#27290799

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11007

Subgroups

	subgroup	G/H
center	C₂²	#16810479
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#536	C₂
socle	C₂²×C₅₃	#379
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#536	D₅₃
hypocenter	C₅₃	#11007
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#536

1270 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)(60,70)(62,76)(63,75)(64,77)(65,79)(67,83)(68,81)(69,84)(71,86)(72,85)(73,87)(74,78)(80,89)(82,88);(54,55)(56,57)(58,60,64,73)(59,65,68,80)(61,70,77,87)(62,72,63,71)(66,79,81,89)(67,82,78,69)(74,84,83,88)(75,86,76,85);(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)(58,59)(60,69)(61,74,77,83)(62,78)(63,67)(64,68)(65,71)(66,76,81,75)(70,79,87,89)(72,80)(73,82)(84,86,88,85)

Polycyclic presentation:

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

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, 76275);