#27281142

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1298 ⋊ C₂
Sylow	C₅₃ ⋊ #9345

Subgroups

	subgroup	G/H
center	C₂²	#16810501
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1298	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂²×#11719945	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1298	D₅₃
hypocenter	C₅₃	#9345
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1298

678 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 97 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:

(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)(62,64)(63,68)(65,71)(66,72)(67,74)(69,77)(70,73)(75,76);(54,56)(55,58,59,61)(57,60)(62,63,65,67,66,69,73,75)(64,68,71,74,72,77,70,76);(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)(56,58,60,61)(57,59)(63,67)(64,70)(65,73)(69,75)(71,72)(74,76)

Polycyclic presentation:

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

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 66618);