#27298801

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12514

Subgroups

	subgroup	G/H
center	C₂²	#16810999
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1710	C₂
socle, prime radical	C₂²×C₅₃	#493
hypercenter	#1710	D₅₃
hypocenter	C₅₃	#12514
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 101 points:

(1,2)(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)(55,60)(59,67)(61,65)(62,68)(70,73,93,90,77,88,75,91,78,87,74,89,76,94,95,72)(71,80,99,98,84,96,82,92,85,86,81,97,83,100,101,79);(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)(54,56,57,63,58,64,66,69)(55,59,61,62,60,67,65,68)(70,72,75,88,76,89,93,73,78,91,95,94,77,90,74,87)(71,79,82,96,83,97,99,80,85,92,101,100,84,98,81,86);(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)(56,62)(57,65)(58,60)(59,63)(61,66)(64,68)(67,69)(70,71,78,85)(72,86,91,80)(73,92,87,79)(74,82,93,101)(75,81,95,99)(76,84,77,83)(88,98,94,97)(89,100,90,96)

Polycyclic presentation:

2^8,53|a^2=h,b^2=dgh,c^2=ef,d^2=fh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gh,i^a=i^52,c^b=ch,i^b=i^52,i^c=i^52

Conjugacy classes & character table

1034 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84277);