C₂₉×#1977 #16815691

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

Aliases

direct product

C₂₉×#1977

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#121
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

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

normal subgroups only

This group has 296 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)(34,35,39,43,40,42,38,36)(37,45,47,49,48,44,46,41);(30,31,32,33)(34,36,39,42,40,43,38,35)(37,44,47,49,48,45,46,41);(34,37,38,46,40,48,39,47)(35,41,43,45,42,49,36,44)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1977, C₂₉×#21248, C₂₉×#21249, …
maximal normal subgroup of	C₂×C₂₉×#1977, C₂₉×#21248, C₂₉×#21249, …
maximal quotient of	C₂₉×#9667, C₂₉×#9672, C₂₉×#9723, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 709);