C₂×#23758312 #34116887

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

Aliases

direct product	C₂×#23758312
unambiguous semidirect product	D₁₅₇⋊#440

Splittings

Fitting	C₂×SD₁₆×C₁₅₇ ⋊ C₄
Sylow	C₁₅₇ ⋊ C₂×#440

Subgroups

	subgroup	G/H
center	C₂²	D₄×#10597845
commutator	C₄×C₁₅₇	C₂³×C₄
Frattini	C₄	C₂³×#10597845
Fitting	C₂×SD₁₆×C₁₅₇	C₄
socle	C₂²×C₁₅₇	C₄×D₄
prime radical	C₂×D₄×D₁₅₇	C₂²
cosocle	C₄×D₁₅₇	C₂⁴
hypercenter	C₂×SD₁₆	#10597845
hypocenter	C₁₅₇	C₂×#440
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

544 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 167
transitive degree	μ_t(G) = 2512

A minimal faithful permutation representation — 4 generators on 167 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 21261);