C₇×#4508 #13051289

order	1344
isoclinism	isoclinic class of #204
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#4508

Splittings

Fitting	C₂×C₃×C₇×C₁₆ ⋊ C₂
Hall	C₃×C₇ ⋊ #350
Sylow	C₃ ⋊ C₇×#350

Subgroups

	subgroup	G/H
center	C₇×C₈	D₁₂
commutator	C₃×C₄	C₂×C₇×C₈
Frattini	C₂×C₈	C₂×D₃×C₇
Fitting	C₂×C₃×C₇×C₁₆	C₂
socle	C₂×C₃×C₇	C₂²⋊C₈
prime radical	C₇×#235	C₄
cosocle	C₂×C₃×C₈	C₂²×C₇
hypercenter	C₂×C₇×C₁₆	D₃
hypocenter	C₃	C₇×#350
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₇×C₈ ◁ C₂×C₇×C₈ ◁ C₂×C₇×C₁₆

112 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 578);