C₂₉×#1768 #16815482

order	3712
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₂₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₂₉×#1768

Subgroups

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

260 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 53
transitive degree	μ_t(G) = 3712

A minimal faithful permutation representation — 3 generators on 53 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)(30,31,32,33)(34,35,37,36)(38,39)(40,43)(41,45)(42,44)(46,50)(47,49)(48,51)(52,53);(31,33)(38,40,41,46,42,47,48,52)(39,43,45,50,44,49,51,53);(34,36,37,35)(39,44)(43,49)(45,51)(50,53)

Conjugacy classes & character table

1624 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₂₉×#1768, C₄×C₂₉×#424, C₂₉×#20427, …
maximal normal subgroup of	C₂×C₂₉×#1768, C₄×C₂₉×#424, C₂₉×#20427, …
maximal quotient of	C₂₉×#8264, C₂₉×#8265, C₂₉×#8270, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 500);