C₂×C₃×C₆₇ #87934

order	402
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃×C₁₁ (order 132) · Inn(G) = C₁ · Out(G) = C₂²×C₃×C₁₁
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₄₀₂
abelian	C₂×C₃×C₆₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 402

A minimal faithful permutation representation — 1 generator on 72 points:

(1,2)(3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)

Conjugacy classes & character table

402 conjugacy classes. The character table is too large to display.

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(402, 6);

center of	C₃×D₄×C₆₇, C₃×Q₈×C₆₇, C₃×Dic₃×C₆₇, …
derived subgroup of	#13107967, #13107968, #13107969, …
Fitting subgroup of	Dic₃×C₆₇, C₃×Dic₆₇, Dic₂₀₁, …
socle of	Dic₃×C₆₇, C₃×Dic₆₇, Dic₂₀₁, …
maximal subgroup of	Dic₃×C₆₇, C₃×Dic₆₇, Dic₂₀₁, …
maximal normal subgroup of	Dic₃×C₆₇, C₃×Dic₆₇, Dic₂₀₁, …
maximal quotient of	C₃×C₄×C₆₇, C₂²×C₃×C₆₇, C₂×C₉×C₆₇, …

C2×C3×C67 #87934