#27281157

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #9347

Subgroups

	subgroup	G/H
center	C₂²	#16810503
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting, prime radical	C₅₃×#3006	C₂
socle	C₂²×C₅₃	#386
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#3006	D₅₃
hypocenter	C₅₃	#9347
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #3006

704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 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)(62,64)(63,69)(65,72)(66,73)(67,74)(68,71)(70,77)(75,76);(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)(54,56,57,60)(55,58)(59,61)(62,63,68,75,66,70,65,67)(64,69,71,76,73,77,72,74);(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)(56,58,60,61)(57,59)(63,67)(64,71)(65,68)(70,75)(72,73)(74,76)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfg,d^2=gh,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,i^a=i^52,d^b=dh,f^b=fh,i^b=i^52,e^c=egh

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, 66633);