C₂×#4800 #78782

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

Aliases

direct product

C₂×#4800

Splittings

Fitting	C₂×C₃×#103 ⋊ C₂
Sylow	C₃ ⋊ C₂×#488

Subgroups

	subgroup	G/H
center	C₂³	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×#103	C₂
socle	C₂³×C₃	C₂×D₄
prime radical	C₂²×D₁₂	C₂²
hypercenter	C₂×#103	D₃
hypocenter	C₃	C₂×#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

412 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 141 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:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10803169, C₄×#4800, #10803180, …
maximal normal subgroup of	#10803169, C₄×#4800, #10803180, …
maximal quotient of	C₂×#68848, C₂×#70254, C₂×#70253, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11358);