#27540978

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #33866

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	D₄×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₄×C₅₃	D₈
hypercenter	D₄×Q₁₆	D₅₃
hypocenter	C₅₃	#33866
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×Q₁₆

548 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)(56,61)(58,60);(62,64,67,75)(63,68,71,85)(65,77,80,93)(66,82,78,74)(69,87,88,92)(70,89,81,84)(72,90,91,76)(73,86,83,79);(54,56)(55,58)(57,61)(59,60);(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)(54,55,57,59)(56,60,61,58)(62,63,67,71)(64,72,75,91)(65,76,80,90)(66,81,78,70)(68,83,85,73)(69,86,88,79)(74,92,82,87)(77,89,93,84)

Polycyclic presentation:

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

Conjugacy classes & character table

965 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326454);