C₂₉×#2888 #16816602

order	3712
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂₉×#2888

Subgroups

	subgroup	G/H
center, socle	C₂×C₂₉	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₄×C₂₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
prime radical	C₂×C₂₉×H(5,𝔽₂)	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₉ ◁ C₂×#41×C₂₉ ◁ G

756 conjugacy classes of subgroups; only the 340 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 340 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) = 464

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

Conjugacy classes & character table

1189 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#13097, C₂₉×#13099, C₂₉×#21778, …
maximal normal subgroup of	C₂₉×#13097, C₂₉×#13099, C₂₉×#21778, …
maximal quotient of	C₂₉×#17350, C₂₉×#17353, C₂₉×#17367, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1620);