#27411534

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #23474

Subgroups

	subgroup	G/H
center	C₄	#16810535
commutator, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₂×C₅₃×#410	C₂
socle	C₂×C₅₃	C₂×#393
prime radical	C₂²×#41×C₅₃	C₂²
hypercenter	C₂×#410	D₅₃
hypocenter	C₅₃	#23474
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#410

1164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

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

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, 197010);