C₂×#5079 #83194

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#5079

Splittings

Sylow

C₃ ⋊ C₂×#487

Subgroups

	subgroup	G/H
center	C₂³	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₃
hypocenter	C₃	C₂×#487
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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) = 384

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

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10690455, #10690457, #10690483, …
maximal normal subgroup of	#10690455, #10690457, #10690483, …
maximal quotient of	C₂×#69574, C₂×#69674, C₂×#71785, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 15770);