#27539619

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #33732

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#499	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×#499	D₅₃
hypocenter	C₅₃	#33732
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

482 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:

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

Polycyclic presentation:

2^8,53|a^2=e,b^2=eh,c^2=fg,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,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, 325095);