#27290842

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11015

Subgroups

	subgroup	G/H
center	C₂²	#16810503
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×#137×C₅₃	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×#137	D₅₃
hypocenter	C₅₃	#11015
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#137

1242 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:

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

Polycyclic presentation:

2^8,53|a^2=g,b^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,i^a=i^52,c^b=cf,f^b=fh,g^b=gh,f^c=fh

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 76318);