C₂×#20266127 #27187585

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product	C₂×#20266127
unambiguous semidirect product	D₁₀₁⋊#388

Splittings

Fitting	C₂×C₂²⋊C₄×C₁₀₁ ⋊ C₄
Sylow	C₁₀₁ ⋊ C₂×#388

Subgroups

	subgroup	G/H
center	C₂³	C₂²×#87938
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂²	C₂³×#87938
Fitting	C₂×C₂²⋊C₄×C₁₀₁	C₄
socle	C₂³×C₁₀₁	C₂²×C₄
prime radical	C₂⁴×D₁₀₁	C₂²
cosocle	C₂²×D₁₀₁	C₂⁴
hypercenter	C₂×C₂²⋊C₄	#87938
hypocenter	C₁₀₁	C₂×#388
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 4 generators on 111 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 20963);

C2×#20266127 #27187585