C₂×C₂₉×#427 #16816623

order	3712
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×C₂₉×#427

Subgroups

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

512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 1856

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)(9,16)(10,14)(11,20)(12,15)(13,18)(17,21)(19,22);(7,9,11,18,12,19,10,17)(8,13,14,16,15,21,20,22);(7,10,12,11)(8,14,15,20)(9,17,19,18)(13,16,21,22)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#10859, C₂₉×#10860, C₂₉×#10866, …
maximal normal subgroup of	C₂₉×#10859, C₂₉×#10860, C₂₉×#10866, …
maximal quotient of	C₂×C₂₉×#1564, C₂×C₂₉×#1568, C₂₉×#17793, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1641);

C2×C29×#427 #16816623