C₇×#909 #10621306

order	672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#909

Splittings

Fitting	C₃×C₇×M₁₆ ⋊ C₂
Hall	C₃×C₇ ⋊ #137
Sylow	C₃ ⋊ C₇×#137

Subgroups

	subgroup	G/H
center	C₂×C₇	C₂×D₁₂
commutator, cosocle	C₃×C₄	C₂³×C₇
Frattini	C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×M₁₆	C₂
socle	C₂×C₃×C₇	C₂×D₄
prime radical	C₇×#235	C₂
hypercenter	C₇×M₁₆	D₃
hypocenter	C₃	C₇×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×M₁₆

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

normal subgroups only

This group has 120 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) = 26
transitive degree	μ_t(G) = 336

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

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

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#13053553, #13053555, #13053593, …
maximal normal subgroup of	#13053553, #13053555, #13053593, …
maximal quotient of	C₇×#4702, C₇×#4714, C₇×#4703, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 888);