C₂₉×#1379 #16815093

order	3712
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1379

Subgroups

	subgroup	G/H
center	C₂×C₈×C₂₉	D₄
commutator	C₄	C₄×C₈×C₂₉
Frattini, cosocle	C₂²×C₈	C₂²×C₂₉
socle	C₂²×C₂₉	C₂²⋊C₈
prime radical	C₂³×C₂₉	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₉ ◁ C₂²×C₈×C₂₉ ◁ G

136 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 2 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)(34,35,38,43,39,44,53,61,40,45,54,62,37,42,51,59)(36,46,48,63,47,57,64,60,49,58,52,65,50,41,55,56);(30,31,32,33)(34,36)(35,41)(37,50)(38,52)(39,47)(40,49)(42,58)(43,60)(44,46)(45,57)(48,54)(51,64)(53,55)(56,62)(59,63)(61,65)

Conjugacy classes & character table

1624 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₂₉×#1379, C₂₉×#12554, C₂₉×#12597, …
maximal normal subgroup of	C₂×C₂₉×#1379, C₂₉×#12554, C₂₉×#12597, …
maximal quotient of	C₂₉×#7141, C₂₉×#7144, C₂₉×#7147, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 111);