C₂₉×#1378 #16815092

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1378

Subgroups

	subgroup	G/H
center	C₈×C₂₉	C₂²⋊C₄
commutator	C₂²	C₄×C₈×C₂₉
Frattini, cosocle	C₂²×C₈	C₂²×C₂₉
socle	C₂×C₂₉	#336
prime radical	C₂³×C₂₉	C₂×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₉ ◁ C₂²×C₈×C₂₉ ◁ G

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

normal subgroups only

This group has 132 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 — 2 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)(30,31,33,37,34,39,44,51,35,38,45,50,46,52,57,60)(32,40,42,53,43,54,56,61,41,36,47,48,55,49,58,59);(30,32)(31,36,38,40)(33,42)(34,43)(35,41)(37,48,50,53)(39,49,52,54)(44,56)(45,47)(46,55)(51,59,60,61)(57,58)

Conjugacy classes & character table

1276 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₂₉×#1378, C₂₉×#12597, C₂₉×#12609, …
maximal normal subgroup of	C₂×C₂₉×#1378, C₂₉×#12597, C₂₉×#12609, …
maximal quotient of	C₂₉×#7140, C₂₉×#7144, C₂₉×#7147, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 110);