Abstract group 990.11: $D_{495}$

• Label: 990.11
• Order: \( 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
• Exponent: \( 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{3} \cdot 5^{2} \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \cdot 5 \)
• Perm deg.: $25$
• Trans deg.: $495$
• Rank: $2$

Group information

Description:	$D_{495}$
Order:	\(990\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Exponent:	\(990\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Automorphism group:	$C_{495}.C_{30}.C_2^3$, of order \(118800\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5^{2} \cdot 11 \)
Composition factors:	$C_2$, $C_3$ x 2, $C_5$, $C_{11}$
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	3	5	9	11	15	33	45	55	99	165	495
Elements	1	495	2	4	6	10	8	20	24	40	60	80	240	990
Conjugacy classes	1	1	1	2	3	5	4	10	12	20	30	40	120	249
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	13
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	13

Dimension	1	2	4	6	8	10	20	24	40	60	80	240
Irr. complex chars.	2	247	0	0	0	0	0	0	0	0	0	0	249
Irr. rational chars.	2	1	1	1	1	1	1	1	1	1	1	1	13

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$495$
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:	$\langle a, b \mid a^{2}=b^{495}=1, b^{a}=b^{494} \rangle$
Permutation group:	Degree $25$ $\langle(2,4)(3,7)(5,8)(6,9)(11,12)(13,14)(16,17)(18,19)(20,21)(22,23)(24,25), (1,2,5,3,6,9,7,8,4) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 36 & 0 \\ 0 & 468 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{991})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{99}$ $\,\rtimes\,$ $D_5$	$C_{55}$ $\,\rtimes\,$ $D_9$	$C_9$ $\,\rtimes\,$ $D_{55}$	$C_5$ $\,\rtimes\,$ $D_{99}$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{165}$ . $S_3$	$C_{33}$ . $D_{15}$	$C_{15}$ . $D_{33}$	$C_3$ . $D_{165}$	more information

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 948 subgroups in 24 conjugacy classes, 13 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $D_{495}$
Commutator:	$G' \simeq$ $C_{495}$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $D_{165}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{495}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{495}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{165}$	$G/\operatorname{soc} \simeq$ $S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
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 990: $D_{495}$

Order 495: $C_{495}$

Order 330: $D_{165}$

Order 198: $D_{99}$

Order 165: $C_{165}$

Order 110: $D_{55}$

Order 99: $C_{99}$

Order 90: $D_{45}$

Order 66: $D_{33}$

Order 55: $C_{55}$

Order 45: $C_{45}$

Order 33: $C_{33}$

Order 30: $D_{15}$

Order 22: $D_{11}$

Order 18: $D_9$

Order 15: $C_{15}$

Order 11: $C_{11}$

Order 10: $D_5$

Order 9: $C_9$

Order 6: $S_3$

Order 5: $C_5$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 990: $D_{495}$

Order 495: $C_{495}$

Order 330: $D_{165}$

Order 198: $D_{99}$

Order 165: $C_{165}$

Order 110: $D_{55}$

Order 99: $C_{99}$

Order 90: $D_{45}$

Order 66: $D_{33}$

Order 55: $C_{55}$

Order 45: $C_{45}$

Order 33: $C_{33}$

Order 30: $D_{15}$

Order 22: $D_{11}$

Order 18: $D_9$

Order 15: $C_{15}$

Order 11: $C_{11}$

Order 10: $D_5$

Order 9: $C_9$

Order 6: $S_3$

Order 5: $C_5$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 990: $D_{495}$ ($C_1$)

Order 495: $C_{495}$ ($C_2$)

Order 165: $C_{165}$ ($S_3$)

Order 99: $C_{99}$ ($D_5$)

Order 55: $C_{55}$ ($D_9$)

Order 45: $C_{45}$ ($D_{11}$)

Order 33: $C_{33}$ ($D_{15}$)

Order 15: $C_{15}$ ($D_{33}$)

Order 11: $C_{11}$ ($D_{45}$)

Order 9: $C_9$ ($D_{55}$)

Order 5: $C_5$ ($D_{99}$)

Order 3: $C_3$ ($D_{165}$)

Order 1: $C_1$ ($D_{495}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 990: $D_{495}$ ($C_1$)

Order 495: $C_{495}$ ($C_2$)

Order 165: $C_{165}$ ($S_3$)

Order 99: $C_{99}$ ($D_5$)

Order 55: $C_{55}$ ($D_9$)

Order 45: $C_{45}$ ($D_{11}$)

Order 33: $C_{33}$ ($D_{15}$)

Order 15: $C_{15}$ ($D_{33}$)

Order 11: $C_{11}$ ($D_{45}$)

Order 9: $C_9$ ($D_{55}$)

Order 5: $C_5$ ($D_{99}$)

Order 3: $C_3$ ($D_{165}$)

Order 1: $C_1$ ($D_{495}$)

Series

Derived series

$D_{495}$

$\rhd$

$C_{495}$

$\rhd$

$C_1$

Chief series

$D_{495}$

$\rhd$

$C_{495}$

$\rhd$

$C_{165}$

$\rhd$

$C_{55}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$D_{495}$

$\rhd$

$C_{495}$

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 $249 \times 249$ 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	3A	5A	9A	11A	15A	33A	45A	55A	99A	165A	495A
	Size	1	495	2	4	6	10	8	20	24	40	60	80	240
	2 P	1A	1A	3A	5A	9A	11A	15A	33A	45A	55A	99A	165A	495A
	3 P	1A	2A	1A	5A	3A	11A	5A	11A	15A	55A	33A	55A	165A
	5 P	1A	2A	3A	1A	9A	11A	3A	33A	9A	11A	99A	33A	99A
	11 P	1A	2A	3A	5A	9A	1A	15A	3A	45A	5A	9A	15A	45A
990.11.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
990.11.1b		$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
990.11.2a		$2$	$0$	$2$	$2$	$-1$	$2$	$2$	$2$	$-1$	$2$	$-1$	$2$	$-1$
990.11.2b		$4$	$0$	$4$	$-1$	$4$	$4$	$-1$	$4$	$-1$	$-1$	$4$	$-1$	$-1$
990.11.2c		$6$	$0$	$-3$	$6$	$0$	$6$	$-3$	$-3$	$0$	$6$	$0$	$-3$	$0$
990.11.2d		$8$	$0$	$8$	$-2$	$-4$	$8$	$-2$	$8$	$1$	$-2$	$-4$	$-2$	$1$
990.11.2e		$10$	$0$	$10$	$10$	$10$	$-1$	$10$	$-1$	$10$	$-1$	$-1$	$-1$	$-1$
990.11.2f		$20$	$0$	$20$	$20$	$-10$	$-2$	$20$	$-2$	$-10$	$-2$	$1$	$-2$	$1$
990.11.2g		$24$	$0$	$-12$	$-6$	$0$	$24$	$3$	$-12$	$0$	$-6$	$0$	$3$	$0$
990.11.2h		$40$	$0$	$40$	$-10$	$40$	$-4$	$-10$	$-4$	$-10$	$1$	$-4$	$1$	$1$
990.11.2i		$60$	$0$	$-30$	$60$	$0$	$-6$	$-30$	$3$	$0$	$-6$	$0$	$3$	$0$
990.11.2j		$80$	$0$	$80$	$-20$	$-40$	$-8$	$-20$	$-8$	$10$	$2$	$4$	$2$	$-1$
990.11.2k		$240$	$0$	$-120$	$-60$	$0$	$-24$	$30$	$12$	$0$	$6$	$0$	$-3$	$0$