C₂₉×#3363 #16817077

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

Aliases

direct product

C₂₉×#3363

Subgroups

	subgroup	G/H
center, socle, prime radical	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

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

normal subgroups only

This group has 336 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) = 53
transitive degree	μ_t(G) = 3712

A minimal faithful permutation representation — 4 generators on 53 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)(31,34)(35,36)(38,39,42,45)(40,46,48,53)(41,49,51,44)(43,50,52,47);(30,31,33,34)(32,36,37,35);(38,40,42,48)(39,43,45,52)(41,50,51,47)(44,53,49,46);(30,32,33,37)(31,35,34,36)

Conjugacy classes & character table

1015 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#19812, C₂₉×#19816, C₂₉×#19817, …
maximal normal subgroup of	C₂₉×#19812, C₂₉×#19816, C₂₉×#19817, …
maximal quotient of	Q₈×C₂₉×#103, C₂₉×#22542, C₂₉×#22740, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2095);