C₂₉×#3395 #16817109

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×#3395

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

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

normal subgroups only

This group has 348 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)(31,35)(36,37)(38,39,42,45)(40,46,48,53)(41,49,51,44)(43,50,52,47);(30,31,34,35)(32,36,33,37);(30,32,34,33)(31,36,35,37)(38,40,42,48)(39,43,45,52)(41,50,51,47)(44,53,49,46);(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×#12330, C₂₉×#12334, C₂₉×#12382, …
maximal normal subgroup of	C₂₉×#12330, C₂₉×#12334, C₂₉×#12382, …
maximal quotient of	C₂₉×#22542, Q₈×C₂₉×#107, C₂₉×#22808, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2127);