C₂×#5147 #83819

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

Aliases

direct product

C₂×#5147

Splittings

Fitting	C₂²×C₃×D₈ ⋊ C₂
Sylow	C₃ ⋊ C₂×#509

Subgroups

	subgroup	G/H
center	C₂²	C₂×#213
commutator, cosocle	C₃×C₈	C₂⁴
Frattini	C₈	C₂³×D₃
Fitting	C₂²×C₃×D₈	C₂
socle	C₂²×C₃	C₂×D₈
hypercenter	C₂²×D₈	D₃
hypocenter	C₃	C₂×#509
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₈

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

normal subgroups only

This group has 400 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) = 21
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10712311, #10712317, #10712332, …
maximal normal subgroup of	#10712311, #10712317, #10712332, …
maximal quotient of	C₂×#72033, C₂×#70598, C₂×#70596, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16395);