C₃²×Dic₁₃₃ #18012200

order	4788
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃²×Dic₁₃₃

Splittings

derived, Hall	C₇×C₁₉ ⋊ C₃²×C₄
Hall	C₃²×C₇×C₁₉ ⋊ C₄
Hall	C₃²×C₁₉ ⋊ Dic₇
Hall	C₃²×C₇ ⋊ Dic₁₉
Sylow	C₁₉ ⋊ C₃²×Dic₇
Sylow	C₇ ⋊ C₃²×Dic₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃²	D₁₃₃
commutator, hypocenter	C₇×C₁₉	C₃²×C₄
Frattini	C₂	C₃²×D₁₃₃
Fitting, socle, prime radical	C₂×C₃²×C₇×C₁₉	C₂
cosocle	C₂×C₇×C₁₉	C₂×C₃²
LCS	G ▷ C₇×C₁₉
UCS	C₁ ◁ C₂×C₃²

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 4788

A minimal faithful permutation representation — 2 generators on 36 points:

(1,2,3)(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);(4,5,6)(7,9,8,10)(12,17)(13,16)(14,15)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)

Conjugacy classes & character table

1224 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃²×Dic₆₆₅, C₃²×C₅×Dic₁₃₃, C₃²×C₇⋊(C₁₉⋊AGL(1,𝔽₅)), …
maximal normal subgroup of	C₃²×C₅×Dic₁₃₃, C₃²×C₇×Dic₁₃₃
maximal quotient of	C₃²×Dic₆₆₅, C₃²×C₅×Dic₁₃₃, C₃²×C₇⋊(C₁₉⋊AGL(1,𝔽₅)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4788, 116);

C32×Dic133 #18012200