#27545128

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₄×Q₈×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #34257

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Fitting	C₂²×C₄×Q₈×C₅₃	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂²×#13133265	C₂
hypercenter	C₂²×C₄×Q₈	D₅₃
hypocenter	C₅₃	#34257
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈

5050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 generators on 77 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)(63,69)(67,70)(68,77)(75,76);(62,65,66,74)(63,68,69,77)(64,71,72,73)(67,75,70,76);(54,56,57,55)(59,61);(62,64,66,72)(63,67,69,70)(65,73,74,71)(68,76,77,75);(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)(54,55,57,56)(58,59,60,61)(62,63)(64,70)(65,68)(66,69)(67,72)(71,76)(73,75)(74,77)

Polycyclic presentation:

2^8,53|a^2=h,b^2=f,c^2=gh,d^2=f,b^a=bf,c^a=cg,e^a=ef,i^a=i^52,d^b=df

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, 330604);