C₂×#5190 #83880

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

Aliases

direct product

C₂×#5190

Splittings

Fitting	C₂²×C₃×Q₁₆ ⋊ C₂
Sylow	C₃ ⋊ C₂×#497

Subgroups

	subgroup	G/H
center	C₂³	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂²×C₃×Q₁₆	C₂
socle	C₂³×C₃	C₂×D₄
prime radical	C₂²×D₁₂	C₂²
hypercenter	C₂²×Q₁₆	D₃
hypocenter	C₃	C₂×#497
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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) = 25
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 25 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10803735, C₄×#5190, #10803756, …
maximal normal subgroup of	#10803735, C₄×#5190, #10803756, …
maximal quotient of	C₂×#71363, C₂×#71216, C₂×#71299, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16456);