#27404837

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1855 ⋊ C₂
Sylow	C₅₃ ⋊ #22887

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1855	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#20283937	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1855	D₅₃
hypocenter	C₅₃	#22887
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1855

1020 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

1046 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 190313);