C₂₉×#1857 #16815571

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#1857

Subgroups

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

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

normal subgroups only

This group has 272 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)(30,31,33,32)(35,39)(37,44)(40,48)(41,45)(42,46)(47,49);(30,32,33,31)(34,35)(36,40)(37,45)(38,41)(39,44)(42,47)(43,49)(46,48);(34,36,38,43)(35,40,41,49)(37,46,44,42)(39,47,45,48)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7376, C₂₉×#12970, C₂₉×#12971, …
maximal normal subgroup of	C₂₉×#7376, C₂₉×#12970, C₂₉×#12971, …
maximal quotient of	C₂₉×#8836, C₂₉×#8838, C₂₉×#8855, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 589);