#27439152

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3040 ⋊ C₂
Sylow	C₅₃ ⋊ #25750

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3040	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#3040	D₅₃
hypocenter	C₅₃	#25750
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3040

1452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

2^8,53|a^2=h,b^2=e,c^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,i^a=i^52,c^b=cg,g^b=gh,d^c=dh,g^c=gh,i^c=i^52

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 224628);