C₂×#20266226 #27187914

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#20266226

Splittings

Sylow

C₁₀₁ ⋊ C₂×C₈×Q₈

Subgroups

	subgroup	G/H
center	C₂³	C₂²×#87938
commutator	C₂×C₁₀₁	C₂³×C₈
Frattini	C₂²	C₂³×#87938
Fitting	C₂×C₄⋊C₄×C₁₀₁	C₄
socle, prime radical	C₂³×C₁₀₁	C₂²×C₄
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	C₂×C₄⋊C₄	#87938
hypocenter	C₁₀₁	C₂×C₈×Q₈
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

440 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 119
transitive degree	μ_t(G) = 12928

A minimal faithful permutation representation — 4 generators on 119 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 21292);