C₂₉×#2133 #16815847

order	3712
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#2133

Subgroups

	subgroup	G/H
center, socle	C₂×C₂₉	C₂×#99
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂×Q₈	C₂³×C₂₉
prime radical	C₂×#41×C₂₉	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₉ ◁ C₂³×C₂₉ ◁ C₂²×Q₈×C₂₉ ◁ G

230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 61
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 3 generators on 61 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)(30,31,34,38,35,39,50,57)(32,40,43,58,41,54,48,60)(33,44,46,56,47,37,49,55)(36,51,53,45,52,59,61,42);(30,32)(31,36)(33,45)(34,48)(35,41)(37,54)(38,53)(39,52)(40,44)(42,47)(43,50)(46,51)(49,59)(55,60)(56,58)(57,61);(32,41)(36,52)(40,54)(42,45)(43,48)(51,59)(53,61)(58,60)

Conjugacy classes & character table

667 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12709, C₂₉×#12710, C₂₉×#12729, …
maximal normal subgroup of	C₂₉×#12709, C₂₉×#12710, C₂₉×#12729, …
maximal quotient of	C₂₉×#11432, C₂₉×#11448, C₂₉×#11480, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 865);