C₂×C₄×Q₁₆×C₂₉ #16816652

order	3712
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₄×Q₁₆×C₂₉

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₂₉	D₄
commutator	C₄	C₂³×C₄×C₂₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
socle, prime radical	C₂³×C₂₉	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 3712

A minimal faithful permutation representation — 4 generators on 51 points:

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

Conjugacy classes & character table

1624 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8946, C₂₉×#8950, C₂₉×#9656, …
maximal normal subgroup of	C₂₉×#8946, C₂₉×#8950, C₂₉×#9656, …
maximal quotient of	C₂×C₈×Q₁₆×C₂₉, C₂×C₂₉×#1584, C₂×C₂₉×#1591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1670);

C2×C4×Q16×C29 #16816652