C₂₉×#3292 #16817006

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

Aliases

direct product

C₂₉×#3292

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

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

normal subgroups only

This group has 476 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)(32,43)(33,39)(34,41)(35,40)(36,42)(37,44)(38,45);(30,32,36,38)(31,37,42,33)(34,39,35,44)(40,45,41,43);(30,33)(31,38)(32,42)(34,45)(35,43)(36,37)(39,40)(41,44);(30,33)(31,39)(32,41)(34,43)(35,45)(36,37)(38,40)(42,44)

Conjugacy classes & character table

841 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12310, C₂₉×#12312, C₂₉×#12366, …
maximal normal subgroup of	C₂₉×#12310, C₂₉×#12312, C₂₉×#12366, …
maximal quotient of	C₂₉×#19885, C₂₉×#19893, C₂₉×#19898, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2024);