#27271879

order	13568
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7502

Subgroups

	subgroup	G/H
center	C₂²	#16810417
commutator	C₄²×C₅₃	C₄²
Frattini	C₂×#124	C₂×D₅₃
Fitting	C₂×C₅₃×#355	C₂
socle	C₂²×C₅₃	#354
prime radical	C₂²×D₄×C₅₃	D₄
cosocle	C₂×#124×C₅₃	C₂²
hypercenter	C₂×#355	D₅₃
hypocenter	C₅₃	#7502
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#124 ◁ C₂×#355

286 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 63 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) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 2 generators on 89 points:

(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(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)(54,55,56,57)(58,60,62,73,64,74,63,72)(59,65,67,79,69,81,68,80)(61,71,77,87,76,89,78,88)(66,70,82,84,83,86,75,85);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(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)(58,59)(60,70,72,84)(61,75,76,82)(62,68,63,67)(64,69)(65,71,79,87)(66,78)(73,85,74,86)(77,83)(80,88,81,89)

Polycyclic presentation:

2^8,53|a^2=d,b^2=e,c^2=g,e^2=h,f^2=h,b^a=bc,c^a=cf,e^a=eg,f^a=fg,g^a=gh,i^a=i^52,d^b=dfgh,i^b=i^52,d^c=dgh,e^d=eh,f^d=fh

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 57355);