C₂₉×#2143 #16815857

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

Aliases

direct product

C₂₉×#2143

Subgroups

	subgroup	G/H
center	C₄×C₂₉	C₂×D₈
commutator	C₈	C₂²×C₄×C₂₉
Frattini, cosocle	C₂×C₈	C₂³×C₂₉
socle	C₂×C₂₉	C₂×#102
prime radical	C₂×C₂₉×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ C₂²×C₈×C₂₉ ◁ G

220 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 100 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)(30,31,34,39,36,40,35,50)(32,41,44,58,42,57,51,55)(33,45,47,56,49,60,48,38)(37,46,53,59,52,61,54,43);(30,32)(31,37)(33,46)(34,44)(35,51)(36,42)(38,55)(39,53)(40,52)(41,45)(43,48)(47,59)(49,61)(50,54)(56,58)(57,60);(32,42)(37,52)(41,57)(43,59)(44,51)(46,61)(53,54)(55,58)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12649, C₂₉×#12651, C₂₉×#12740, …
maximal normal subgroup of	C₂₉×#12649, C₂₉×#12651, C₂₉×#12740, …
maximal quotient of	C₂₉×#11592, C₂₉×#11593, C₂₉×#11594, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 875);