C₂×#5068 #83009

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#5068

Splittings

Fitting	C₂×C₃×#125 ⋊ C₂
Sylow	C₃ ⋊ C₂×#488

Subgroups

	subgroup	G/H
center	C₂³	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×#125	C₂
socle	C₂³×C₃	C₂×D₄
prime radical	C₂²×D₁₂	C₂²
hypercenter	C₂×#125	D₃
hypocenter	C₃	C₂×#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

412 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 29
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 29 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10694739, #10694741, #10694746, …
maximal normal subgroup of	#10694739, #10694741, #10694746, …
maximal quotient of	C₂×#69547, C₂×#69614, C₂×#69615, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 15585);