C₇×Dic₅₁ #13083607

order	1428
isoclinism	isoclinic class of D₅₁
automorphism	?
homology	Abelianization (H₁) = C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×Dic₅₁
unambiguous semidirect product	C₃⋊(C₇×Dic₁₇)

Splittings

derived, Hall	C₃×C₁₇ ⋊ C₄×C₇
Hall	C₃×C₇×C₁₇ ⋊ C₄
Hall	C₇×C₁₇ ⋊ Dic₃
Hall	C₃×C₇ ⋊ Dic₁₇
Sylow	C₁₇ ⋊ C₇×Dic₃
Sylow	C₃ ⋊ C₇×Dic₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇	D₅₁
commutator, hypocenter	C₃×C₁₇	C₄×C₇
Frattini	C₂	C₇×D₅₁
Fitting, socle, prime radical	C₂×C₃×C₇×C₁₇	C₂
cosocle	C₂×C₃×C₁₇	C₂×C₇
LCS	G ▷ C₃×C₁₇
UCS	C₁ ◁ C₂×C₇

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);(8,10,9,11)(13,14)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×Dic₁₅₃, ΓL(1,𝔽₈)×Dic₅₁, C₃×C₇×Dic₅₁, …
maximal normal subgroup of	ΓL(1,𝔽₈)×Dic₅₁, C₃×C₇×Dic₅₁, C₇×C₁₁×Dic₅₁, …
maximal quotient of	C₇×Dic₁₅₃, C₃×C₇×Dic₅₁, C₇×C₃⋊Dic₅₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1428, 9);

C7×Dic51 #13083607