C₁₅₁×#11478 #50316403

order	38656
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₅₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₅₁×#11478

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₅₁	#99
commutator	C₂×C₄	C₂²×C₈×C₁₅₁
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₁₅₁
socle	C₂²×C₁₅₁	C₂×#99
prime radical	C₂³×C₁₅₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₅₁ ◁ C₂²×C₄×C₁₅₁ ◁ C₂×C₄×Q₈×C₁₅₁ ◁ G

428 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 9664

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

(1,2,4,7,5,8,3,6)(9,10)(11,19)(12,16)(13,17,22,23)(14,18)(15,20,24,21)(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);(9,11,14,12)(10,15,18,24)(13,21,22,20)(16,23,19,17);(1,3,5,4)(2,6,8,7)(9,12,14,11)(10,16,18,19)(13,21,22,20)(15,23,24,17)

Conjugacy classes & character table

7852 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38656, 4466);