C₂×#20274687 #27383430

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#20274687

Splittings

Fitting	C₂×C₅₃×#419 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1796

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#419	C₂
socle	C₂²×C₅₃	#379
prime radical	C₂²×#13133265	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#419	D₅₃
hypocenter	C₅₃	C₂×#1796
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#419

3602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 87 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)(56,59)(57,63)(58,67)(60,71)(61,72)(62,76)(64,78)(65,70)(66,80)(68,82)(69,83)(73,85)(74,86)(75,87)(77,84)(79,81);(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)(56,58,61,69)(57,62,65,77)(59,67,72,83)(60,74,75,68)(63,76,70,84)(64,79,73,66)(71,86,87,82)(78,81,85,80);(56,57)(58,66)(59,70,72,63)(60,73)(61,65)(62,68)(64,75)(67,81,83,80)(69,79)(71,78,87,85)(74,77)(76,82,84,86);(54,55)

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 168906);

C2×#20274687 #27383430