C₂₉×#2121 #16815835

order	3712
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#2121

Subgroups

	subgroup	G/H
center, socle	C₂×C₂₉	C₂×#99
commutator	C₂³	C₂²×C₄×C₂₉
Frattini, cosocle	C₂×D₄	C₂³×C₂₉
prime radical	C₂₉×#534	C₂
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₉ ◁ C₂³×C₂₉ ◁ C₂²×D₄×C₂₉ ◁ G

280 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 84 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)(33,41,42,36)(34,37)(35,40,43,45)(39,44);(30,32)(31,35)(33,40)(34,39)(36,38)(37,43)(41,44)(42,45);(32,39)(35,43)(38,44)(40,45)

Conjugacy classes & character table

667 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12705, C₂₉×#12706, C₂₉×#12725, …
maximal normal subgroup of	C₂₉×#12705, C₂₉×#12706, C₂₉×#12725, …
maximal quotient of	C₂₉×#11426, C₂₉×#11453, C₂₉×#11473, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 853);