C₁₉×#547 #11926223

order	1216
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#547

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ G

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

normal subgroups only

This group has 292 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) = 31
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 4 generators on 31 points:

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

Conjugacy classes & character table

475 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#14562724, #14562786, #14562870, …
maximal subgroup of	C₁₉×#1625, C₁₉×#1635, C₁₉×#1647, …
maximal normal subgroup of	C₁₉×#1625, C₁₉×#1635, C₁₉×#1647, …
maximal quotient of	C₁₉×#2339, C₁₉×#2344, D₄×C₄⋊C₄×C₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 228);