C₂₉×#1759 #16815473

order	3712
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₂₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂₉×#1759

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂×D₄
commutator	C₄	C₂×C₄²×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂×#41×C₂₉	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

320 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 49
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 3 generators on 49 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)(30,31,33,32)(34,35)(36,42)(37,45)(38,41)(39,44)(40,46)(43,48)(47,49);(34,36)(35,39,41,47)(37,44)(38,43)(40,42,45,48)(46,49);(30,32,33,31)(36,43)(39,47)(42,48)(44,49)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#8412, C₂₉×#8413, C₂₉×#8414, …
maximal normal subgroup of	C₂₉×#8412, C₂₉×#8413, C₂₉×#8414, …
maximal quotient of	C₂₉×#8170, C₂₉×#8171, C₂₉×#8174, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 491);