C₂₉×#1795 #16815509

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1795
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂₉×#1795

Subgroups

	subgroup	G/H
center, socle	C₂×C₂₉	#379
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂×C₂₉×H(5,𝔽₂)	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

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

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 45
transitive degree	μ_t(G) = 464

A minimal faithful permutation representation — 3 generators on 45 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)(32,39,41,36)(33,35)(34,38)(37,42)(40,44,45,43);(31,35)(33,42)(36,43)(37,38)(39,44)(40,45);(30,32)(31,36)(33,40)(34,41)(35,43)(37,44)(38,39)(42,45)

Conjugacy classes & character table

841 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12711, C₂₉×#12713, C₂₉×#12716, …
maximal normal subgroup of	C₂₉×#12711, C₂₉×#12713, C₂₉×#12716, …
maximal quotient of	C₂₉×#8322, C₂₉×#8324, C₂₉×#8325, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 527);