C₃×#89472 #13055133

order	1344
isoclinism	isoclinic class of #1132
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#89472

Splittings

Fitting	C₃×C₄×C₇×C₈ ⋊ C₂
Hall	C₃×C₇ ⋊ #443
Sylow	C₇ ⋊ C₃×#443

Subgroups

	subgroup	G/H
center	C₃×C₈	D₂₈
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₄×C₇×C₈	C₂
socle	C₂×C₃×C₇	C₄×D₄
prime radical	C₃×#6467	C₂
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₃×C₄×C₈	D₇
hypocenter	C₇	C₃×#443
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₃×C₄×C₈

212 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4422);