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Q₈×#237 #86743

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

Splittings

Fitting	C₃×D₄×Q₈ ⋊ C₂
Sylow	C₃ ⋊ Q₈×#41

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁵
Fitting	C₃×D₄×Q₈	C₂
prime radical	C₂×#237	C₂²
hypercenter	D₄×Q₈	D₃
hypocenter	C₃	Q₈×#41
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

1068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 5 generators on 19 points:

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10771164, #10771172, #10771179, …
maximal normal subgroup of	#10771164, #10771172, #10771179, …
maximal quotient of	C₄⋊C₄×#237, Q₈×#877, Q₈×#892, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19319);