#34115254

order	20096
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₁₅₇ ⋊ #1401

Subgroups

	subgroup	G/H
center	C₈	C₂²⋊#10597845
commutator	C₄×C₁₅₇	C₂×C₁₆
Frattini	C₂×C₈	C₂×#10597845
Fitting	M₃₂×C₁₅₇	C₄
socle	C₂×C₁₅₇	C₂²⋊C₁₆
prime radical	#14578954	C₈
cosocle	C₂×#14578929	C₂²
hypercenter	M₃₂	#10597845
hypocenter	C₁₅₇	#1401
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 221
transitive degree	μ_t(G) = 10048

A minimal faithful permutation representation — 2 generators on 221 points:

(1,2)(3,11)(4,5)(6,7)(8,24)(9,10)(12,20)(13,14)(15,16)(17,41)(18,19)(21,37)(22,23)(25,33)(26,27)(28,29)(30,62)(31,32)(34,58)(35,36)(38,54)(39,40)(42,50)(43,44)(45,46)(47,87)(48,49)(51,83)(52,53)(55,79)(56,57)(59,75)(60,61)(63,71)(64,65)(66,67)(68,116)(69,70)(72,112)(73,74)(76,108)(77,78)(80,104)(81,82)(84,100)(85,86)(88,96)(89,90)(91,92)(93,142)(94,95)(97,140)(98,99)(101,138)(102,103)(105,136)(106,107)(109,133)(110,111)(113,129)(114,115)(117,125)(118,119)(120,121)(122,139)(123,124)(126,137)(127,128)(130,134)(131,132)(141,149)(143,148)(144,145)(146,147)(150,155)(151,152)(153,154)(156,157)(158,160)(159,166)(161,177)(162,180)(163,175)(164,173)(165,176)(167,190)(168,193)(169,188)(170,172)(171,189)(174,183)(178,201)(179,205)(181,208)(182,186)(184,202)(185,204)(187,196)(191,199)(192,214)(194,216)(195,207)(197,200)(198,213)(203,210)(206,220)(209,211)(212,218)(215,219)(217,221);(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,91)(53,72,73,87)(57,76,77,83)(61,80,81,79)(65,84,85,75)(67,88,89,71)(70,93,94,120)(74,97,98,116)(78,101,102,112)(82,105,106,108)(86,109,110,104)(90,113,114,100)(92,117,118,96)(95,122,123,146)(99,126,127,142)(103,130,131,140)(107,134,135,138)(111,137,132,136)(115,139,128,133)(119,141,124,129)(121,143,144,125)(145,150,151,149)(147,152,153,148)(154,156,157,155)(158,159,162,168,163,169,181,194,164,170,182,195,184,197,209,217,165,171,183,196,185,198,210,218,161,167,178,191,179,192,206,215)(160,172,174,199,175,200,203,219,173,189,201,193,202,213,220,216,176,190,180,207,204,214,208,221,177,166,186,187,205,188,211,212)

Polycyclic presentation:

2^7,157|a^2=d,c^2=g,d^2=e,e^2=f,f^2=g,b^a=bc,h^a=h^129,c^b=cg,d^b=dg,h^b=h^156,h^d=h^156

Conjugacy classes & character table

836 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 19628);