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C₃×C₁₇×Dic₇ #13083604

order	1428
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₁₇ · Schur multiplier (H₂) = C₁

Aliases

C₃×C₁₇×Dic₇

Splittings

derived, Sylow	C₇ ⋊ C₃×C₄×C₁₇
Hall	C₃×C₇×C₁₇ ⋊ C₄
Hall	C₇×C₁₇ ⋊ C₃×C₄
Hall	C₃×C₇ ⋊ C₄×C₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₇	D₇
commutator, hypocenter	C₇	C₃×C₄×C₁₇
Frattini	C₂	C₃×D₇×C₁₇
Fitting, socle, prime radical	C₂×C₃×C₇×C₁₇	C₂
cosocle	C₂×C₇	C₂×C₃×C₁₇
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₁₇

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 1428

A minimal faithful permutation representation — 2 generators on 31 points:

(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22)(23,24)(25,26,27,28,29,30,31);(21,23,22,24)(26,31)(27,30)(28,29)

Conjugacy classes & character table

510 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₇×Dic₇⋊C₉, C₉×C₁₇×Dic₇, C₃×C₁₇×ΓL(1,𝔽₈)⋊C₄, …
maximal normal subgroup of	C₁₇×Dic₇⋊C₉, C₉×C₁₇×Dic₇, C₃×C₁₇×ΓL(1,𝔽₈)⋊C₄, …
maximal quotient of	C₉×C₁₇×Dic₇, C₃²×C₁₇×Dic₇, C₃×C₁₇×Dic₂₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1428, 6);