C₂₉×#1782 #16815496

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂₉×#1782

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₂₉
Frattini, cosocle	C₂⁴	C₂³×C₂₉
prime radical	C₂²×#41×C₂₉	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂³×C₄×C₂₉ ◁ G

612 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 49
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 3 generators on 49 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)(32,33)(34,35)(36,42,44,39)(37,40)(38,41)(43,48,49,46)(45,47);(34,36)(35,39)(37,43)(38,44)(40,46)(41,42)(45,49)(47,48);(31,33)(34,37,38,45)(35,40,41,47)(36,43,44,49)(39,46,42,48)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7305, C₂₉×#7309, C₂₉×#7312, …
maximal normal subgroup of	C₂₉×#7305, C₂₉×#7309, C₂₉×#7312, …
maximal quotient of	C₂₉×#8300, C₂₉×#8302, C₂₉×#8303, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 514);