#34115305

order	20096
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₁₅₇ ⋊ M₁₂₈

Subgroups

	subgroup	G/H
center, Frattini	C₁₆	C₂×#10597845
commutator, socle	C₂×C₁₅₇	C₂×C₃₂
Fitting	C₂×C₁₆×C₁₅₇	C₄
prime radical	C₂²×C₁₅₇	C₃₂
cosocle	#19097677	C₂²
hypercenter	C₂×C₁₆	#10597845
hypocenter	C₁₅₇	M₁₂₈
LCS	G ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆

38 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

1328 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 19679);