Finite Group Databasebeta

← Prev (#63710) Next (#63712) →

C₂×Q₈×D₉ #63711

order	288
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂×Q₈×D₉

Splittings

Fitting	C₂×Q₈×C₉ ⋊ C₂
Sylow	C₉ ⋊ C₂²×Q₈

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₉
commutator, cosocle	C₂×C₉	C₂⁴
Frattini	C₂×C₃	C₂³×D₃
Fitting	C₂×Q₈×C₉	C₂
socle	C₂²×C₃	C₂²×D₃
prime radical	C₂²×D₉	C₂²
hypercenter	C₂×Q₈	D₉
hypocenter	C₉	C₂²×Q₈
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₉×#101, D₉×#103, #10588482, …
maximal normal subgroup of	D₉×#101, D₉×#103, #10588482, …
maximal quotient of	C₂×#63449, C₂×#63450, C₂×#63451, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 359);