C₁₃₁×#7325 #45693209

order	33536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₃₁ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₃₁×#7325

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#342
commutator	C₂⁴	C₄²×C₁₃₁
Frattini, cosocle	C₂×#117	C₂²×C₁₃₁
prime radical	C₂⁴×C₁₃₁	C₄²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂×#117×C₁₃₁ ◁ G

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

normal subgroups only

This group has 244 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) = 179
transitive degree	μ_t(G) = 8384

A minimal faithful permutation representation — 2 generators on 179 points:

(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,21,25,22,26,34,41)(19,27,29,42,30,38,43,47)(20,24,32,39,33,40,45,48)(23,35,36,31,37,44,46,28)(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);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,23)(20,31,33,28)(21,29)(22,30)(24,38,40,27)(25,36)(26,37)(32,44,45,35)(34,43)(39,47,48,42)(41,46)

Conjugacy classes & character table

4454 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33536, 313);