#27291273

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11086

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133203
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#416	C₂
socle	C₂²×C₅₃	#397
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#416	D₅₃
hypocenter	C₅₃	#11086
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#416

866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 89 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)(58,61)(59,66)(60,70)(62,78)(63,77)(64,76)(65,79)(67,74)(68,85)(69,84)(71,86)(72,87)(73,75)(80,88)(81,83)(82,89);(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)(59,65)(61,75,76,87)(62,72)(63,73)(64,71)(66,83,74,88)(67,82)(68,80)(69,81)(70,77,86,78)(79,84,89,85);(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)(58,59,63,69,64,67,62,68)(60,65,73,81,71,82,72,80)(61,74,78,84,76,66,77,85)(70,79,87,88,86,89,75,83)

Polycyclic presentation:

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

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 76749);