C₂₉×#2975 #16816689

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

Aliases

direct product

C₂₉×#2975

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	C₂²×D₄×C₂₉	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ G

452 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 268 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) = 928

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)(32,36)(33,35)(34,37)(38,39,41,43,40,42,44,45);(30,32)(31,34)(33,37)(35,36)(39,42)(43,45);(32,37)(34,36)(39,42)(43,45);(31,35)(34,36)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#8837, C₂₉×#8860, C₂₉×#8961, …
maximal normal subgroup of	C₂₉×#8837, C₂₉×#8860, C₂₉×#8961, …
maximal quotient of	C₂₉×#20524, C₂₉×#20536, C₂₉×#20568, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1707);