C₂₉×#1490 #16815204

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

Aliases

direct product

C₂₉×#1490

Subgroups

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

316 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 928

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)(32,36)(33,35)(34,37)(38,39,40,43,41,42,44,45);(31,35)(32,33)(36,37);(30,32,34,33)(31,36,37,35)(39,42)(43,45)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#8367, C₂₉×#8370, C₂₉×#8372, …
maximal normal subgroup of	C₂₉×#8367, C₂₉×#8370, C₂₉×#8372, …
maximal quotient of	C₂₉×#7606, C₂₉×#7607, C₂₉×#7608, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 222);