C₂×#20273961 #27359889

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#20273961

Splittings

Fitting	C₂×C₅₃×#500 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1649

Subgroups

	subgroup	G/H
center	C₂³	#13133185
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle	C₂³×C₅₃	#120
prime radical	C₂²×D₂₁₂	D₄
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	C₂×#1649
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

1206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 79 points:

(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)(64,67,70,66)(65,71,74,79)(68,76,69,77)(72,78,73,75);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(57,60)(58,63)(61,62)(64,66,70,67)(65,71,74,79)(68,77,69,76)(72,78,73,75);(56,57)(58,60)(59,62)(61,63)(64,65,70,74)(66,75,67,78)(68,73,69,72)(71,77,79,76);(54,55)

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 145365);

C2×#20273961 #27359889