#27433876

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #25314

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2047	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂³×C₅₃	C₂×#41
hypercenter	#2047	D₅₃
hypocenter	C₅₃	#25314
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2047

680 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

719 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 219352);