#27537348

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₄×M₃₂×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #33443

Subgroups

	subgroup	G/H
center	C₈	C₂²×D₂₁₂
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂×C₅₃	C₂×C₈×D₄
prime radical	C₂×#13133239	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#33443
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ C₄×M₃₂

648 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)(55,62)(57,71)(61,77)(63,65)(64,78)(66,80)(72,85)(81,82);(54,57,59,66,58,72,73,77,60,71,75,80,74,85,79,61)(55,56,64,68,65,69,82,83,62,70,78,84,63,67,81,76);(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)(55,61)(57,64)(58,69)(59,68)(60,70)(62,77)(63,80)(65,66)(67,74)(71,78)(72,82)(73,83)(75,84)(76,79)(81,85);(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)(54,55,59,64,58,65,73,82,60,62,75,78,74,63,79,81)(56,66,68,72,69,77,83,71,70,80,84,85,67,61,76,57)

Polycyclic presentation:

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

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, 322824);