C₃×#90219 #13060873

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#90219

Splittings

Fitting	C₃×C₇×#117 ⋊ C₂
Hall	C₃×C₇ ⋊ #529
Sylow	C₇ ⋊ C₃×#529

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₇×#117	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₃×#6447	C₂
hypercenter	C₃×#117	D₇
hypocenter	C₇	C₃×#529
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×#117

468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2,3)(4,5,7,9,10,8,6)(12,16)(15,18)(21,25)(23,26);(11,13,14,17)(12,15,16,18)(19,21)(20,23)(22,25)(24,26);(19,20,22,24)(21,23,25,26);(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10162);

C3×#90219 #13060873