#27299897

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#352 ⋊ C₂
Sylow	C₅₃ ⋊ #12760

Subgroups

	subgroup	G/H
center	C₂²	#16810811
commutator	C₂×D₄×C₅₃	C₂²×C₄
Frattini	C₂²×D₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#352	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
prime radical	C₂×#13133213	C₂²
cosocle	C₂²×D₄×C₅₃	C₂³
hypercenter	C₂×#352	D₅₃
hypocenter	C₅₃	#12760
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×#352

876 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 107 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) = 73
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 73 points:

(5,7,9,15)(6,11,13,20)(8,17,16,14)(10,18,12,19)(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)(69,70)(71,72);(6,10)(8,16)(11,18)(12,13)(14,17)(19,20)(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)(70,71)(72,73);(1,2,3,4)(5,6)(7,11)(8,12,16,10)(9,13)(14,19,17,18)(15,20)

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85373);