C₂₉×#1531 #16815245

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1531

Subgroups

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

212 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 3 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)(31,35,36,37)(38,39,41,45,42,44,50,52)(40,46,47,53,48,43,49,51);(30,31)(32,35)(33,37)(34,36)(38,40,42,48)(39,43,44,46)(41,49,50,47)(45,53,52,51);(30,32,34,33)(31,35,36,37)(39,44)(43,46)(45,52)(51,53)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8417, C₂₉×#8418, C₂₉×#8457, …
maximal normal subgroup of	C₂₉×#8417, C₂₉×#8418, C₂₉×#8457, …
maximal quotient of	C₂₉×#7579, C₂₉×#7588, C₂₉×#7591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 263);