#27411354

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #23446

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133186
commutator, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3449	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#3449	D₅₃
hypocenter	C₅₃	#23446
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #3449

1994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

2^8,53|b^2=h,d^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=dh,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, 196830);