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C₉×C₂²⋊C₄ #3678

order	144
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×C₃×#120 (order 192) · Inn(G) = C₂² · Out(G) = C₂×C₃×D₄
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₂²

Aliases

C₉×C₂²⋊C₄

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂²
commutator	C₂	C₂×C₄×C₉
Frattini, socle, cosocle	C₂²×C₃	C₂²×C₃
prime radical	C₂³×C₃	C₂×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ G

51 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#63365, #63439, #63440, …
maximal subgroup of	#63365, C₉×#99, #63439, …
maximal normal subgroup of	#63365, C₉×#99, #63439, …
maximal quotient of	C₉×#95, C₉×C₂²⋊C₈, C₉×#99, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 21);