C₂×#4915 #82023

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

Aliases

direct product

C₂×#4915

Splittings

Fitting	C₂×C₃×SD₃₂ ⋊ C₂
Sylow	C₃ ⋊ C₂×#510

Subgroups

	subgroup	G/H
center	C₂²	D₃×D₈
commutator, cosocle	C₃×C₈	C₂⁴
Frattini	C₈	C₂³×D₃
Fitting	C₂×C₃×SD₃₂	C₂
socle	C₂²×C₃	C₂×D₈
prime radical	C₂×#912	C₂
hypercenter	C₂×SD₃₂	D₃
hypocenter	C₃	C₂×#510
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×SD₃₂

360 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 117 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:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#4915, #10948519, #10948533, …
maximal normal subgroup of	C₄×#4915, #10948519, #10948533, …
maximal quotient of	C₂×#69222, C₂×#69189, C₂×#69346, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14599);