C₇×Dic₆₉ #14515374

order	1932
isoclinism	isoclinic class of D₆₉
automorphism	?
homology	Abelianization (H₁) = C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×Dic₆₉
unambiguous semidirect product	C₃⋊(C₇×Dic₂₃)
unambiguous semidirect product	C₂₃⋊(C₇×Dic₃)

Splittings

derived, Hall	C₃×C₂₃ ⋊ C₄×C₇
Hall	C₃×C₇×C₂₃ ⋊ C₄
Hall	C₇×C₂₃ ⋊ Dic₃
Sylow	C₂₃ ⋊ C₇×Dic₃
Hall	C₃×C₇ ⋊ Dic₂₃
Sylow	C₃ ⋊ C₇×Dic₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇	D₆₉
commutator, hypocenter	C₃×C₂₃	C₄×C₇
Frattini	C₂	C₇×D₆₉
Fitting, socle, prime radical	C₂×C₃×C₇×C₂₃	C₂
cosocle	C₂×C₃×C₂₃	C₂×C₇
LCS	G ▷ C₃×C₂₃
UCS	C₁ ◁ C₂×C₇

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 1932

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37);(8,10,9,11)(13,14)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄₉×Dic₆₉, C₇²×Dic₆₉, C₇×Dic₄₈₃, …
maximal normal subgroup of	C₄₉×Dic₆₉, C₇²×Dic₆₉, C₇×C₁₁×Dic₆₉, …
maximal quotient of	C₄₉×Dic₆₉, C₇²×Dic₆₉, C₇×Dic₄₈₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1932, 9);

C7×Dic69 #14515374