#27538149

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×B(4,𝔽₂) ⋊ C₂
Sylow	C₅₃ ⋊ #33551

Subgroups

	subgroup	G/H
center	C₂	C₂×#16810809
commutator, cosocle	C₂×D₄×C₅₃	C₂⁴
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₂×C₅₃×B(4,𝔽₂)	C₂
socle	C₂×C₅₃	C₂×B(4,𝔽₂)
hypercenter	C₂×B(4,𝔽₂)	D₅₃
hypocenter	C₅₃	#33551
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×B(4,𝔽₂)

1326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 4 generators on 69 points:

(3,10)(6,13)(9,15)(11,16)(17,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19);(4,12)(11,16);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(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)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

866 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323625);