#27296157

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1318 ⋊ C₂
Sylow	C₅₃ ⋊ #12096

Subgroups

	subgroup	G/H
center	C₂²	#16810811
commutator, cosocle	C₂×C₄⋊C₄×C₅₃	C₂³
Frattini	C₂×C₄⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#1318	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
hypercenter	#1318	D₅₃
hypocenter	C₅₃	#12096
LCS	G ▷ C₂×C₄⋊C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #1318

726 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 69 points:

(10,13)(11,16)(14,15)(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);(2,5)(3,8)(6,7)(11,16)(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)(3,5)(4,7)(6,8)(9,10)(11,13)(12,15)(14,16)(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)

Polycyclic presentation:

2^8,53|d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gh,i^a=i^52,d^b=df,e^b=egh,i^b=i^52,d^c=dg,e^c=eh,e^d=eh

Conjugacy classes & character table

704 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81633);