$	$-1$	$4$	$4$	$-1$	$-1$	$4$	$-1$
1610.7.2b		$6$	$0$	$6$	$-1$	$6$	$-1$	$6$	$-1$	$-1$
1610.7.2c		$22$	$0$	$22$	$22$	$-1$	$22$	$-1$	$-1$	$-1$
1610.7.2d		$24$	$0$	$-6$	$-4$	$24$	$1$	$-6$	$-4$	$1$
1610.7.2e		$88$	$0$	$-22$	$88$	$-4$	$-22$	$1$	$-4$	$1$
1610.7.2f		$132$	$0$	$132$	$-22$	$-6$	$-22$	$-6$	$1$	$1$
1610.7.2g		$528$	$0$	$-132$	$-88$	$-24$	$22$	$6$	$4$	$-1$