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Q₈×C₂₇ #6174

order	216
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×C₉×S₄ (order 432) · Inn(G) = C₂² · Out(G) = C₂×D₃×C₉
homology	Abelianization (H₁) = C₂²×C₂₇ · Schur multiplier (H₂) = C₁

Aliases

Q₈×C₂₇

Subgroups

	subgroup	G/H
center	C₂×C₂₇	C₂²
commutator	C₂	C₂²×C₂₇
Frattini, cosocle	C₂×C₉	C₂²×C₃
socle, prime radical	C₂×C₃	C₂²×C₉
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₇ ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 2 generators on 35 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35);(1,5,3,7)(2,6,4,8)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13047027, #13047028, SL(2,𝔽₃)×D₂₇, …
Fitting subgroup of	#88370, #88371, Q₈×D₂₇, …
maximal subgroup of	#88370, #88371, SD₁₆×C₂₇, …
maximal normal subgroup of	#88370, #88371, SD₁₆×C₂₇, …
maximal quotient of	C₄⋊C₄×C₂₇, C₂×Q₈×C₂₇, Q₈×C₈₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 11);