C₂₉×#554 #14270849

order	1856
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂₉×#554

Subgroups

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

228 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 928

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)(30,31,34,33)(32,36,37,35)(38,39,42,41)(40,44,45,43);(31,33)(35,36)(38,40,42,45)(39,43,41,44);(30,32)(31,35)(33,36)(34,37);(30,33,34,31)(32,36,37,35)(38,41,42,39)(40,43,45,44)

Conjugacy classes & character table

725 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16818564, #16818626, #16818699, …
maximal subgroup of	C₂₉×#1622, C₂₉×#1632, C₂₉×#1643, …
maximal normal subgroup of	C₂₉×#1622, C₂₉×#1632, C₂₉×#1643, …
maximal quotient of	C₂₉×#2345, D₄×C₄⋊C₄×C₂₉, C₂₉×#2349, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1856, 235);