SD₁₆×#6732 #72276873

order	3840
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

non-solvable

Aliases

direct product

SD₁₆×#6732

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄×A₅
commutator	C₄×SL(2,𝔽₅)	C₂³
Frattini, cosocle	C₂×C₄	C₂³×A₅
Fitting, hypercenter	C₄×SD₁₆	A₅
socle	C₂²	C₂×D₄×A₅
prime radical	D₄×#6732	C₂
hypocenter	SL(2,𝔽₅)	C₂×SD₁₆
LCS	G ▷ C₄×SL(2,𝔽₅) ▷ C₂×SL(2,𝔽₅) ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

1036 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 32
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 32 points:

(1,6,2,4)(3,8,5,7)(9,13)(10,27,21,26,32,18,29,11,14,17)(12,22,31,16,30,25,24,15,19,23)(20,28);(1,2)(4,7)(6,8)(9,32,13,17)(10,30,18,23)(11,29,21,27)(12,24,25,22)(14,15,26,31)(16,28,19,20);(1,4,2,6)(3,7,5,8)(9,19,29,20,32,25,13,16,27,28,17,12)(10,14,24,23,15,21,18,26,22,30,31,11)

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

GL(2,𝔽₃)×#6732

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,6,2,4)(3,8,5,7)(9,13)(10,27,21,26,32,18,29,11,14,17)(12,22,31,16,30,25,24,15,19,23)(20,28), (1,2)(4,7)(6,8)(9,32,13,17)(10,30,18,23)(11,29,21,27)(12,24,25,22)(14,15,26,31)(16,28,19,20), (1,4,2,6)(3,7,5,8)(9,19,29,20,32,25,13,16,27,28,17,12)(10,14,24,23,15,21,18,26,22,30,31,11));