C₄×C₂₉×#102 #16815474

order	3712
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₂₉ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₄×C₂₉×#102

Subgroups

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

396 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 184 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 — 3 generators on 45 points:

(1,2,3,4)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(5,6,7,8)(9,10)(11,13)(12,15)(14,16);(10,13)(11,16)(14,15)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₈×C₂₉×#102, C₂₉×#8200, C₂₉×#8202, …
maximal normal subgroup of	C₈×C₂₉×#102, C₂₉×#8200, C₂₉×#8202, …
maximal quotient of	C₄×C₂₉×D₄⋊C₈, C₂₉×#8174, C₂₉×#8177, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 492);