C₂₉×#2251 #16815965

order	3712
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₂₉×#2251

Subgroups

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

178 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 65
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 3 generators on 65 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)(31,33)(34,35,39,43)(36,44,46,58)(37,49,51,64)(38,52,50,42)(40,53,54,47)(41,56,57,45)(48,63,62,60)(55,61,59,65);(34,36,45,61,38,48,51,53,39,46,56,65,50,62,37,47)(35,40,49,60,42,55,57,58,43,54,64,63,52,59,41,44);(30,31,32,33)(36,46)(40,54)(44,58)(47,53)(48,62)(55,59)(60,63)(61,65)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12326, C₂₉×#12327, C₂₉×#12338, …
maximal normal subgroup of	C₂₉×#12326, C₂₉×#12327, C₂₉×#12338, …
maximal quotient of	C₂₉×#12507, C₂₉×#12508, C₂₉×#12512, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 983);