C₃×Dic₁₁₂ #13051110

order	1344
isoclinism	isoclinic class of D₂₂₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×Dic₁₁₂

Splittings

Hall	C₃×C₇ ⋊ Q₆₄
Sylow	C₇ ⋊ C₃×Q₆₄

Subgroups

	subgroup	G/H
center	C₂×C₃	D₁₁₂
commutator, cosocle	C₇×C₁₆	C₂²×C₃
Frattini	C₁₆	C₂×C₃×D₇
Fitting	C₃×C₇×C₃₂	C₂
socle, prime radical	C₂×C₃×C₇	D₁₆
hypercenter	C₃×C₃₂	D₇
hypocenter	C₇	C₃×Q₆₄
LCS	G ▷ C₇×C₁₆ ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈ ◁ C₃×C₁₆ ◁ C₃×C₃₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 74
transitive degree	μ_t(G) = 1344

A minimal faithful permutation representation — 2 generators on 74 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)(36,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37)(68,69,70,71,72,73,74);(4,36,20,52)(5,37,21,53)(6,38,22,54)(7,39,23,55)(8,40,24,56)(9,41,25,57)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)(17,49,33,65)(18,50,34,66)(19,51,35,67)(69,74)(70,73)(71,72)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 399);

C3×Dic112 #13051110