#27406615

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1876 ⋊ C₂
Sylow	C₅₃ ⋊ #23017

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1876	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#20284166	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1876	D₅₃
hypocenter	C₅₃	#23017
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1876

864 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)(54,56,57,55)(59,67)(65,79)(66,81)(68,76)(73,82)(78,88)(80,86)(83,89);(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)(58,61)(59,66)(60,69)(62,77)(63,74)(64,75)(65,78)(67,81)(68,82)(70,87)(71,84)(72,85)(73,76)(79,88)(80,89)(83,86);(55,56)(58,60,63,71,64,72,62,70)(59,65,68,80,67,79,76,86)(61,69,74,84,75,85,77,87)(66,78,82,89,81,88,73,83);(54,55,57,56)(58,59,63,68,64,67,62,76)(60,65,71,80,72,79,70,86)(61,73,74,66,75,82,77,81)(69,83,84,78,85,89,87,88)

Polycyclic presentation:

2^8,53|a^2=g,b^2=eg,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=de,f^c=fh,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, 192091);