C₂₉×#3138 #16816852

order	3712
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂₉×#3138

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
prime radical	C₂×C₂₉×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

484 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 928

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

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#9424, C₂₉×#9425, C₂₉×#9438, …
maximal normal subgroup of	C₂₉×#9424, C₂₉×#9425, C₂₉×#9438, …
maximal quotient of	C₂₉×#20764, C₂₉×#21095, C₂₉×#21203, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1870);