C₂₉×#1529 #16815243

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×#1529

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₄²	C₂³×C₂₉
prime radical	C₂₉×#121	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

216 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 88 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)(31,34,35,36)(38,39,42,45,43,46,41,50)(40,47,48,52,49,44,51,53);(30,31)(32,36)(33,35)(34,37)(38,40)(39,44)(41,48)(42,51)(43,49)(45,52)(46,47)(50,53);(38,41,43,42)(39,45,46,50)(40,48,49,51)(44,52,47,53)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#8412, C₂₉×#8413, C₂₉×#8414, …
maximal normal subgroup of	C₂₉×#8412, C₂₉×#8413, C₂₉×#8414, …
maximal quotient of	C₂₉×#7574, C₂₉×#7577, C₂₉×#7582, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 261);