SD₁₆×#444 #72309681

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

SD₁₆×#444

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄²
commutator	C₄²	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂²	C₄×D₄²
prime radical	D₄×#135	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(9,10,13,15,14,16,11,17)(12,20,22,18,21,23,19,24);(2,5)(3,7)(6,8)(9,11,14,13)(10,15,16,17)(12,19,21,22)(18,23,24,20);(1,2,4,6)(3,8,7,5)(10,16)(15,17)(18,24)(20,23);(10,17,16,15)(12,21)(18,23,24,20)(19,22);(9,12,13,22,14,21,11,19)(10,18,17,20,16,24,15,23)

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309705, #72309909, #72309999, …
maximal normal subgroup of	#72309705, #72309909, #72309999, …

GAP

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

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

SD16×#444 #72309681