#27381514

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#341 ⋊ C₂
Sylow	C₅₃ ⋊ #20351

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#341	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂²×#11719945	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#341	D₅₃
hypocenter	C₅₃	#20351
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#341

1978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

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)(61,75)(66,81)(71,85)(76,77)(78,88)(82,83)(84,89)(86,87);(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)(58,61)(59,66)(60,71)(62,76)(63,77)(64,75)(65,78)(67,82)(68,83)(69,81)(70,84)(72,86)(73,87)(74,85)(79,89)(80,88);(54,55,56,57)(58,60,63,73,64,74,62,72)(59,65,68,79,69,80,67,70)(61,71,77,87,75,85,76,86)(66,78,83,89,81,88,82,84);(58,59,62,68,64,69,63,67)(60,70,72,65,74,79,73,80)(61,66,76,83,75,81,77,82)(71,84,86,78,85,89,87,88)

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,e^2=h,f^2=h,g^2=h,b^a=be,e^a=eh,g^a=gh,d^c=dh,i^d=i^52

Conjugacy classes & character table

1544 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 166990);