C₂×#4916 #82021

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

Aliases

direct product

C₂×#4916

Splittings

Fitting	C₂×C₃×SD₃₂ ⋊ C₂
Sylow	C₃ ⋊ C₂×#508

Subgroups

	subgroup	G/H
center	C₂²	D₃×D₈
commutator, cosocle	C₃×C₈	C₂⁴
Frattini	C₈	C₂³×D₃
Fitting	C₂×C₃×SD₃₂	C₂
socle	C₂²×C₃	C₂×D₈
hypercenter	C₂×SD₃₂	D₃
hypocenter	C₃	C₂×#508
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×SD₃₂

368 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 117 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:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#4916, #10948492, #10948569, …
maximal normal subgroup of	C₄×#4916, #10948492, #10948569, …
maximal quotient of	C₂×#69188, C₂×#69220, C₂×#69345, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14597);