C₂₉×#3150 #16816864

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂₉×#3150

Subgroups

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

564 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 49
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 4 generators on 49 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)(31,33)(34,35)(36,43)(37,41)(38,47)(39,42)(40,46)(44,49)(45,48);(34,36,38,46,39,44,37,45)(35,40,41,43,42,48,47,49);(30,31,32,33)(36,44)(40,48)(43,49)(45,46);(34,37,39,38)(35,41,42,47)(36,45,44,46)(40,43,48,49)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#9007, C₂₉×#10032, C₂₉×#10034, …
maximal normal subgroup of	C₂₉×#9007, C₂₉×#10032, C₂₉×#10034, …
maximal quotient of	C₂₉×#19608, C₂₉×#19613, C₂₉×#19616, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1882);