C₂₉×#3293 #16817007

order	3712
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂₉×#3293

Subgroups

	subgroup	G/H
center, socle	C₂×C₂₉	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₉ ◁ C₂×C₄×C₂₉ ◁ G

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

normal subgroups only

This group has 444 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) = 61
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 4 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)(30,31,37,42)(32,43,46,48)(33,45,38,52)(34,39,51,57)(35,41,36,40)(44,59,61,55)(47,60,54,49)(50,58,53,56);(30,32,37,46)(31,38,42,33)(34,49,51,60)(35,52,36,45)(39,55,57,59)(40,48,41,43)(44,53,61,50)(47,56,54,58);(30,33,37,38)(31,32,42,46)(34,47,51,54)(35,48,36,43)(39,44,57,61)(40,45,41,52)(49,58,60,56)(50,59,53,55);(30,34)(31,39)(32,44)(33,47)(35,53)(36,50)(37,51)(38,54)(40,58)(41,56)(42,57)(43,59)(45,60)(46,61)(48,55)(49,52)

Conjugacy classes & character table

841 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12311, C₂₉×#12313, C₂₉×#12367, …
maximal normal subgroup of	C₂₉×#12311, C₂₉×#12313, C₂₉×#12367, …
maximal quotient of	C₂₉×#19887, C₂₉×#19894, C₂₉×#19902, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2025);