#27385535

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1878 ⋊ C₂
Sylow	C₅₃ ⋊ #20935

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1878	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂³×D₅₃	C₂×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1878	D₅₃
hypocenter	C₅₃	#20935
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1878

1690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 93 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)(63,71)(64,76)(70,86)(72,73)(75,89)(77,78)(79,87)(90,91);(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)(62,65,68,82)(63,70,71,86)(64,75,76,89)(66,81,67,80)(69,83,85,93)(72,87,73,79)(74,88,84,92)(77,91,78,90);(54,56,57,55)(59,61)(62,64)(63,69)(65,75)(66,78)(67,77)(68,76)(70,83)(71,85)(72,84)(73,74)(79,88)(80,91)(81,90)(82,89)(86,93)(87,92);(54,55,57,56)(58,59,60,61)(62,63,67,73,68,71,66,72)(64,74,78,69,76,84,77,85)(65,79,81,70,82,87,80,86)(75,83,91,88,89,93,90,92)

Polycyclic presentation:

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

Conjugacy classes & character table

1052 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 171011);