C₂₉×#1886 #16815600

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂₉×#1886

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂³×C₂₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

324 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 112 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)(31,33)(34,35,40,43,39,44,38,42)(36,45,47,41,48,49,46,37);(30,31,32,33)(34,36)(35,41)(37,44)(38,47)(39,48)(40,46)(42,49)(43,45);(34,37,39,41)(35,36,44,48)(38,45,40,49)(42,47,43,46)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7483, C₂₉×#7491, C₂₉×#12775, …
maximal normal subgroup of	C₂₉×#7483, C₂₉×#7491, C₂₉×#12775, …
maximal quotient of	C₂₉×#8958, C₂₉×#8962, C₂₉×#9000, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 618);