C₂₉×#1585 #16815299

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂₉×#1585

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₄²	C₂³×C₂₉
socle	C₂²×C₂₉	C₄×D₄
prime radical	C₂×D₄×C₂₉	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₄×C₈×C₂₉ ◁ G

168 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 45
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 3 generators on 45 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)(32,34)(33,35)(36,37)(38,39,41,40,42,43,45,44);(31,34)(32,37)(35,36);(31,35)(34,36)(38,40,41,43,42,44,45,39)

Conjugacy classes & character table

1102 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₂₉×#1585, C₂₉×#17868, C₂₉×#17869, …
maximal normal subgroup of	C₂×C₂₉×#1585, C₂₉×#17868, C₂₉×#17869, …
maximal quotient of	C₂₉×#8204, C₂₉×#8213, C₂₉×#8289, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 317);