#27394416

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1855 ⋊ C₂
Sylow	C₅₃ ⋊ #21950

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1855	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂²×#11719945	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1855	D₅₃
hypocenter	C₅₃	#21950
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1855

894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 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)(55,59)(58,60)(64,74)(69,83)(73,87)(75,76)(82,92)(84,90)(86,93)(88,89);(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)(56,61)(58,60)(62,65,66,78,68,80,67,79)(63,70,71,77,72,85,81,91)(64,73,75,88,74,87,76,89)(69,82,84,86,83,92,90,93);(54,56)(55,58)(57,61)(59,60)(62,64)(63,69)(65,73)(66,75)(67,76)(68,74)(70,82)(71,84)(72,83)(77,86)(78,88)(79,89)(80,87)(81,90)(85,92)(91,93);(54,55)(56,60)(57,59)(58,61)(62,63,67,71,68,72,66,81)(64,69,76,84,74,83,75,90)(65,77,79,85,80,91,78,70)(73,86,89,92,87,93,88,82)

Polycyclic presentation:

2^8,53|a^2=g,c^2=f,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,i^d=i^52

Conjugacy classes & character table

1040 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 179892);