#27298922

order	13568
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12541

Subgroups

	subgroup	G/H
center	C₂×C₈	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#12541
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,c^2=f,d^2=g,g^2=h,b^a=bg,e^a=eh,c^b=ch,d^b=dh,i^b=i^52

Conjugacy classes & character table

2168 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84398);