Q₈×D₈×C₂₉ #16817092

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

Q₈×D₈×C₂₉

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
prime radical	C₂×D₈×C₂₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄×Q₈×C₂₉ ◁ G

404 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 208 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) = 45
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 4 generators on 45 points:

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

Conjugacy classes & character table

1015 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12352, C₂₉×#12356, C₂₉×#12404, …
maximal normal subgroup of	C₂₉×#12352, C₂₉×#12356, C₂₉×#12404, …
maximal quotient of	Q₈×C₂₉×#102, C₄⋊C₄×D₈×C₂₉, Q₈×C₂₉×#107, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2110);