C₁₅₁×#13565 #50318490

order	38656
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₅₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₅₁×#13565

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₅₁	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₁₅₁
Frattini, cosocle	#124	C₂³×C₁₅₁
prime radical	C₂×H(5,𝔽₂)×C₁₅₁	C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₅₁ ◁ C₂³×C₁₅₁ ◁ C₂²×C₄×C₁₅₁ ◁ C₂×#124×C₁₅₁ ◁ G

496 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 183 points:

(1,2,4,9,6,11,18,27)(3,12,14,25,13,28,29,32)(5,10,21,26,20,8,17,24)(7,16,23,19,22,30,31,15)(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)(2,7)(4,16)(5,19)(6,13)(8,12)(9,25)(10,28)(11,22)(14,21)(15,20)(17,29)(18,30)(23,26)(24,31)(27,32);(3,13)(7,22)(12,28)(14,29)(15,19)(16,30)(23,31)(25,32)

Conjugacy classes & character table

4228 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38656, 6553);