#27280199

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1492 ⋊ C₂
Sylow	C₅₃ ⋊ #9210

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1492	C₂
socle	C₂²×C₅₃	#386
prime radical	#20284164	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1492	D₅₃
hypocenter	C₅₃	#9210
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1492

510 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 65675);