#27454088

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1772 ⋊ C₂
Sylow	C₅₃ ⋊ #26851

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₂₁₂
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1772	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#13133245	C₂²
hypercenter	#1772	D₅₃
hypocenter	C₅₃	#26851
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772

720 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

1502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 239564);