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ΓL(1,𝔽8)×C191⋊C5 #34095382

order20055
automorphism?
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×C191⋊C5
unambiguous semidirect productC7⋊(C3×C191⋊C5)
unambiguous semidirect productC191⋊(C5×ΓL(1,𝔽8))
unambiguous semidirect product(ΓL(1,𝔽8)×C191)⋊C5
unambiguous semidirect product(C7×C191⋊C5)⋊C3

Splittings

Fitting, socle, derived, HallC7×C191 ⋊ C3×C5
HallC7×C191⋊C5 ⋊ C3
HallΓL(1,𝔽8)×C191 ⋊ C5
SylowC191 ⋊ C5×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C191⋊C5

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C191C3×C5
LCSG ▷ C7×C191
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 198
transitive degreeμt(G) = 1337

A minimal faithful permutation representation — 2 generators on 198 points:

(2,3,5,9,17)(4,7,13,25,45)(6,11,21,38,33)(8,15,29,53,89)(10,19,24,18,34)(12,23,42,72,77)(14,27,49,83,60)(16,31,57,96,32)(20,37,65,46,78)(22,40,69,67,61)(26,47,80,35,62)(28,51,86,135,121)(30,55,92,87,100)(36,52,39,59,44)(41,71,113,115,122)(43,74,118,130,101)(48,82,128,79,123)(50,85,133,108,102)(54,90,138,63,103)(56,94,141,166,167)(58,98,146,151,149)(64,95,73,116,125)(66,106,84,131,76)(68,109,105,81,126)(70,111,110,99,88)(75,120,165,181,168)(91,139,183,124,169)(93,140,142,154,150)(97,144,107,104,136)(112,157,158,161,171)(114,159,163,176,137)(117,162,173,127,172)(119,164,178,148,143)(129,174,132,177,152)(134,179,156,155,153)(145,184,160,170,189)(147,185,175,186,182)(180,191,187,188,190)(193,194,195)(196,198,197);(1,2,4,8,16,32,59,99,148,133,178,98,147,86,136,182,83,130,55,93,49,84,132,139,169,27,50,25,46,79,124,170,11,22,41,65,105,152,157,3,6,12,24,44,76,109,155,146,179,17,33,60,100,149,96,131,176,94,142,85,134,180,165,189,51,87,47,81,127,118,163,23,43,75,106,153,72,115,7,14,28,52,88,137,128,173,185,191,34,61,101,80,125,144,177,42,73,117,45,77,121,167,21,39,68,82,129,175,164,9,18,35,63,104,151,116,161,15,30,56,95,143,141,174,190,135,181,62,102,92,126,171,184,183,74,119,53,78,122,168,138,40,70,112,113,158,187,89,19,36,64,69,110,156,162,188,31,58,38,67,108,154,186,57,97,145,103,150,140,172,29,54,91,120,166,90,123,13,26,48,71,114,160,159,5,10,20,37,66,107,111)(192,193,194,196,195,197,198)

Conjugacy classes & character table

215 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20055, 4);