C₁₂₇×#12760 #44576433

order	32512
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₂₇×#12760

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₂₇	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×D₄	C₂³×C₁₂₇
prime radical	C₂×#127×C₁₂₇	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂³×C₁₂₇ ◁ C₂²×D₄×C₁₂₇ ◁ G

876 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 147
transitive degree	μ_t(G) = 4064

A minimal faithful permutation representation — 3 generators on 147 points:

(1,2)(3,4)(5,6)(7,11)(8,12,16,10)(9,13)(14,19,17,18)(15,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(6,10)(8,16)(11,18)(12,13)(14,17)(19,20);(2,4)(5,7,9,15)(6,11,13,20)(8,17,16,14)(10,18,12,19)

Conjugacy classes & character table

5080 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 5748);