C₂₉×#3199 #16816913

order	3712
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂₉×#3199

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×#41×C₂₉ ◁ G

442 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 53
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 4 generators on 53 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)(32,36)(33,35)(34,37)(38,39)(40,43)(41,44)(42,46)(45,49)(47,51)(48,50)(52,53);(30,32)(31,34)(33,37)(35,36)(38,40,42,47)(39,43,46,51)(41,48,49,53)(44,50,45,52);(32,37)(34,36)(39,44)(41,49)(43,50)(45,46)(48,53)(51,52);(31,35)(34,36)(40,47)(43,51)(48,53)(50,52)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12108, C₂₉×#12111, C₂₉×#12117, …
maximal normal subgroup of	C₂₉×#12108, C₂₉×#12111, C₂₉×#12117, …
maximal quotient of	C₂₉×#21770, C₂₉×#21947, C₂₉×#22056, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1931);