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Q₈×D₁₂ #5575

order	192
automorphism	Aut(G) order 18432
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

Q₈×D₁₂

Splittings

Fitting	C₃×C₄×Q₈ ⋊ C₂
Sylow	C₃ ⋊ D₄×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×C₄×Q₈	C₂
prime radical	C₂×D₁₂	C₂²
hypercenter	C₄×Q₈	D₃
hypocenter	C₃	D₄×Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#68251, #68266, #68269, …
maximal normal subgroup of	#68251, #68266, #68269, …
maximal quotient of	#73650, #73675, #73682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1134);