#27413215

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #23674

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133189
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2911	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂⁴×C₅₃	#41
hypercenter	#2911	D₅₃
hypocenter	C₅₃	#23674
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2911

766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

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, 198691);