C₂₉×#2971 #16816685

order	3712
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂₉×#2971

Subgroups

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

402 conjugacy classes of subgroups; only the 276 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 276 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) = 45
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 4 generators on 45 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)(32,35)(33,34)(36,37)(38,39,43,40,42,44,41,45);(31,34)(35,37)(39,44)(40,45);(38,40,43,44,42,45,41,39);(30,32,33,36)(31,35,34,37)(38,41,42,43)(39,40,44,45)

Conjugacy classes & character table

1450 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#9160, C₂₉×#9162, C₂₉×#9350, …
maximal normal subgroup of	C₂₉×#9160, C₂₉×#9162, C₂₉×#9350, …
maximal quotient of	C₂₉×#20480, C₂₉×#20494, Q₈×C₂₉×#97, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1703);