C₁₇×Q₈⋊C₃₂ #16865136

order	4352
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₇×C₃₂ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₁₇×Q₈⋊C₃₂
unambiguous semidirect product	Q₈⋊(C₁₇×C₃₂)

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₁₇	D₄
commutator	C₄	C₂×C₁₇×C₃₂
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₇
socle, prime radical	C₂²×C₁₇	C₂²⋊C₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₇ ◁ C₄×C₁₆×C₁₇ ◁ G

118 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 118 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 57
transitive degree	μ_t(G) = 4352

A minimal faithful permutation representation — 2 generators on 57 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(19,22)(20,24)(23,25)(26,27,28,32,29,33,36,42,30,34,37,43,39,45,48,52,31,35,38,44,40,46,49,53,41,47,50,54,51,55,56,57);(18,19,21,23)(20,25,24,22)

Conjugacy classes & character table

1904 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 380);