#27410704

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #23367

Subgroups

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#1840	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1840	D₅₃
hypocenter	C₅₃	#23367
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1840

600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 13568

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

Polycyclic presentation:

2^8,53|a^2=g,b^2=e,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de,i^b=i^52,d^c=dh

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 196180);