C₂₉×#3152 #16816866

order	3712
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂₉×#3152

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
prime radical	C₂₉×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

552 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 224 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) = 49
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 4 generators on 49 points:

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

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#9421, C₂₉×#9429, C₂₉×#9435, …
maximal normal subgroup of	C₂₉×#9421, C₂₉×#9429, C₂₉×#9435, …
maximal quotient of	C₂₉×#19622, C₂₉×#19626, C₂₉×#19630, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1884);