C₁₅₁×#12080 #50317005

order	38656
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₅₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₅₁×#12080

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₅₁	#442
commutator	C₂×C₈	C₂²×C₄×C₁₅₁
Frattini, cosocle	C₄×C₈	C₂³×C₁₅₁
prime radical	C₂×D₈×C₁₅₁	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₄²×C₁₅₁ ◁ #322×C₁₅₁ ◁ G

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

normal subgroups only

This group has 218 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) = 175
transitive degree	μ_t(G) = 19328

A minimal faithful permutation representation — 3 generators on 175 points:

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

Conjugacy classes & character table

6040 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38656, 5068);