C₂×#5179 #83870

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

Aliases

direct product

C₂×#5179

Splittings

Fitting	C₂²×C₃×Q₁₆ ⋊ C₂
Sylow	C₃ ⋊ C₂×#510

Subgroups

	subgroup	G/H
center	C₂²	C₂×#213
commutator, cosocle	C₃×C₈	C₂⁴
Frattini	C₈	C₂³×D₃
Fitting	C₂²×C₃×Q₁₆	C₂
socle	C₂²×C₃	C₂×D₈
prime radical	C₂×#904	C₂
hypercenter	C₂²×Q₁₆	D₃
hypocenter	C₃	C₂×#510
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

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

normal subgroups only

This group has 360 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) = 37
transitive degree	μ_t(G) = 192

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

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10712318, #10712324, #10712339, …
maximal normal subgroup of	#10712318, #10712324, #10712339, …
maximal quotient of	C₂×#72061, C₂×#70598, C₂×#70596, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16446);