C₁₅₁×#12230 #50317155

order	38656
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₅₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₅₁×#12230

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₅₁	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₅₁
prime radical	#392×C₁₅₁	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₂³×C₁₅₁ ◁ C₂×C₂²⋊C₄×C₁₅₁ ◁ G

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

normal subgroups only

This group has 422 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) = 183
transitive degree	μ_t(G) = 4832

A minimal faithful permutation representation — 3 generators on 183 points:

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

Conjugacy classes & character table

3775 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38656, 5218);