C₂₉×#1278 #16814992

order	3712
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1278

Subgroups

	subgroup	G/H
center	C₄²×C₂₉	D₄
commutator	C₄	C₄×C₈×C₂₉
Frattini, cosocle	C₂×C₄²	C₂²×C₂₉
socle	C₂²×C₂₉	#95
prime radical	C₂³×C₂₉	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

172 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 2 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)(32,34)(33,36)(35,37)(38,39,41,44,43,45,42,40);(31,34,36,35)(38,40,42,45,43,44,41,39)

Conjugacy classes & character table

1624 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₂₉×#1278, C₂₉×#7882, C₂₉×#7883, …
maximal normal subgroup of	C₂×C₂₉×#1278, C₂₉×#7882, C₂₉×#7883, …
maximal quotient of	C₂₉×#7015, C₂₉×#7021, C₂₉×#7023, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 10);