#27355828

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1451 ⋊ C₂
Sylow	C₅₃ ⋊ #18063

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1451	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133239	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1451	D₅₃
hypocenter	C₅₃	#18063
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 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)(55,56)(60,61)(62,66,68,64)(63,71,73,75)(65,74,67,76)(69,72,77,70);(54,56,57,61,58,55,59,60)(62,65,68,67)(63,70,73,72)(64,74,66,76)(69,71,77,75);(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)(62,64,68,66)(63,69,73,77)(65,74,67,76)(70,71,72,75);(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)(54,55,57,60,58,56,59,61)(62,63,68,73)(64,71,66,75)(65,72,67,70)(69,76,77,74)

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cg,d^a=dgh,e^a=eg,i^a=i^52,e^b=eg,i^b=i^52,d^c=dgh

Conjugacy classes & character table

1514 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 141304);