C₂×C₂₉×#405 #16816586

order	3712
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄²×C₂₉ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×C₂₉×#405

Subgroups

	subgroup	G/H
center	C₂²×C₈×C₂₉	C₂²
commutator	C₂	C₂²×C₄²×C₂₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
socle	C₂³×C₂₉	C₂²×C₄
prime radical	C₂⁴×C₂₉	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₉ ◁ G

592 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);(7,8,10,13,11,14,17,20)(9,12,16,19,15,18,21,22);(3,4,5,6)(7,9)(8,12)(10,16)(11,15)(13,19)(14,18)(17,21)(20,22);(9,15)(12,18)(16,21)(19,22)

Conjugacy classes & character table

2320 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8156, C₂₉×#8157, C₂₉×#8601, …
maximal normal subgroup of	C₂₉×#8156, C₂₉×#8157, C₂₉×#8601, …
maximal quotient of	C₂×C₈×M₁₆×C₂₉, C₂×C₂₉×#1450, C₂×C₂₉×#1451, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1604);

C2×C29×#405 #16816586