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D₂₁⋊Q₈ #66674

order	336
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

unambiguous semidirect product

D₂₁⋊Q₈

Splittings

Hall	C₃×C₇ ⋊ C₂×Q₈
Sylow	C₇ ⋊ D₃×Q₈
Sylow	C₃ ⋊ Q₈×D₇

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₇
commutator, socle, cosocle	C₂×C₃×C₇	C₂³
Fitting	C₃×C₄×C₇	C₂²
prime radical	C₂×D₂₁	C₂²
hypercenter	C₄	D₃×D₇
hypocenter	C₃×C₇	C₂×Q₈
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 3 generators on 18 points:

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

Polycyclic presentation:

2^4,3,7|a^2=d,b^2=d,c^2=d,c^a=cd,f^a=f^6,c^b=cd,e^b=e^2

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10620817, #10620826, #10620840, …
maximal normal subgroup of	#10620817, #10620826, #10620840, …
maximal quotient of	#10620887, #10620889, #10620891, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 143);