#27537236

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#364 ⋊ C₂
Sylow	C₅₃ ⋊ #33428

Subgroups

	subgroup	G/H
center	C₈	#13133175
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂×C₅₃	C₂×#432
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#33428
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#364

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

2^8,53|a^2=f,c^2=e,f^2=g,g^2=h,b^a=be,c^a=cg,e^a=eh,d^b=dh,f^b=fh,i^b=i^52,d^c=dh,f^c=fh,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, 322712);