SD₁₆⋊#11719908 #27295904

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

unambiguous semidirect product

SD₁₆⋊#11719908

Splittings

Sylow

C₅₃ ⋊ SD₁₆⋊C₁₆

Subgroups

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₈×SD₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×D₄×C₅₃	C₂×C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₈×SD₁₆	D₅₃
hypocenter	C₅₃	SD₁₆⋊C₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×SD₁₆

232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|a^2=e,c^2=e,d^2=f,e^2=g,g^2=h,b^a=bd,c^a=cf,d^a=df,i^a=i^52,d^b=df,i^c=i^52

Conjugacy classes & character table

1544 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81380);

SD16⋊#11719908 #27295904