C₂₉×#1810 #16815524

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

Aliases

direct product

C₂₉×#1810

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
socle	C₂²×C₂₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₉	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₂³×C₄×C₂₉ ◁ G

292 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 128 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)(32,33)(34,35,38,41,39,42,37,47)(36,43,45,48,46,49,44,40);(34,36,37,44,39,46,38,45)(35,40,41,43,42,48,47,49);(31,33)

Conjugacy classes & character table

1276 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#7231, C₂₉×#7232, C₂₉×#7234, …
maximal normal subgroup of	C₂₉×#7231, C₂₉×#7232, C₂₉×#7234, …
maximal quotient of	C₂₉×#8595, C₂₉×#8596, C₂₉×#8600, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 542);