#27290822

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#421 ⋊ C₂
Sylow	C₅₃ ⋊ #11012

Subgroups

	subgroup	G/H
center	C₂²	#16810499
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#421	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂²×#13133242	C₂
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#421	D₅₃
hypocenter	C₅₃	#11012
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#421

1422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 73 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)(55,57)(60,67)(61,70)(63,71)(65,73)(66,72)(68,69);(58,60)(59,63)(61,67)(62,69)(64,66)(65,71)(68,70)(72,73);(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)(56,57)(58,59)(60,66,69,63)(61,65)(62,64)(67,72,68,71)(70,73)

Polycyclic presentation:

2^8,53|a^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dh,i^a=i^52,c^b=cf,f^b=fh,g^b=gh,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, 76298);