C₁₇×SD₁₆⋊C₈ #14524584

order	2176
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₁₇×SD₁₆⋊C₈
unambiguous semidirect product	SD₁₆⋊(C₈×C₁₇)

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₇	C₂×D₄
commutator	C₄	C₂²×C₈×C₁₇
Frattini, cosocle	C₄²	C₂³×C₁₇
socle	C₂²×C₁₇	C₄×D₄
prime radical	C₂×D₄×C₁₇	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₄×C₈×C₁₇ ◁ G

176 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 33
transitive degree	μ_t(G) = 1088

A minimal faithful permutation representation — 3 generators on 33 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,22)(21,23)(24,25)(26,27,29,32,30,28,31,33);(19,22)(20,25)(23,24);(19,23)(22,24)(26,28,29,33,30,27,31,32)

Conjugacy classes & character table

748 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₁₇×SD₁₆⋊C₈, #17003725, #17003735, …
maximal subgroup of	C₂×C₁₇×SD₁₆⋊C₈, C₁₇×#17849, C₈×C₁₇×AGL(1,ℤ/8), …
maximal normal subgroup of	C₂×C₁₇×SD₁₆⋊C₈, C₁₇×#17849, C₈×C₁₇×AGL(1,ℤ/8), …
maximal quotient of	C₁₇×#8200, C₁₇×#8201, C₁₇×#8285, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2176, 310);