C₂×#458×C₁₅₇ #34097382

order	20096
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₅₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#458×C₁₅₇

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₅₇	#120
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₅₇
prime radical	C₂×H(5,𝔽₂)×C₁₅₇	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₇ ◁ C₂⁴×C₁₅₇ ◁ G

794 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 175
transitive degree	μ_t(G) = 5024

A minimal faithful permutation representation — 4 generators on 175 points:

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

Conjugacy classes & character table

5024 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 1756);