C₂₉×#1824 #16815538

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂₉×#1824

Subgroups

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂⁴×C₂₉	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₂³×C₄×C₂₉ ◁ G

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

normal subgroups only

This group has 344 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) = 1856

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)(36,39)(37,38)(40,41)(42,43)(44,46)(45,47)(48,49);(35,38)(39,41)(43,46)(47,48);(34,36,37,40)(35,39,38,41)(43,47)(45,49)(46,48)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#8378, C₂₉×#8380, C₂₉×#9021, …
maximal normal subgroup of	C₂₉×#8378, C₂₉×#8380, C₂₉×#9021, …
maximal quotient of	C₂₉×#8707, C₂₉×#8710, C₂₉×#8713, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 556);