C₁₂₇×#13647 #44577320

order	32512
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₁₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₂₇×#13647

Subgroups

	subgroup	G/H
center	C₃₂×C₁₂₇	D₄
commutator	C₄	C₂²×C₁₆×C₁₂₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
socle	C₂×C₁₂₇	D₄×C₁₆
prime radical	#135×C₁₂₇	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₂×C₁₂₇ ◁ C₂×C₃₂×C₁₂₇ ◁ G

186 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 191
transitive degree	μ_t(G) = 8128

A minimal faithful permutation representation — 3 generators on 191 points:

(1,2,6,12,7,13,28,42,5,14,25,43,26,45,56,63,8,15,29,44,30,46,58,64,27,11,38,40,57,41,32,59)(3,16,18,47,19,22,36,53,20,34,49,60,51,24,37,55,21,39,50,62,52,54,35,61,17,9,31,33,48,4,10,23)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191);(1,3)(2,9)(4,13)(5,20)(6,18)(7,19)(8,21)(10,32)(11,39)(12,33)(14,16)(15,34)(17,27)(22,45)(23,42)(24,46)(25,49)(26,51)(28,36)(29,50)(30,52)(31,38)(35,58)(37,56)(40,62)(41,54)(43,47)(44,60)(48,57)(53,63)(55,64)(59,61);(1,4,6,23,7,16,28,47,5,22,25,53,26,34,56,60,8,24,29,55,30,39,58,62,27,54,38,61,57,9,32,33)(2,10,12,3,13,18,42,19,14,36,43,20,45,49,63,51,15,37,44,21,46,50,64,52,11,35,40,17,41,31,59,48)

Conjugacy classes & character table

14224 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6635);