Q₈×M₁₆×C₂₉ #16816677

order	3712
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

Q₈×M₁₆×C₂₉

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
socle	C₂²×C₂₉	C₂³×C₄
prime radical	C₂³×C₂₉	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ G

388 conjugacy classes of subgroups.

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);(10,14)(12,16)

Conjugacy classes & character table

1450 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8907, C₂₉×#8909, C₂₉×#8990, …
maximal normal subgroup of	C₂₉×#8907, C₂₉×#8909, C₂₉×#8990, …
maximal quotient of	C₄⋊C₄×M₁₆×C₂₉, C₂₉×#20520, Q₈×C₂₉×#97, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1695);

Q8×M16×C29 #16816677