#27296808

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2029 ⋊ C₂
Sylow	C₅₃ ⋊ #12202

Subgroups

	subgroup	G/H
center	C₂²	#16810809
commutator, cosocle	C₂×C₂²⋊C₄×C₅₃	C₂³
Frattini	C₂×C₂²⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#2029	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
prime radical	#20278028	C₂
hypercenter	#2029	D₅₃
hypocenter	C₅₃	#12202
LCS	G ▷ C₂×C₂²⋊C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #2029

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

normal subgroups only

This group has 976 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) = 848

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

(11,15)(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)(10,13)(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);(1,2,4,7)(3,5,8,6)(9,10)(11,13)(12,14)(15,16)

Polycyclic presentation:

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

Conjugacy classes & character table

707 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 82284);