#27314750

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#388 ⋊ C₂
Sylow	C₅₃ ⋊ #14623

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Fitting	C₂×C₅₃×#388	C₂
socle, cosocle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂×#16810532	C₂
hypercenter	C₂×#388	D₅₃
hypocenter	C₅₃	#14623
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#388

1910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 77 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)(64,68)(66,69)(70,72,73,76)(71,75,74,77);(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)(59,61)(63,67)(66,69)(72,76)(75,77);(58,59)(60,61)(62,64)(63,66)(65,68)(67,69)(71,74)(72,76);(54,55,56,57)(62,63)(64,66)(65,67)(68,69)(70,71)(72,75)(73,74)(76,77)

Polycyclic presentation:

2^8,53|a^2=h,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de,i^c=i^52

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, 100226);