C₂₉×#2197 #16815911

order	3712
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#2197

Subgroups

	subgroup	G/H
center, socle	C₂×C₂₉	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂³×C₂₉
prime radical	C₂₉×B(4,𝔽₂)	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₉ ◁ C₂²×C₂₉ ◁ C₂×D₄×C₂₉ ◁ G

258 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 3 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)(32,38)(33,36)(34,37)(35,39)(40,44)(41,43)(42,45);(30,32,34,40)(31,35,37,42)(33,39,41,45)(36,38,43,44);(33,41)(39,45)

Conjugacy classes & character table

580 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#2197, C₂₉×#33546, C₂₉×#33551, …
maximal normal subgroup of	C₂×C₂₉×#2197, C₂₉×#33546, C₂₉×#33551, …
maximal quotient of	C₂₉×#12091, C₂₉×#12092, C₂₉×#12098, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 929);