#27294940

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1341 ⋊ C₂
Sylow	C₅₃ ⋊ #11837

Subgroups

	subgroup	G/H
center	C₂	#20273260
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂×M₁₆	C₂²×D₅₃
Fitting	C₅₃×#1341	C₂
socle	C₂×C₅₃	#1514
prime radical	#20285448	C₂
cosocle	C₂×M₁₆×C₅₃	C₂³
hypercenter	#1341	D₅₃
hypocenter	C₅₃	#11837
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #1341

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

normal subgroups only

This group has 324 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) = 69
transitive degree	μ_t(G) = 848

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

Polycyclic presentation:

2^8,53|a^2=e,c^2=fgh,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,g^a=gh,i^a=i^52,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,d^c=dh,i^c=i^52,e^d=eh

Conjugacy classes & character table

548 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 80416);