Finite Group Databasebeta

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C₇×#828 #10620626

order	672
isoclinism	isoclinic class of #826
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Fitting	C₃×C₇×Q₁₆ ⋊ C₂
Hall	C₃×C₇ ⋊ SD₃₂
Sylow	C₃ ⋊ C₇×SD₃₂

Subgroups

	subgroup	G/H
center	C₂×C₇	#213
commutator, cosocle	C₃×C₈	C₂²×C₇
Frattini	C₈	C₂×D₃×C₇
Fitting	C₃×C₇×Q₁₆	C₂
socle	C₂×C₃×C₇	D₈
prime radical	C₇×D₂₄	C₂
hypercenter	C₇×Q₁₆	D₃
hypocenter	C₃	C₇×SD₃₂
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈ ◁ C₇×Q₁₆

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

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#828, #13053668, #13053671, …
maximal normal subgroup of	D₇×#828, #13053668, #13053671, …
maximal quotient of	C₇×#4490, C₇×#4491, C₇×#4564, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 208);