#27292919

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11364

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#427	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#427	D₅₃
hypocenter	C₅₃	#11364
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#427

606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(54,57,58,56)(55,59,60,61)(62,65)(63,69)(64,73)(66,81)(67,79)(68,83)(70,76)(71,89)(72,90)(74,92)(75,91)(77,80)(78,82)(84,93)(85,88)(86,87);(54,56,58,57)(55,59,60,61)(62,64)(63,68)(65,77)(66,80)(67,75)(69,85)(70,72)(71,88)(73,78)(74,82)(76,93)(79,92)(81,91)(83,86)(84,87)(89,90);(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)(64,72,75,68)(65,76,79,69)(66,71)(67,70)(73,83)(74,88,80,84)(77,85)(78,89,81,86)(82,87)(90,91)(92,93)

Polycyclic presentation:

2^8,53|a^2=g,b^2=de,c^2=de,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,i^a=i^52,c^b=cf,f^b=fh,g^b=gh,f^c=fh,g^c=gh

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78395);