#27290746

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#421 ⋊ C₂
Sylow	C₅₃ ⋊ #10999

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133185
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#421	C₂
socle	C₂²×C₅₃	#379
prime radical	#20288044	C₂
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#421	D₅₃
hypocenter	C₅₃	#10999
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#421

1310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(55,57)(58,61)(59,66)(60,70)(62,77)(63,73)(64,74)(65,78)(67,82)(68,83)(69,84)(71,88)(72,87)(75,81)(76,86)(79,89)(80,85);(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)(54,55)(56,57)(58,60)(59,65)(61,70)(62,76)(63,71)(64,72)(66,78)(67,80)(68,79)(69,75)(73,88)(74,87)(77,86)(81,84)(82,85)(83,89);(58,59,63,67,64,68,62,75)(60,69,71,65,72,80,76,79)(61,66,73,82,74,83,77,81)(70,85,88,89,87,84,86,78)

Polycyclic presentation:

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

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 76222);