C₃×#89298 #13051169

order	1344
isoclinism	isoclinic class of D₁₁₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#89298

Splittings

Hall	C₃×C₇ ⋊ #358
Sylow	C₇ ⋊ C₃×#358

Subgroups

	subgroup	G/H
center	C₂²×C₃	D₅₆
commutator	C₇×C₈	C₂×C₃×C₄
Frattini	C₂×C₈	C₂×C₃×D₇
Fitting	C₂×C₃×C₇×C₁₆	C₂
socle, prime radical	C₂²×C₃×C₇	D₈
cosocle	C₂×C₇×C₈	C₂²×C₃
hypercenter	C₂×C₃×C₁₆	D₇
hypocenter	C₇	C₃×#358
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×C₁₆

116 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 116 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 30
transitive degree	μ_t(G) = 1344

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,3)(4,5,7,9,10,8,6)(11,13,14,12)(15,16,21,30,19,23,17,20,18,22,24,29,25,27,26,28);(5,6)(7,8)(9,10)(11,12,14,13)(16,20)(17,24)(19,25)(21,26)(22,28)(23,30)(27,29)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 458);