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C₂³×D₃×Q₈ #87560

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

C₂³×D₃×Q₈

Splittings

Fitting	C₂³×C₃×Q₈ ⋊ C₂
Sylow	C₃ ⋊ C₂⁴×Q₈

Subgroups

	subgroup	G/H
center	C₂⁴	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂⁶
Frattini	C₂	C₂⁵×D₃
Fitting	C₂³×C₃×Q₈	C₂
socle	C₂⁴×C₃	C₂³
prime radical	C₂⁴×D₃	C₂²
hypercenter	C₂³×Q₈	D₃
hypocenter	C₃	C₂⁴×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

6264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 6 generators on 17 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1446, #10717571, #10717574, …
maximal normal subgroup of	D₃×#1446, #10717571, #10717574, …
maximal quotient of	C₂³×D₃×C₄⋊C₄, C₂²×C₄×D₃×Q₈, C₂³×#887, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20136);