C₂×#4517 #71446

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#4517

Splittings

Sylow

C₃ ⋊ C₂×#362

Subgroups

	subgroup	G/H
center	C₂²	#821
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₂×C₃×M₃₂	C₂
socle	C₂²×C₃	#102
prime radical	C₂³×C₃	SD₁₆
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂×M₃₂	D₃
hypocenter	C₃	C₂×#362
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

216 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 37
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 37 points:

(3,4,5)(6,8,9,21,22,33,28,34,11,18,24,17,10,19,23,37)(7,12,13,29,25,31,36,32,15,26,30,16,14,27,20,35);(4,5)(6,7,11,15)(8,16,18,29)(9,20,24,36)(10,25,22,14)(12,21,26,17)(13,28,30,23)(19,35,33,32)(27,34,31,37);(1,2)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10710444, C₄×#4517, #10710465, …
maximal normal subgroup of	#10710444, C₄×#4517, #10710465, …
maximal quotient of	C₂×#67517, #10704967, #10705198, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4022);