C₂₉×#2175 #16815889

order	3712
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#2175

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×D₈
commutator	C₈	C₂²×C₄×C₂₉
Frattini, cosocle	C₂×C₈	C₂³×C₂₉
prime radical	C₂×D₈×C₂₉	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ C₄×C₈×C₂₉ ◁ G

160 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 1856

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)(30,31,33,32)(34,35)(36,39)(37,46)(38,40)(41,43)(42,44)(45,47)(48,49);(35,39)(36,43)(37,44)(40,47)(41,42)(45,49)(46,48);(30,32,33,31)(35,40)(39,47)(41,48)(42,46)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12065, C₂₉×#12077, C₂₉×#12080, …
maximal normal subgroup of	C₂₉×#12065, C₂₉×#12077, C₂₉×#12080, …
maximal quotient of	C₂₉×#12043, C₂₉×#12074, C₂₉×#12075, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 907);