C₃×#6505 #10621253

order	672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#6505

Splittings

Fitting	C₂×C₃×C₇×Q₈ ⋊ C₂
Hall	C₃×C₇ ⋊ #137
Sylow	C₇ ⋊ C₃×#137

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×C₂²⋊D₇
commutator, cosocle	C₄×C₇	C₂³×C₃
Frattini	C₄	C₂²×C₃×D₇
Fitting	C₂×C₃×C₇×Q₈	C₂
socle	C₂×C₃×C₇	C₂×D₄
prime radical	C₃×#1157	C₂
hypercenter	C₂×C₃×Q₈	D₇
hypocenter	C₇	C₃×#137
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×Q₈

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

Conjugacy classes & character table

123 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#13054759, #13054761, #13054768, …
maximal normal subgroup of	#13054759, #13054761, #13054768, …
maximal quotient of	C₃×#89743, C₃×#89736, C₃×#89738, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 835);