C₂×#5192 #83891

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

Aliases

direct product

C₂×#5192

Splittings

Fitting	C₂²×C₃×Q₁₆ ⋊ C₂
Sylow	C₃ ⋊ C₂×#473

Subgroups

	subgroup	G/H
center	C₂²	#938
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂²×C₃×Q₁₆	C₂
socle	C₂²×C₃	#121
prime radical	C₂×#235	C₂²
hypercenter	C₂²×Q₁₆	D₃
hypocenter	C₃	C₂×#473
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

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

normal subgroups only

This group has 448 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) = 37
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 37 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10712156, #10712162, #10712205, …
maximal normal subgroup of	#10712156, #10712162, #10712205, …
maximal quotient of	C₂×#71305, C₂×#71371, C₂×#71232, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16467);