C₂₉×#1888 #16815602

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂₉×#1888

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂²×D₄×C₂₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

344 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 112 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 — 3 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)(34,35,40,43,39,36,38,42)(37,41,47,49,46,44,45,48);(30,31,32,33)(34,36,39,35)(37,44,46,41)(38,42,40,43)(45,48,47,49);(34,37,39,46)(35,41,36,44)(38,47,40,45)(42,49,43,48)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7481, C₂₉×#7489, C₂₉×#12774, …
maximal normal subgroup of	C₂₉×#7481, C₂₉×#7489, C₂₉×#12774, …
maximal quotient of	C₂₉×#8960, C₂₉×#8964, C₂₉×#9002, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 620);