#27299566

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12690

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133132
commutator	C₂⁴×C₅₃	C₂²×C₄
Frattini	C₂²×D₄	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#534	C₂
socle	C₂²×C₅₃	C₂×#99
cosocle	C₂²×D₄×C₅₃	C₂³
hypercenter	C₂×#534	D₅₃
hypocenter	C₅₃	#12690
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ C₂×#534

1228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(60,67)(63,71)(66,72)(68,73);(58,60)(59,63)(61,68)(62,67)(64,66)(65,71)(69,72)(70,73);(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)(58,59)(60,66)(61,69,70,64)(62,65)(63,68,71,73)(67,72)

Polycyclic presentation:

2^8,53|a^2=f,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^52,c^b=ch,f^b=fg,f^d=fh

Conjugacy classes & character table

1178 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85042);