#27300035

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#364 ⋊ C₂
Sylow	C₅₃ ⋊ #12789

Subgroups

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#12789
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

454 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:

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=d,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=ch,i^c=i^52

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85511);