#27415697

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×B(4,𝔽₂) ⋊ C₂
Sylow	C₅₃ ⋊ #23903

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₂×C₅₃×B(4,𝔽₂)	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	C₂×B(4,𝔽₂)	D₅₃
hypocenter	C₅₃	#23903
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×B(4,𝔽₂)

2096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

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

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, 201173);