C₂₉×#3261 #16816975

order	3712
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂₉×#3261

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 ◁ C₂₉×#117 ◁ G

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

normal subgroups only

This group has 322 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) = 69
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 4 generators on 69 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)(31,34)(32,36)(38,39,43,47)(40,44,49,58)(41,51,53,61)(42,54,56,46)(45,55,60,52)(48,62,63,69)(50,66,65,59)(57,67,68,64);(30,31)(32,35)(33,34)(36,37)(38,40,43,49)(39,44,47,58)(41,48,53,63)(42,50,56,65)(45,57,60,68)(46,59,54,66)(51,62,61,69)(52,64,55,67);(31,34)(35,37)(38,41,43,53)(39,45,47,60)(40,48,49,63)(42,55,56,52)(44,57,58,68)(46,61,54,51)(50,67,65,64)(59,69,66,62);(30,32,33,36)(31,35,34,37)(40,49)(44,58)(48,63)(50,65)(57,68)(59,66)(62,69)(64,67)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12165, C₂₉×#12168, C₂₉×#12171, …
maximal normal subgroup of	C₂₉×#12165, C₂₉×#12168, C₂₉×#12171, …
maximal quotient of	C₂₉×#21898, C₂₉×#21968, C₂₉×#22124, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1993);