C₂×#20264937 #27170838

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#20264937

Splittings

Sylow

C₁₀₁ ⋊ C₂×#339

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₄₀₄
commutator	C₄×C₁₀₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂²×C₄²×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂²⋊C₄
prime radical	C₂⁴×C₁₀₁	D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂²×C₄²	D₁₀₁
hypocenter	C₁₀₁	C₂×#339
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄²

488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 3 generators on 115 points:

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

Conjugacy classes & character table

3256 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 4216);

C2×#20264937 #27170838