#27393504

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1984 ⋊ C₂
Sylow	C₅₃ ⋊ #21819

Subgroups

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#1984	C₂
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1984	D₅₃
hypocenter	C₅₃	#21819
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1984

1042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

1532 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 178980);