C₂²×#909 #87157

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#909

Splittings

Fitting	C₂²×C₃×M₁₆ ⋊ C₂
Sylow	C₃ ⋊ C₂²×#137

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₁₂
commutator, cosocle	C₃×C₄	C₂⁵
Frattini	C₄	C₂⁴×D₃
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂³×C₃	C₂×D₄
prime radical	C₂²×#235	C₂
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	C₂²×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

1464 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 565 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) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 5 generators on 23 points:

(5,6,7)(8,11,12,19,13,20,10,17)(9,15,14,18,16,23,21,22);(1,2)(3,4)(8,10,13,12)(9,14,16,21)(11,19,20,17)(15,22,23,18);(6,7)(8,9)(10,14)(11,18)(12,21)(13,16)(15,19)(17,23)(20,22);(3,4);(1,2)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10701922, #10701926, #10701936, …
maximal normal subgroup of	#10701922, #10701926, #10701936, …
maximal quotient of	C₂²×#5120, C₂²×#5133, C₂²×#5121, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19733);