C₂₉×#1772 #16815486

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

Aliases

direct product

C₂₉×#1772

Subgroups

	subgroup	G/H
center	C₂×C₈×C₂₉	D₄
commutator	C₄	C₂×C₄²×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
socle	C₂²×C₂₉	C₄×D₄
prime radical	C₂³×C₂₉	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₉ ◁ C₂²×C₈×C₂₉ ◁ G

232 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 49
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 3 generators on 49 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)(35,39,41,44)(40,46,47,48);(30,31,32,33)(34,35)(36,40)(37,44)(38,41)(39,45)(42,48)(43,47)(46,49);(34,36,37,42,38,43,45,49)(35,40,44,48,41,47,39,46)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#7179, C₂₉×#7180, C₂₉×#7181, …
maximal normal subgroup of	C₂₉×#7179, C₂₉×#7180, C₂₉×#7181, …
maximal quotient of	C₂₉×#8266, C₂₉×#8267, C₂₉×#8270, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 504);