C₂₉×#1459 #16815173

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂₉×#1459

Subgroups

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

332 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 124 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)(32,36,37,34)(33,35)(38,39,41,44,42,45,40,43);(30,32)(31,34)(33,37)(35,36);(38,40,42,41)(39,43,45,44)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#7203, C₂₉×#7205, C₂₉×#7285, …
maximal normal subgroup of	C₂₉×#7203, C₂₉×#7205, C₂₉×#7285, …
maximal quotient of	C₂₉×#7565, C₂₉×#7566, C₂₉×#7582, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 191);