C₁₅₁×#7397 #50312322

order	38656
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₅₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₅₁×#7397

Subgroups

	subgroup	G/H
center	C₄×C₁₅₁	#353
commutator	C₂×M₁₆	C₂×C₄×C₁₅₁
Frattini, cosocle	#407	C₂²×C₁₅₁
socle	C₂×C₁₅₁	#1343
prime radical	#2010×C₁₅₁	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₅₁ ◁ C₂×C₄×C₁₅₁ ◁ C₂²×C₄×C₁₅₁ ◁ #407×C₁₅₁ ◁ G

268 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 183
transitive degree	μ_t(G) = 4832

A minimal faithful permutation representation — 2 generators on 183 points:

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

Conjugacy classes & character table

4681 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38656, 385);