C₁₂₇×#13708 #44577381

order	32512
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#13708

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₂₇	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ C₄×C₈×C₁₂₇ ◁ C₄×C₁₆×C₁₂₇ ◁ G

158 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 199
transitive degree	μ_t(G) = 32512

A minimal faithful permutation representation — 3 generators on 199 points:

(1,2,4,6)(3,7,8,5)(9,10,12,16)(11,21,22,50)(13,29,26,51)(14,34,27,59)(15,36,28,20)(17,41,38,23)(18,44,39,32)(19,47,40,31)(24,55,53,72)(25,56,37,52)(30,48,57,70)(33,66,58,62)(35,45,60,69)(42,67,68,64)(43,54,49,63)(46,65,61,71)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199);(1,3,4,8)(2,5,6,7)(10,16)(17,38)(18,39)(19,40)(20,36)(21,50)(29,51)(34,59)(42,68)(43,49)(45,69)(46,61)(48,70)(52,56)(55,72)(62,66);(9,11,12,22)(10,17,16,38)(13,30,26,57)(14,35,27,60)(15,37,28,25)(18,45,39,69)(19,48,40,70)(20,49,36,43)(21,51,50,29)(23,44,41,32)(24,47,53,31)(33,67,58,64)(34,42,59,68)(46,72,61,55)(52,62,56,66)(54,65,63,71)

Conjugacy classes & character table

6223 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 6696);