#27400083

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1908 ⋊ C₂
Sylow	C₅₃ ⋊ #22464

Subgroups

	subgroup	G/H
center	C₂²	#16811257
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1908	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#16811432	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1908	D₅₃
hypocenter	C₅₃	#22464
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1908

1346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

884 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 185559);