C₂×#20275074 #27385563

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#20275074

Splittings

Fitting	C₂×C₅₃×#421 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1881

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133185
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#421	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂×#16811421	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#421	D₅₃
hypocenter	C₅₃	C₂×#1881
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#421

1682 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)(57,64)(58,68)(60,72)(61,76)(62,73)(63,77)(65,80)(66,81)(67,82)(69,84)(70,86)(71,75)(74,85)(78,87)(79,83);(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)(56,58,60,74,62,70,61,69)(57,63,65,78,66,79,75,67)(59,68,72,85,73,86,76,84)(64,77,80,87,81,83,71,82);(56,57)(58,67,70,78)(59,71)(60,65)(61,75)(62,66)(63,69,79,74)(64,72)(68,83,86,77)(73,80)(76,81)(82,84,87,85);(54,55)

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 171039);

C2×#20275074 #27385563