C₂×#5095 #83225

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#5095

Splittings

Fitting	C₂×C₃×C₄⋊Q₈ ⋊ C₂
Sylow	C₃ ⋊ C₂×#456

Subgroups

	subgroup	G/H
center	C₂²	#937
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄⋊Q₈	C₂
socle	C₂²×C₃	#120
prime radical	C₂×#1007	C₂
hypercenter	C₂×C₄⋊Q₈	D₃
hypocenter	C₃	C₂×#456
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈

708 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 21
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10712532, #10712534, #10712542, …
maximal normal subgroup of	#10712532, #10712534, #10712542, …
maximal quotient of	C₂×#69705, C₂×#69711, C₂×#69706, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 15801);