C₂₉×#3004 #16816718

order	3712
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂₉×#3004

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
prime radical	C₂₉×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂³×C₄×C₂₉ ◁ G

738 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 224 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) = 928

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)(30,31)(32,33)(34,35)(36,42)(37,45)(38,40)(39,43)(41,47)(44,49)(46,48);(34,36,38,44)(35,39,40,48)(37,46,47,43)(41,49,45,42);(31,33);(35,40)(39,48)(41,45)(42,49)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#9519, C₂₉×#18136, C₂₉×#18146, …
maximal normal subgroup of	C₂₉×#9519, C₂₉×#18136, C₂₉×#18146, …
maximal quotient of	C₂₉×#18133, C₂₉×#18138, C₂₉×#18143, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1736);