C₂×#4909 #82010

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#4909

Splittings

Fitting	C₂×C₃×M₃₂ ⋊ C₂
Sylow	C₃ ⋊ C₂×#510

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₂₄
commutator, cosocle	C₃×C₈	C₂⁴
Frattini	C₈	C₂³×D₃
Fitting	C₂×C₃×M₃₂	C₂
socle	C₂²×C₃	C₂×D₈
prime radical	C₂×#904	C₂
hypercenter	C₂×M₃₂	D₃
hypocenter	C₃	C₂×#510
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

360 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 133 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)(6,9,18,33,19,21,10,22,11,23,30,36,8,17,25,37)(7,13,28,34,12,26,14,20,16,29,24,32,15,27,31,35);(1,2)(6,8,11,19)(7,12,16,15)(9,21,23,17)(10,18,25,30)(13,27,29,26)(14,24,31,28)(20,34,35,32)(22,36,37,33);(4,5)(6,7)(8,12)(9,20)(10,24)(11,16)(13,22)(14,30)(15,19)(17,32)(18,31)(21,34)(23,35)(25,28)(26,33)(27,36)(29,37);(1,2)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10712320, #10712322, #10712341, …
maximal normal subgroup of	#10712320, #10712322, #10712341, …
maximal quotient of	C₂×#71445, C₂×#71439, C₂×#71428, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14586);