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C₇×#1204 #11719767

order	840
isoclinism	isoclinic class of #1201
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Hall	C₃×C₅×C₇ ⋊ Q₈
Hall	C₅×C₇ ⋊ Dic₆
Hall	C₃×C₇ ⋊ Dic₁₀
Hall	C₃×C₅ ⋊ C₇×Q₈
Sylow	C₅ ⋊ C₇×Dic₆
Sylow	C₃ ⋊ C₇×Dic₁₀

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇	D₃×D₅
commutator, cosocle	C₂×C₃×C₅	C₂²×C₇
Frattini	C₂	D₃×C₇×D₅
Fitting, socle, prime radical	C₂×C₃×C₅×C₇	C₂²
hypocenter	C₃×C₅	C₇×Q₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₇

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 840

A minimal faithful permutation representation — 2 generators on 23 points:

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

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#1204, #13132510, #13132512, …
maximal normal subgroup of	D₇×#1204, #13132510, #13132512, …
maximal quotient of	C₇×Dic₃⋊Dic₅, C₇×#6669, C₇×#6668, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 69);