C₂₉×#3185 #16816899

order	3712
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂₉×#3185

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
prime radical	C₂³×C₂₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

358 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 1856

A minimal faithful permutation representation — 4 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)(32,35)(33,34)(36,37)(38,39,42,45)(40,46,47,53)(41,50,48,44)(43,51,52,49);(31,34)(35,37)(40,47)(43,52)(46,53)(49,51);(38,40,48,51,42,47,41,49)(39,43,50,53,45,52,44,46);(30,32,33,36)(31,35,34,37)(40,47)(43,52)(46,53)(49,51)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#9416, C₂₉×#11838, C₂₉×#18557, …
maximal normal subgroup of	C₂₉×#9416, C₂₉×#11838, C₂₉×#18557, …
maximal quotient of	C₂₉×#20824, C₂₉×#21287, C₂₉×#21409, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1917);