C₂×C₁₉×C₂²⋊Q₈ #11926199

order	1216
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₁₉×C₂²⋊Q₈

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₉
prime radical	C₂⁴×C₁₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

322 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 33
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 4 generators on 33 points:

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

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#14561999, C₂×#11926281, #14562155, …
maximal subgroup of	C₁₉×#1472, C₂×C₁₉×#328, C₁₉×#1508, …
maximal normal subgroup of	C₁₉×#1472, C₂×C₁₉×#328, C₁₉×#1508, …
maximal quotient of	C₂×C₁₉×#380, C₂×C₁₉×#385, C₂×C₁₉×#387, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 204);