#27365302

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #18991

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133186
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3531	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#3531	D₅₃
hypocenter	C₅₃	#18991
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3531

840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150778);