C₂×#20272803 #27298960

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#20272803

Splittings

Sylow

C₅₃ ⋊ C₂×#1379

Subgroups

	subgroup	G/H
center	C₂²×C₈	D₂₁₂
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂²⋊C₈
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	C₂×#1379
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂³×C₁₆

606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 91 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)(56,57,58,59)(60,62)(61,67)(63,76)(64,78)(65,73)(66,75)(68,84)(69,86)(70,72)(71,83)(74,80)(77,90)(79,81)(82,88)(85,89)(87,91);(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)(60,61,64,69,65,70,79,87,66,71,80,88,63,68,77,85)(62,72,74,89,73,83,90,86,75,84,78,91,76,67,81,82);(54,55)

Conjugacy classes & character table

3440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84436);

C2×#20272803 #27298960