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C₇×#800 #10620598

order	672
isoclinism	isoclinic class of D₄₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Fitting	C₃×C₇×C₁₆ ⋊ C₂
Hall	C₃×C₇ ⋊ SD₃₂
Sylow	C₃ ⋊ C₇×SD₃₂

Subgroups

	subgroup	G/H
center	C₂×C₇	D₂₄
commutator, cosocle	C₃×C₈	C₂²×C₇
Frattini	C₈	C₂×D₃×C₇
Fitting	C₃×C₇×C₁₆	C₂
socle	C₂×C₃×C₇	D₈
prime radical	C₇×D₂₄	C₂
hypercenter	C₇×C₁₆	D₃
hypocenter	C₃	C₇×SD₃₂
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈ ◁ C₇×C₁₆

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 2 generators on 26 points:

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

Conjugacy classes & character table

189 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#800, #13052195, #13052201, …
maximal normal subgroup of	D₇×#800, #13052195, #13052201, …
maximal quotient of	C₇×#4505, C₇×#4509, C₇×#4503, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 180);