Q₈×SD₁₆×C₂₃ #15670998

order	2944
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

Q₈×SD₁₆×C₂₃

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂×D₄×C₂₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×Q₈×C₂₃ ◁ G

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

normal subgroups only

This group has 370 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) = 39
transitive degree	μ_t(G) = 1472

A minimal faithful permutation representation — 4 generators on 39 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);(1,5,3,7)(2,6,4,8);(9,10,11,12,13,14,15,16);(10,12)(11,15)(14,16)

Conjugacy classes & character table

805 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₃×#19765, C₂₃×#19769, C₂₃×#19796, …
maximal normal subgroup of	C₂₃×#19765, C₂₃×#19769, C₂₃×#19796, …
maximal quotient of	Q₈×C₂₃×#102, Q₈×C₂₃×#103, C₄⋊C₄×SD₁₆×C₂₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 2111);