#27537315

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 ⋊ #33438

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₄×M₃₂×C₅₃	C₂
prime radical	#16811173	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#33438
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 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)(58,61)(59,66)(60,69)(62,73)(63,74)(64,72)(65,77)(67,79)(68,80)(70,82)(71,83)(75,86)(76,85)(78,87)(81,88)(84,89);(58,60,62,70,63,71,75,84,64,59,65,67,76,68,78,81)(61,66,73,79,74,80,86,88,72,69,77,82,85,83,87,89);(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)(61,72)(66,69)(73,77)(74,85)(79,82)(80,83)(86,87)(88,89);(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)(56,57)(58,59,62,67,63,68,75,81,64,60,65,70,76,71,78,84)(61,66,73,79,74,80,86,88,72,69,77,82,85,83,87,89)

Polycyclic presentation:

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

Conjugacy classes & character table

2180 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322791);