C₂₉×#1388 #16815102

order	3712
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1388

Subgroups

	subgroup	G/H
center	C₄×C₂₉	#102
commutator	C₈	C₄²×C₂₉
Frattini, cosocle	C₂²×C₈	C₂²×C₂₉
socle	C₂×C₂₉	#340
prime radical	C₂³×C₂₉	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ C₂²×C₈×C₂₉ ◁ G

124 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 61
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 2 generators on 61 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)(30,31,34,40,37,42,36,53)(32,43,45,55,48,57,47,61)(33,49,51,58,52,41,35,39)(38,46,54,50,56,60,59,44);(30,32,35,46,36,47,52,44,37,48,51,60,34,45,33,50)(31,38,41,55,40,54,39,57,42,56,49,61,53,59,58,43)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1388, C₂₉×#12646, C₂₉×#12651, …
maximal normal subgroup of	C₂×C₂₉×#1388, C₂₉×#12646, C₂₉×#12651, …
maximal quotient of	C₂₉×#7158, C₂₉×#7176, C₂₉×#7177, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 120);