Finite Group Databasebeta

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C₇×#826 #10620624

order	672
isoclinism	isoclinic class of #826
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

Aliases

Splittings

Fitting	C₃×C₇×D₈ ⋊ C₂
Hall	C₃×C₇ ⋊ D₁₆
Sylow	C₃ ⋊ C₇×D₁₆

Subgroups

	subgroup	G/H
center	C₂×C₇	#213
commutator, cosocle	C₃×C₈	C₂²×C₇
Frattini	C₈	C₂×D₃×C₇
Fitting	C₃×C₇×D₈	C₂
socle	C₂×C₃×C₇	D₈
hypercenter	C₇×D₈	D₃
hypocenter	C₃	C₇×D₁₆
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈ ◁ C₇×D₈

56 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)(12,16)(13,20)(14,21)(17,18)(19,25)(22,26)(23,24);(9,10)(11,12)(13,16)(14,23)(15,19)(17,20)(18,21)(22,25)(24,26)

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#826, #13053644, #13053647, …
maximal normal subgroup of	D₇×#826, #13053644, #13053647, …
maximal quotient of	C₇×#4489, C₇×#4491, C₇×#4519, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 206);