#27361914

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #18685

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133186
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3544	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#3544	D₅₃
hypocenter	C₅₃	#18685
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3544

930 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 147390);