C₂₉×#1510 #16815224

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂₉×#1510

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂×C₂₉×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂³×C₄×C₂₉ ◁ G

326 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 92 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)(40,42)(41,43)(44,45);(30,32)(31,34)(33,37)(35,36)(39,42)(40,45)(43,44);(39,43)(42,44)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8487, C₂₉×#8516, C₂₉×#8517, …
maximal normal subgroup of	C₂₉×#8487, C₂₉×#8516, C₂₉×#8517, …
maximal quotient of	C₂₉×#7592, C₂₉×#7593, C₂₉×#7599, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 242);