C₁₅₁×#9409 #50314334

order	38656
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₅₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₅₁×#9409

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₅₁	#386
commutator	C₂²×C₄	C₂²×C₄×C₁₅₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₅₁
prime radical	#2539×C₁₅₁	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₂³×C₄×C₁₅₁ ◁ G

840 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 4832

A minimal faithful permutation representation — 3 generators on 167 points:

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

Conjugacy classes & character table

7399 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38656, 2397);