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C5⋊((C163⋊C3)⋊D7) #46803863

order34230
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊((C163⋊C3)⋊D7)
unambiguous semidirect product(C163⋊C3)⋊D35

Splittings

Fitting, socle, derived, HallC5×C7×C163 ⋊ C2×C3
HallC5×C7×C163⋊C3 ⋊ C2
HallD5705 ⋊ C3
HallC7×C163⋊C3 ⋊ D5
HallC5×C163⋊C3 ⋊ D7
HallC7×C163 ⋊ C3×D5
HallC5×C163 ⋊ C3×D7
HallC163⋊C3 ⋊ D35
SylowC163 ⋊ C3×D35
HallC5×C7 ⋊ D163⋊C3
SylowC7 ⋊ C5⋊(D163⋊C3)
SylowC5 ⋊ (C163⋊C3)⋊D7

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C163C2×C3
LCSG ▷ C5×C7×C163
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 175
transitive degreeμt(G) = 5705

A minimal faithful permutation representation — 2 generators on 175 points:

(2,3)(4,5)(7,8)(9,10)(11,12)(14,18,19,15,16,20)(17,26,27,21,22,36)(23,39,40,29,30,53)(24,38,34,32,28,25)(31,58,59,42,43,76)(33,56,57,45,41,50)(35,37,55,47,48,51)(44,81,82,61,62,105)(46,79,80,64,60,73)(49,54,78,66,67,74)(52,94,92,70,71,69)(63,110,111,84,85,139)(65,108,109,87,83,102)(68,77,107,89,90,103)(72,123,124,95,96,98)(75,129,121,99,100,93)(86,144,145,113,114,166)(88,142,143,116,112,136)(91,106,141,118,119,137)(97,151,152,125,126,133)(101,156,115,130,131,128)(104,161,150,134,135,122)(117,167,168,147,146,164)(120,140,163,149,138,165)(127,169,170,153,154,162)(132,172,148,157,158,160)(155,173,174,171,159,175);(1,2,4,5,3)(6,7,9,11,12,10,8)(13,14,17,23,31,44,63,86,115,124,80,40,55,56,39,54,75,101,132,159,174,168,111,141,161,156,151,142,110,140,119,62,83,96,100,103,105,136,162,171,153,125,95,70,48,22,28,36,50,69,93,122,149,118,89,66,47,34,19,18,24,33,46,65,88,117,148,152,109,59,78,94,79,58,77,104,114,146,154,158,145,163,165,166,172,169,167,144,134,90,43,60,71,74,76,102,133,160,147,116,87,64,45,32,16,20,25,35,49,68,91,120,150,121,92,57,27,38,26,37,52,72,97,127,155,170,143,82,107,129,123,108,81,106,138,85,112,126,131,135,137,139,164,175,173,157,130,99,67,30,41,51,53,73,98,128,113,84,61,42,29,21,15)

Polycyclic presentation:

2,3,5,7,163|c^a=c^4,d^a=d^6,e^a=e^162,e^b=e^104

Conjugacy classes & character table

1002 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34230, 45);