C₂×#4901 #82000

order	384
isoclinism	isoclinic class of Dic₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#4901

Splittings

Fitting	C₂²×C₃×C₁₆ ⋊ C₂
Sylow	C₃ ⋊ C₂×#504

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₃
commutator	C₂×C₃	C₂³×C₈
Frattini	C₈	C₂³×D₃
Fitting	C₂²×C₃×C₁₆	C₂
socle	C₂²×C₃	C₂²×C₈
prime radical	C₂×#235	C₄
cosocle	C₃×C₈	C₂⁴
hypercenter	C₂²×C₁₆	D₃
hypocenter	C₃	C₂×#504
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

368 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 213 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 — 4 generators on 37 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10710284, #10710290, #10710292, …
maximal normal subgroup of	#10710284, #10710290, #10710292, …
maximal quotient of	C₂×#69051, C₂×#71399, C₄×#4901, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14576);