#27271972

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7538

Subgroups

	subgroup	G/H
center	C₂²	#16810458
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₅₃×#1423	C₂
socle, prime radical	C₂²×C₅₃	D₃₂
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	#1423	D₅₃
hypocenter	C₅₃	#7538
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #1423

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 2 generators on 121 points:

(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)(58,59,64,69)(60,71,75,100)(61,76,79,111)(62,80,82,114)(63,83,81,68)(65,84,87,70)(66,89,91,72)(67,93,94,77)(73,104,106,120)(74,108,105,99)(78,113,101,110)(85,117,118,103)(86,98,95,107)(88,115,119,96)(90,102,109,112)(92,116,121,97);(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)(54,55,56,57)(59,65)(60,70)(61,72)(62,77)(63,81)(66,88)(67,92)(68,95)(69,87)(71,76)(73,103)(74,107)(75,84)(78,112)(79,89)(80,85)(82,93)(83,86)(90,120)(91,119)(94,121)(96,97)(98,105)(99,113)(100,111)(101,102)(104,109)(106,117)(108,110)(114,118)(115,116)

Polycyclic presentation:

2^8,53|a^2=d,b^2=h,c^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=ce,e^b=ef,f^b=fg,g^b=gh,i^b=i^52

Conjugacy classes & character table

1058 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57448);