SD₁₆⋊#11715906 #27158865

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

unambiguous semidirect product

SD₁₆⋊#11715906

Splittings

Fitting	SD₁₆×C₉₇ ⋊ C₈
Sylow	C₉₇ ⋊ SD₁₆⋊C₈

Subgroups

	subgroup	G/H
center	C₂	D₄×#11715906
commutator	C₄×C₉₇	C₂²×C₈
Frattini	C₄	C₂²×#11715906
Fitting	SD₁₆×C₉₇	C₈
socle	C₂×C₉₇	C₈×D₄
prime radical	D₄×D₉₇	C₂×C₄
cosocle	C₄×#87604	C₂³
hypercenter	SD₁₆	#11715906
hypocenter	C₉₇	SD₁₆⋊C₈
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

176 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 105
transitive degree	μ_t(G) = 776

A minimal faithful permutation representation — 3 generators on 105 points:

(1,2,12,58,95,90,29,11)(3,19,71,50,96,77,56,44)(4,13,61,28,59,32,5,26)(6,10,48,85,75,82,37,18)(7,30,57,94,92,87,45,9)(8,20,14,63,53,83,81,41)(15,39,67,79,51,74,23,55)(16,40,24,47,68,54,34,60)(17,38,33,70,22,49,65,36)(21,73,42,35,91,27,86,62)(25,84,93,52,69,66,72,46)(31,43,78,88,76,97,89,80)(99,103)(102,104);(99,102)(100,105)(103,104);(2,3,5,9,6,4,8,11)(7,14,16,37,17,15,36,18)(10,21,23,46,24,22,45,26)(12,27,29,20,30,28,56,44)(13,31,33,42,34,32,19,41)(25,50,52,68,53,51,82,40)(35,62,48,77,59,60,63,54)(38,43,65,39,66,64,72,55)(47,69,67,88,75,74,73,70)(49,78,79,83,80,61,57,87)(58,89,81,71,84,90,86,91)(76,92,94,97,95,93,96,85)(98,99)(100,102)(101,103)(104,105)

Polycyclic presentation:

2^7,97|a^2=e,c^2=e,d^2=f,e^2=g,b^a=bd,c^a=cf,d^a=df,h^a=h^33,d^b=df,h^c=h^64,h^e=h^22,h^g=h^96

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21639);