C₂₉×#2312 #16816026

order	3712
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₂₉ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂₉×#2312

Subgroups

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₉
Frattini, cosocle	C₂³	C₂⁴×C₂₉
prime radical	C₂⁵×C₂₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ G

672 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 280 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 — 4 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)(36,40)(37,38)(39,41)(43,46)(44,48);(35,38)(39,40)(42,43,45,46)(44,47,48,49);(34,35)(36,39)(37,38)(40,41)(42,43)(44,47)(45,46)(48,49);(34,36)(35,39)(37,41)(38,40)(42,44)(43,47)(45,48)(46,49)

Conjugacy classes & character table

1276 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8736, C₂₉×#8750, C₂₉×#8759, …
maximal normal subgroup of	C₂₉×#8736, C₂₉×#8750, C₂₉×#8759, …
maximal quotient of	C₂₉×#13775, C₂₉×#13778, C₂₉×#13788, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1044);