Abstract group 1004497044480.a: $\PSp(4,17)$

• Label: 100...480.a
• Order: \( 2^{10} \cdot 3^{4} \cdot 5 \cdot 17^{4} \cdot 29 \)
• Exponent: \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 17 \cdot 29 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{4} \cdot 5 \cdot 17^{4} \cdot 29 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $5220$
• Trans deg.: $5220$
• Rank: $2$

Group information

Description:	$\PSp(4,17)$
Order:	\(1004497044480\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 5 \cdot 17^{4} \cdot 29 \)
Exponent:	\(354960\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 17 \cdot 29 \)
Automorphism group:	Group of order \(2008994088960\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 5 \cdot 17^{4} \cdot 29 \)
Composition factors:	$\PSp(4,17)$
Derived length:	$0$

This group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

Group statistics

Order	1	2	3	4	5	6	8	9	12	16	17	18	24	29	34	36	48	51	68	72	102	136	144	145	153	272	306
Elements	1	12864835	45592640	3975108300	6927565824	6610932800	31493116080	37340372160	13951347840	61139730240	6975757440	57036392640	27902695680	48492960768	7192909440	41854043520	55805391360	13130680320	14772015360	83708087040	6565340160	29544030720	167416174080	193971843072	39392040960	29544030720	19696020480	1004497044480
Conjugacy classes	1	2	2	3	1	3	9	12	2	14	6	15	4	7	6	6	8	3	3	12	2	6	24	28	9	8	6	202
Divisions	1	2	2	3	1	3	5	4	2	4	4	5	2	1	4	2	2	2	2	2	1	2	2	1	2	1	1	63
Autjugacy classes	1	2	2	3	1	3	9	12	2	14	4	15	4	7	4	6	8	2	2	12	1	4	24	28	6	4	3	183

Minimal presentations

Permutation degree:	$5220$
Transitive degree:	$5220$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	145	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\PSp(4,17)$, $\Omega(5,17)$, $\POmega(5,17)$, $\PSigmaSp(4,17)$
Permutation group:	Degree $5220$ $\langle(5,1291,4069,4104,1389,4896,3170,528,820,1190,3261,4992,1039,657,513,640) \!\cdots\! \rangle$
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

Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\Sp(4,17)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 2 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $\PSp(4,17)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8^2.C_2^3.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}^3.C_{17}$
29-Sylow subgroup:	$P_{ 29 } \simeq$ $C_{29}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 7546368 (order at least 133110) are shown, as well as all normal subgroups of any index.

Order 1004497044480: $\PSp(4,17)$

Order 192432384: $\He_{17}.C_{16}.\PSL(2,17)$, $C_{17}^3.C_8.\PSL(2,17).C_2$

Order 96216192: $\He_{17}.C_8.\PSL(2,17)$, $C_{17}^3.C_8.\PSL(2,17)$

Order 48108096: $\He_{17}.C_4.\PSL(2,17)$, $C_{17}^3.C_4.\PSL(2,17)$

Order 24137280: $\POMinus(4,17)$

Order 24054048: $\He_{17}.C_2.\PSL(2,17)$, $C_{17}^3.C_2.\PSL(2,17)$

Order 23970816: $C_2.\PSL(2,17)^2.C_2$

Order 12068640: $\PSL(2,289)$

Order 12027024: $C_{17}^3.\PSL(2,17)$

Order 11985408: $C_2.\PSL(2,17).\PSL(2,17)$

Order 10690688: $\He_{17}.C_{34}.C_8^2$

Order 5345344: $\He_{17}.D_{34}.C_4^2$, $\He_{17}.C_{68}.C_4^2$, $\He_{17}.C_{34}.C_4^2.C_2$

Order 2672672: $\He_{17}.D_{17}:C_8.C_2$, $\He_{17}.D_{17}.C_4^2$, $\He_{17}.C_{68}.C_2^2.C_2$, $\He_{17}.C_{136}.C_2^2$, $C_{17}^3.D_{17}.C_4^2$, $C_{17}^3.D_{17}.C_4.C_2^2$, $C_{17}^3.C_{34}.C_4^2$

Order 1886592: $\He_{17}.Q_8.C_{24}.C_2$, $\He_{17}.Q_8.(C_8\times S_3)$

Order 1414944: $\He_{17}.C_{72}.C_2^2$, $C_{17}^3.C_{72}.C_2^2$

Order 1336336: $C_{17}^3.F_{17}$ x 5, $\He_{17}.C_{68}.C_4$ x 2, $\He_{17}.D_{34}.C_2^2$, $\He_{17}.D_{17}:C_8$, $\He_{17}.D_{17}.C_2^2.C_2$, $\He_{17}.C_{68}.C_2^2$, $\He_{17}.C_{272}$, $C_{17}^3.D_{17}:C_8$, $C_{17}^3.D_{17}.C_2^2.C_2$, $C_{17}^3.C_{136}.C_2$

Order 1257728: $\He_{17}.C_8.C_4.C_2^3$, $C_{17}^3.C_8.C_{16}.C_2$

Order 943296: $\He_{17}.Q_8.D_6.C_2$ x 2, $\He_{17}.Q_8.C_{24}$ x 2, $\He_{17}.Q_8.C_6.C_4$ x 2, $C_{17}^3.(C_8\times S_4)$

