#27412659

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #23603

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133189
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#419	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂⁴×C₅₃	#41
hypercenter	C₂×#419	D₅₃
hypocenter	C₅₃	#23603
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#419

970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 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)(60,71)(65,80)(69,86)(70,87)(72,78)(73,88)(79,89)(81,85);(55,57)(58,61,63,75)(59,66,68,83)(60,70,71,87)(62,77,64,74)(65,79,80,89)(67,84,76,82)(69,85,86,81)(72,88,78,73);(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)(58,60,62,72,63,71,64,78)(59,65,67,69,68,80,76,86)(61,73,74,87,75,88,77,70)(66,81,82,89,83,85,84,79);(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)(58,59,64,67,63,68,62,76)(60,69,78,80,71,86,72,65)(61,66,77,82,75,83,74,84)(70,85,88,79,87,81,73,89)

Polycyclic presentation:

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

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 198135);