C₂₉×#2141 #16815855

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

Aliases

direct product

C₂₉×#2141

Subgroups

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

220 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 104 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)(30,31,32,33)(34,35)(36,45)(37,42)(38,51)(39,55)(40,44)(41,47)(43,53)(46,59)(48,65)(49,62)(50,60)(52,58)(54,61)(56,63)(57,64);(34,36,40,49)(35,41,44,57)(37,48,39,46)(38,52,54,63)(42,58,55,56)(43,59,60,65)(45,61,62,51)(47,50,64,53);(34,37,40,39)(35,42,44,55)(36,46,49,48)(38,50,54,53)(41,56,57,58)(43,51,60,61)(45,59,62,65)(47,63,64,52)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12648, C₂₉×#12742, C₂₉×#12751, …
maximal normal subgroup of	C₂₉×#12648, C₂₉×#12742, C₂₉×#12751, …
maximal quotient of	C₂₉×#11594, C₂₉×#11650, C₂₉×#11662, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 873);