C₂₉×#3391 #16817105

order	3712
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂₉×#3391

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄×Q₈×C₂₉ ◁ G

460 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 1856

A minimal faithful permutation representation — 4 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)(31,35)(36,37)(38,39)(40,42)(41,44)(43,45);(30,31,34,35)(32,36,33,37);(30,32,34,33)(31,36,35,37)(39,42)(40,45)(43,44);(30,33,34,32)(31,36,35,37)

Conjugacy classes & character table

1015 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12328, C₂₉×#12332, C₂₉×#12380, …
maximal normal subgroup of	C₂₉×#12328, C₂₉×#12332, C₂₉×#12380, …
maximal quotient of	C₂₉×#22540, Q₈×C₂₉×#107, C₂₉×#22807, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2123);