#34102449

order	20096
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₂×D₁₆×C₁₅₇ ⋊ C₂
Sylow	C₁₅₇ ⋊ #2263

Subgroups

	subgroup	G/H
center	C₂	C₂×#19097712
commutator, cosocle	C₁₆×C₁₅₇	C₂³
Frattini	C₁₆	C₂²×D₁₅₇
Fitting	C₂×D₁₆×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂×D₁₆
hypercenter	C₂×D₁₆	D₁₅₇
hypocenter	C₁₅₇	#2263
LCS	G ▷ C₁₆×C₁₅₇ ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(164,177)(166,180)(167,181)(168,176)(173,189)(178,182)(179,183);(159,164)(160,169)(161,170)(162,174)(165,177)(166,167)(168,183)(171,188)(172,186)(173,178)(175,187)(176,179)(180,181)(182,189)(184,185);(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)(160,164)(161,173)(162,176)(163,165)(166,169)(167,170)(168,174)(171,183)(172,177)(175,189)(178,185)(179,184)(180,186)(181,187)(182,188)

Polycyclic presentation:

2^7,157|d^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=de,e^a=ef,f^a=fg,h^a=h^156,d^b=de,e^b=ef,f^b=fg

Conjugacy classes & character table

1742 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 6823);