C₁₅₁×#7491 #50312416

order	38656
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₅₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₅₁×#7491

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₅₁	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₄²×C₁₅₁
Frattini, cosocle	C₂×C₄⋊Q₈	C₂²×C₁₅₁
prime radical	C₂³×C₁₅₁	#99
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₂³×C₁₅₁ ◁ C₂²×C₄×C₁₅₁ ◁ C₂×C₄⋊Q₈×C₁₅₁ ◁ G

276 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 171
transitive degree	μ_t(G) = 4832

A minimal faithful permutation representation — 2 generators on 171 points:

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

Conjugacy classes & character table

5134 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38656, 479);