C₂₉×#2123 #16815837

order	3712
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#2123

Subgroups

	subgroup	G/H
center, socle	C₂×C₂₉	C₂×#99
commutator	C₂³	C₂²×C₄×C₂₉
Frattini, cosocle	C₂×D₄	C₂³×C₂₉
prime radical	C₂×#41×C₂₉	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₉ ◁ C₂×C₄×C₂₉ ◁ C₂×#41×C₂₉ ◁ G

234 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 3 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)(32,40)(33,37)(34,47,48,38)(35,39)(36,42,50,57)(41,54)(43,55)(44,58)(45,60,61,52)(46,53)(49,56,51,59);(30,32,35,43)(31,36,39,50)(33,44,46,41)(34,42,48,57)(37,51,53,49)(38,40,47,55)(45,59,61,56)(52,58,60,54);(30,33,35,46)(31,37,39,53)(32,41,43,44)(34,45,48,61)(36,49,50,51)(38,52,47,60)(40,54,55,58)(42,56,57,59)

Conjugacy classes & character table

667 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12716, C₂₉×#12717, C₂₉×#12723, …
maximal normal subgroup of	C₂₉×#12716, C₂₉×#12717, C₂₉×#12723, …
maximal quotient of	C₂₉×#11428, C₂₉×#11454, C₂₉×#11474, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 855);