C₂×#1323×C₁₃₁ #45697326

order	33536
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₃₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#1323×C₁₃₁

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
socle	C₂³×C₁₃₁	C₂²⋊C₈
prime radical	C₂⁴×C₁₃₁	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂³×C₈×C₁₃₁ ◁ G

394 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 173
transitive degree	μ_t(G) = 16768

A minimal faithful permutation representation — 3 generators on 173 points:

(1,2)(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);(3,4)(5,9,10,7)(6,8)(11,12,15,20,16,21,30,38,17,22,31,39,14,19,28,36)(13,23,25,40,24,34,41,37,26,35,29,42,27,18,32,33);(3,5,6,10)(4,7,8,9)(11,13)(12,18)(14,27)(15,29)(16,24)(17,26)(19,35)(20,37)(21,23)(22,34)(25,31)(28,41)(30,32)(33,39)(36,40)(38,42)

Conjugacy classes & character table

9956 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33536, 4430);