#27466052

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#427 ⋊ C₂
Sylow	C₅₃ ⋊ #27673

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133188
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#427	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#16811134	C₂²
hypercenter	C₂×#427	D₅₃
hypocenter	C₅₃	#27673
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#427

696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 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)(62,66,67,79,69,78,68,64)(63,72,73,70,75,76,74,86)(65,81,82,92,83,93,84,77)(71,87,80,85,88,90,89,91);(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)(56,60)(58,61)(62,65)(63,71)(64,77)(66,81)(67,82)(68,84)(69,83)(70,85)(72,87)(73,80)(74,89)(75,88)(76,90)(78,93)(79,92)(86,91);(54,56)(55,58)(57,60)(59,61)(62,64,68,78,69,79,67,66)(63,70,74,72,75,86,73,76)(65,77,84,93,83,92,82,81)(71,85,89,87,88,91,80,90);(54,55,57,59)(56,58,60,61)(62,63)(64,76)(65,80)(66,70)(67,74)(68,73)(69,75)(71,82)(72,79)(77,91)(78,86)(81,90)(83,89)(84,88)(85,92)(87,93)

Polycyclic presentation:

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

Conjugacy classes & character table

1502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 251528);