#27539604

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #33730

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3164	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	#3164	D₅₃
hypocenter	C₅₃	#33730
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|a^2=e,b^2=e,c^2=fgh,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,i^c=i^52

Conjugacy classes & character table

884 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 325080);