C₃×#89305 #13051170

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

Aliases

direct product

C₃×#89305

Splittings

Fitting	C₂×C₃×C₇×C₁₆ ⋊ C₂
Hall	C₃×C₇ ⋊ #359
Sylow	C₇ ⋊ C₃×#359

Subgroups

	subgroup	G/H
center	C₃×C₄	D₅₆
commutator	C₇×C₈	C₂×C₃×C₄
Frattini	C₂×C₈	C₂×C₃×D₇
Fitting	C₂×C₃×C₇×C₁₆	C₂
socle	C₂×C₃×C₇	#102
prime radical	C₃×#6467	C₂
cosocle	C₂×C₇×C₈	C₂²×C₃
hypercenter	C₂×C₃×C₁₆	D₇
hypocenter	C₇	C₃×#359
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×C₁₆

112 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 42
transitive degree	μ_t(G) = 672

A minimal faithful permutation representation — 2 generators on 42 points:

(1,2,3)(4,5,7,9,10,8,6)(11,13,14,22,26,36,41,42,16,24,28,38,15,23,27,37)(12,17,30,34,29,33,25,35,19,31,18,21,20,32,40,39);(5,6)(7,8)(9,10)(11,12)(13,21)(14,25)(15,29)(16,19)(17,22)(18,27)(20,26)(23,39)(24,34)(28,40)(30,41)(31,38)(32,42)(33,37)(35,36)

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, 459);