Abstract group 9792.b: $C_2.\PGL(2,17)$

• Label: 9792.b
• Order: \( 2^{6} \cdot 3^{2} \cdot 17 \)
• Exponent: \( 2^{5} \cdot 3^{2} \cdot 17 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $576$
• Trans deg.: $576$
• Rank: $2$

Group information

Description:	$C_2.\PGL(2,17)$
Order:	\(9792\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 17 \)
Exponent:	\(4896\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 17 \)
Automorphism group:	$C_2\times \PGL(2,17)$, of order \(9792\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 17 \)
Composition factors:	$C_2$ x 2, $\PSL(2,17)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	8	9	12	16	17	18	32	34	36
Elements	1	1	272	578	272	612	816	544	1224	288	816	2448	288	1632	9792
Conjugacy classes	1	1	1	2	1	2	3	2	4	1	3	8	1	6	36
Divisions	1	1	1	2	1	1	1	1	1	1	1	1	1	1	15
Autjugacy classes	1	1	1	2	1	2	3	1	4	1	3	4	1	3	28

Dimension	1	16	17	18	32	36	48	72	96	144
Irr. complex chars.	2	17	2	15	0	0	0	0	0	0	36
Irr. rational chars.	2	3	2	1	1	1	2	1	1	1	15

Minimal presentations

Permutation degree:	$576$
Transitive degree:	$576$
Rank:	$2$
Inequivalent generating pairs:	$6864$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	32	32
Arbitrary	16	32	32

Constructions

Groups of Lie type:	$\CSU(2,17)$
Permutation group:	Degree $576$ $\langle(1,2,8,36,102,334,562,444,269,362,115,180,332,101,24,5)(3,13,58,135,150,39,149,279,272,452,190,474,276,77,66,15) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{ll}\alpha^{170} & \alpha^{179} \\ \alpha^{171} & \alpha^{211} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{170} & \alpha^{264} \\ \alpha^{64} & \alpha^{35} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{144} & 0 \\ 0 & \alpha^{144} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{78} & \alpha^{98} \\ \alpha^{90} & \alpha^{224} \\ \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{289}) = \GL_{2}(\F_{17}[\alpha]/(\alpha^{2} + 16 \alpha + 3))$
Direct product:	not isomorphic to a non-trivial direct product
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:	$\SL(2,17)$ . $C_2$	$C_2$ . $\PGL(2,17)$			more information

Elements of the group are displayed as matrices in $\CSU(2,17)$.

Homology

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

Subgroups

There are 5603 subgroups in 36 conjugacy classes, 4 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $\PGL(2,17)$
Commutator:	$G' \simeq$ $\SL(2,17)$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $\PGL(2,17)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2$	$G/\operatorname{Fit} \simeq$ $\PGL(2,17)$
Radical:	$R \simeq$ $C_2$	$G/R \simeq$ $\PGL(2,17)$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $\PGL(2,17)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{64}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
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

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 9792: $C_2.\PGL(2,17)$

Order 4896: $\SL(2,17)$

Order 544: $C_{17}:C_{32}$

Order 272: $C_{17}:C_{16}$

Order 136: $C_{17}:C_8$

Order 72: $C_9:Q_8$

Order 68: $C_{17}:C_4$

Order 64: $Q_{64}$

Order 48: $C_2.S_4$

Order 36: $C_9:C_4$ x 2, $C_{36}$

Order 34: $C_{34}$

Order 32: $Q_{32}$ x 2, $C_{32}$

Order 24: $C_3:Q_8$, $\SL(2,3)$

Order 18: $C_{18}$

Order 17: $C_{17}$

Order 16: $Q_{16}$ x 2, $C_{16}$

Order 12: $C_3:C_4$ x 2, $C_{12}$

Order 9: $C_9$

Order 8: $Q_8$ x 2, $C_8$

Order 6: $C_6$

Order 4: $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 9792: $C_2.\PGL(2,17)$

