#34113239

order	20096
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₁₅₇ ⋊ #3407

Subgroups

	subgroup	G/H
center, Frattini	C₈	C₂³×D₁₅₇
commutator, socle	C₂×C₁₅₇	C₂³×C₈
Fitting	C₂×#131×C₁₅₇	C₂
prime radical	C₂×#41×C₁₅₇	C₄
cosocle	C₈×C₁₅₇	C₂⁴
hypercenter	C₂×#131	D₁₅₇
hypocenter	C₁₅₇	#3407
LCS	G ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₈ ◁ C₂×#131

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 189
transitive degree	μ_t(G) = 5024

A minimal faithful permutation representation — 4 generators on 189 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

3188 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 17613);