C₂₉×#2242 #16815956

order	3712
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₂₉×#2242

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₂₉
prime radical	C₂×D₈×C₂₉	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ C₄×C₈×C₂₉ ◁ G

138 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 3 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)(30,31)(32,35)(33,34)(36,37)(38,39)(40,52)(41,46)(42,60)(43,63)(44,64)(45,51)(47,56)(48,53)(49,62)(50,61)(54,66)(55,65)(57,69)(58,68)(59,67);(31,34)(35,37)(38,40,53,58,44,55,62,59,45,56,43,54,61,57,42,41)(39,46,60,69,50,66,63,47,51,67,49,65,64,68,48,52);(30,32,33,36)(31,35,34,37)(38,41,42,57,61,54,43,56,45,59,62,55,44,58,53,40)(39,47,48,66,64,69,49,46,51,52,63,68,50,65,60,67)

Conjugacy classes & character table

1102 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₂₉×#2242, C₂₉×#33697, C₂₉×#33702, …
maximal normal subgroup of	C₂×C₂₉×#2242, C₂₉×#33697, C₂₉×#33702, …
maximal quotient of	C₂₉×#12889, C₂₉×#13147, C₂₉×#13148, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 974);