#27532832

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1785 ⋊ C₂
Sylow	C₅₃ ⋊ #32893

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133187
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1785	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂×#16811459	C₂
hypercenter	#1785	D₅₃
hypocenter	C₅₃	#32893
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1785

3630 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)(55,57)(58,61,63,59)(60,67,69,64)(62,70,72,65)(66,68,73,71);(62,72)(65,70)(66,73)(68,71);(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)(61,67)(62,71)(63,69)(65,73)(66,70)(68,72);(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)(54,55)(56,57)(58,59,63,61)(60,66,69,73)(62,70,72,65)(64,71,67,68)

Polycyclic presentation:

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

Conjugacy classes & character table

881 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 318308);