C₂²×#16810406 #27536788

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#16810406

Splittings

Fitting	C₂³×C₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂²×#350

Subgroups

	subgroup	G/H
center	C₂²×C₈	D₂₁₂
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂²⋊C₈
prime radical	C₂²×#11719945	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	C₂²×#350
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂³×C₁₆

1174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 89 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)(58,60)(59,65)(61,74)(62,76)(63,71)(64,73)(66,82)(67,84)(68,70)(69,81)(72,78)(75,88)(77,79)(80,86)(83,87)(85,89);(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)(58,59,62,67,63,68,77,85,64,69,78,86,61,66,75,83)(60,70,72,87,71,81,88,84,73,82,76,89,74,65,79,80);(56,57);(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, 322264);

C22×#16810406 #27536788