#27270802

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1322 ⋊ C₂
Sylow	C₅₃ ⋊ #7088

Subgroups

	subgroup	G/H
center	C₂²	#16810358
commutator	C₂²×C₈×C₅₃	C₂×C₄
Frattini	C₂×C₄⋊C₈	C₂×D₅₃
Fitting	C₅₃×#1322	C₂
socle	C₂²×C₅₃	#327
prime radical	C₂²×D₄₂₄	C₄
cosocle	C₂×C₄⋊C₈×C₅₃	C₂²
hypercenter	#1322	D₅₃
hypocenter	C₅₃	#7088
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

248 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 49 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 — 2 generators on 77 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)(55,58)(59,60)(63,67)(64,71)(65,72)(68,69)(70,76)(73,77)(74,75);(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)(54,55,56,59,57,60,61,58)(62,63)(64,67)(65,74)(66,70)(68,71)(69,72)(73,76)(75,77)

Polycyclic presentation:

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

Conjugacy classes & character table

1178 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 56278);