#27296335

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12128

Subgroups

	subgroup	G/H
center	C₂²	#16810809
commutator, cosocle	C₂×C₄⋊C₄×C₅₃	C₂³
Frattini	C₂×C₄⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#1616	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
prime radical	C₅₃×#537	C₂²
hypercenter	#1616	D₅₃
hypocenter	C₅₃	#12128
LCS	G ▷ C₂×C₄⋊C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #1616

408 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 77 points:

(9,11,13,18)(10,14,16,23)(12,21,22,19)(15,24,20,17)(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)(73,74)(75,76);(1,2,4,6)(3,8,7,5)(15,20)(17,24)(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)(74,75)(76,77);(2,5)(3,7)(6,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,24)(21,23)

Polycyclic presentation:

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

Conjugacy classes & character table

626 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81811);