Order 4896: $\SL(2,17)$

Order 544: $C_{17}:C_{32}$

Order 272: $C_{17}:C_{16}$

Order 136: $C_{17}:C_8$

Order 72: $C_9:Q_8$

Order 68: $C_{17}:C_4$

Order 64: $Q_{64}$

Order 48: $C_2.S_4$

Order 36: $C_9:C_4$ x 2, $C_{36}$

Order 34: $C_{34}$

Order 32: $Q_{32}$ x 2, $C_{32}$

Order 24: $C_3:Q_8$, $\SL(2,3)$

Order 18: $C_{18}$

Order 17: $C_{17}$

Order 16: $Q_{16}$ x 2, $C_{16}$

Order 12: $C_3:C_4$ x 2, $C_{12}$

Order 9: $C_9$

Order 8: $Q_8$ x 2, $C_8$

Order 6: $C_6$

Order 4: $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 9792: $C_2.\PGL(2,17)$ ($C_1$)

Order 4896: $\SL(2,17)$ ($C_2$)

Order 2: $C_2$ ($\PGL(2,17)$)

Order 1: $C_1$ ($C_2.\PGL(2,17)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 9792: $C_2.\PGL(2,17)$ ($C_1$)

Order 4896: $\SL(2,17)$ ($C_2$)

Order 2: $C_2$ ($\PGL(2,17)$)

Order 1: $C_1$ ($C_2.\PGL(2,17)$)

Series

Derived series	$C_2.\PGL(2,17)$	$\rhd$	$\SL(2,17)$
Chief series	$C_2.\PGL(2,17)$	$\rhd$	$\SL(2,17)$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$C_2.\PGL(2,17)$	$\rhd$	$\SL(2,17)$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

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

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

Character theory

Complex character table

See the $36 \times 36$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	4A	4B	6A	8A	9A	12A	16A	17A	18A	32A	34A	36A
	Size	1	1	272	272	306	272	612	816	544	1224	288	816	2448	288	1632
	2 P	1A	1A	3A	2A	2A	3A	4B	9A	6A	8A	17A	9A	16A	17A	18A
	3 P	1A	2A	1A	4A	4B	2A	8A	3A	4A	16A	17A	6A	32A	34A	12A
	17 P	1A	2A	3A	4A	4B	6A	8A	9A	12A	16A	1A	18A	32A	2A	36A
	Schur
9792.b.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
9792.b.1b		$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$
9792.b.16a		$16$	$16$	$-2$	$2$	$0$	$-2$	$0$	$1$	$2$	$0$	$-1$	$1$	$0$	$-1$	$-1$
9792.b.16b		$16$	$16$	$-2$	$-2$	$0$	$-2$	$0$	$1$	$-2$	$0$	$-1$	$1$	$0$	$-1$	$1$
9792.b.16c	2	$16$	$-16$	$-2$	$0$	$0$	$2$	$0$	$-2$	$0$	$0$	$-1$	$2$	$0$	$1$	$0$
9792.b.16d	2	$32$	$-32$	$-4$	$0$	$0$	$4$	$0$	$2$	$0$	$0$	$-2$	$-2$	$0$	$2$	$0$
9792.b.16e		$48$	$48$	$3$	$6$	$0$	$3$	$0$	$0$	$-3$	$0$	$-3$	$0$	$0$	$-3$	$0$
9792.b.16f		$48$	$48$	$3$	$-6$	$0$	$3$	$0$	$0$	$3$	$0$	$-3$	$0$	$0$	$-3$	$0$
9792.b.16g	2	$96$	$-96$	$6$	$0$	$0$	$-6$	$0$	$0$	$0$	$0$	$-6$	$0$	$0$	$6$	$0$
9792.b.17a		$17$	$17$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$0$	$-1$	$1$	$0$	$-1$
9792.b.17b		$17$	$17$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$0$	$-1$	$-1$	$0$	$1$
9792.b.18a		$18$	$18$	$0$	$0$	$2$	$0$	$2$	$0$	$0$	$-2$	$1$	$0$	$0$	$1$	$0$
9792.b.18b		$36$	$36$	$0$	$0$	$4$	$0$	$-4$	$0$	$0$	$0$	$2$	$0$	$0$	$2$	$0$
9792.b.18c		$72$	$72$	$0$	$0$	$-8$	$0$	$0$	$0$	$0$	$0$	$4$	$0$	$0$	$4$	$0$
9792.b.18d	2	$144$	$-144$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$8$	$0$	$0$	$-8$	$0$