#34108747

order	20096
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₁₅₇ ⋊ #2916

Subgroups

	subgroup	G/H
center	C₄	C₂²×D₆₂₈
commutator	C₄×C₁₅₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	C₂×#131×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₁₅₇	C₂²
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	C₂×#131	D₁₅₇
hypocenter	C₁₅₇	#2916
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ C₂×#131

440 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)(161,174)(166,180)(170,184)(175,176)(178,188)(181,187)(182,189)(185,186);(158,161)(159,166)(160,170)(162,175)(163,176)(164,174)(165,178)(167,181)(168,180)(169,182)(171,185)(172,186)(173,184)(177,187)(179,188)(183,189);(158,160,162,171,164,173,163,172)(159,165,167,169,168,179,177,183)(161,170,175,185,174,184,176,186)(166,178,181,182,180,188,187,189);(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)(158,159,163,167,164,168,162,177)(160,169,172,179,173,183,171,165)(161,166,176,181,174,180,175,187)(170,182,186,188,184,189,185,178)

Polycyclic presentation:

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

Conjugacy classes & character table

3164 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 13121);