Order 707472: $C_{17}^3.C_{72}.C_2$ x 2, $\He_{17}.C_{72}.C_2$, $\He_{17}.C_{144}$, $\He_{17}.(C_8\times D_9)$, $C_{17}^3.C_{144}$

Order 668168: $(C_{17}^3.C_{17}):C_8$ x 12, $(C_{17}^3.C_{17}):(C_2\times C_4)$ x 3

Order 665856: $C_{17}:C_4.C_4.\PSL(2,17)$

Order 628864: $\He_{17}.C_4.C_4.C_2^3$ x 2, $C_{17}^3.C_8^2.C_2$ x 2, $\He_{17}.D_8:C_2.C_2^2$, $\He_{17}.C_8^2.C_2$, $\He_{17}.C_4.C_4:C_8$, $\He_{17}.C_2.D_4:C_8$, $\He_{17}.C_2.C_8^2$, $C_{17}^3.C_4^2.C_2^3$

Order 471648: $\He_{17}.Q_8.D_6$ x 2, $\He_{17}.Q_8.C_{12}$ x 2, $\He_{17}.(C_4.S_4)$ x 2, $C_{17}^3.A_4.C_8$ x 2, $\He_{17}.C_{48}.C_2$, $C_{17}^3.C_{24}.C_2^2$, $C_{17}^3.(C_4\times S_4)$

Order 353736: $\He_{17}.D_{18}.C_2$, $\He_{17}.C_{72}$, $\He_{17}.C_9.C_8$, $C_{17}^3.C_{72}$, $C_{17}^3.C_{36}.C_2$, $C_{17}^3.C_9.C_8$

Order 334084: $(C_{17}^3.C_{17}):C_4$ x 6, $(C_{17}^3.C_{17}):C_2^2$

Order 332928: $C_{34}.C_4.\PSL(2,17)$

Order 314432: $\He_{17}.\OD_{32}.C_2$ x 3, $\He_{17}.C_4.C_4^2$ x 2, $\He_{17}.(D_8.C_4)$ x 2, $\He_{17}.(D_4.C_8)$ x 2, $\He_{17}.\OD_{16}:C_2.C_2$, $\He_{17}.D_{16}:C_2$, $\He_{17}.C_8^2$, $\He_{17}.C_8.C_2^3$, $\He_{17}.C_4.C_4:C_4$, $\He_{17}.(C_8.C_4).C_2$, $C_{17}^3.D_4:C_8$, $C_{17}^3.C_8^2$, $C_{17}^3.C_4:C_{16}$, $C_{17}^3.C_4.C_4^2$, $C_{17}^3.C_2.C_4:C_8$, $C_{17}^3.(C_8\times D_4)$, $C_{17}^3.(C_4\times D_8)$

Order 235824: $\He_{17}.Q_8.S_3$ x 4, $\He_{17}.Q_8.C_6$ x 2, $C_{17}^3.A_4.C_4$ x 2, $\He_{17}.C_{48}$, $\He_{17}.C_6.C_8$, $\He_{17}.(C_8\times S_3)$, $C_{17}^3.C_{48}$, $C_{17}^3.C_{24}.C_2$, $C_{17}^3.C_6.C_8$, $C_{17}^3.A_4.C_2^2$

Order 176868: $\He_{17}:D_{18}$, $\He_{17}:C_{36}$, $\He_{17}:(C_9:C_4)$, $C_{17}^3:D_{18}$, $C_{17}^3:C_{36}$, $C_{17}^3:(C_9:C_4)$

Order 167042: $(C_{17}^3.C_{17}):C_2$ x 3

Order 166464: $C_{17}:C_4.\PSL(2,17)$

Order 157216: $\He_{17}.C_8.C_2^2$ x 4, $C_{17}^3.C_8.C_2^2$ x 4, $\He_{17}.\SD_{32}$ x 2, $\He_{17}.\OD_{32}$ x 2, $\He_{17}.D_8:C_2$ x 2, $\He_{17}.C_4\wr C_2$ x 2, $C_{17}^3.C_4^2.C_2$ x 2, $\He_{17}.\OD_{16}.C_2$, $\He_{17}.Q_{32}$, $\He_{17}.D_{16}$, $\He_{17}.C_{16}.C_2$, $\He_{17}.C_4^2.C_2$, $\He_{17}.C_4.C_2^3$, $\He_{17}.C_2.C_4^2$, $\He_{17}.(C_8.C_4)$, $C_{17}^3.D_4:C_4$, $C_{17}^3.C_4:C_8$, $C_{17}^3.C_4.C_2^3$, $C_{17}^3.C_2^2:C_8$, $C_{17}^3.C_2.C_4^2$, $C_{17}^3.(C_4\times D_4)$, $C_{17}^3.(C_2.D_8)$

Order 88128: $D_9.\PSL(2,17).C_2$

Order 83521: $C_{17}^3.C_{17}$

Order 78336: $D_8.\PSL(2,17).C_2$

Order 2448: $\PSL(2,17)$

Order 1920: $C_2^4:S_5$ x 2

Order 1024: $C_8^2.C_2^3.C_2$

Order 360: $A_6$

Order 81: $C_9^2$

Order 29: $C_{29}$

Order 5: $C_5$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 1004497044480: $\PSp(4,17)$ ($C_1$)

Order 1: $C_1$ ($\PSp(4,17)$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Character theory

Complex character table

The $202 \times 202$ character table is not available for this group.

Rational character table

The $63 \times 63$ rational character table is not available for this group.