C₁₉×#554 #11926230

order	1216
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₉×#554

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₉
prime radical	C₂×D₄×C₁₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ 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) = 35
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 4 generators on 35 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,23)(22,26,27,25)(28,29,32,31)(30,34,35,33);(21,23)(25,26)(28,30,32,35)(29,33,31,34);(20,22)(21,25)(23,26)(24,27);(20,23,24,21)(22,26,27,25)(28,31,32,29)(30,33,35,34)

Conjugacy classes & character table

475 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#14562707, #14562769, #14562842, …
maximal subgroup of	C₁₉×#1622, C₁₉×#1632, C₁₉×#1643, …
maximal normal subgroup of	C₁₉×#1622, C₁₉×#1632, C₁₉×#1643, …
maximal quotient of	C₁₉×#2345, D₄×C₄⋊C₄×C₁₉, C₁₉×#2349, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 235);