C₂₉×#3291 #16817005

order	3712
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂₉×#3291

Subgroups

	subgroup	G/H
center, socle	C₂×C₂₉	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₉ ◁ C₂×C₄×C₂₉ ◁ G

460 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 45
transitive degree	μ_t(G) = 464

A minimal faithful permutation representation — 4 generators on 45 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)(30,31,36,41)(32,42,43,44)(33,38,37,45)(34,40,35,39);(30,32)(31,37)(33,41)(34,45)(35,38)(36,43)(39,44)(40,42);(30,33)(31,32)(34,44)(35,42)(36,37)(38,39)(40,45)(41,43);(30,33)(31,38)(32,39)(34,42)(35,44)(36,37)(40,43)(41,45)

Conjugacy classes & character table

841 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12314, C₂₉×#12315, C₂₉×#12364, …
maximal normal subgroup of	C₂₉×#12314, C₂₉×#12315, C₂₉×#12364, …
maximal quotient of	C₂₉×#19884, C₂₉×#19888, C₂₉×#19890, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2023);