C₂₉×#536 #14270831

order	1856
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂₉×#536

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₂₉
prime radical	C₂×#41×C₂₉	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ G

270 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 49
transitive degree	μ_t(G) = 928

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

Conjugacy classes & character table

638 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16820015, #16820030, #16820045, …
maximal subgroup of	C₂₉×#1909, C₂₉×#1912, C₂₉×#1915, …
maximal normal subgroup of	C₂₉×#1909, C₂₉×#1912, C₂₉×#1915, …
maximal quotient of	C₂₉×#2310, C₂₉×#2311, C₂₉×#2317, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1856, 217);