C₇×D₁₅₃ #14523874

order	2142
isoclinism	isoclinic class of D₁₅₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₁₅₃
unambiguous semidirect product	C₉⋊(C₇×D₁₇)
unambiguous semidirect product	C₁₇⋊(C₇×D₉)

Splittings

Fitting, Hall	C₇×C₉×C₁₇ ⋊ C₂
derived, Hall	C₉×C₁₇ ⋊ C₂×C₇
Hall	C₇×C₁₇ ⋊ D₉
Hall	C₇×C₉ ⋊ D₁₇
Sylow	C₁₇ ⋊ C₇×D₉
Sylow	C₉ ⋊ C₇×D₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₁₅₃
commutator, cosocle, hypocenter	C₉×C₁₇	C₂×C₇
Frattini	C₃	C₇×D₅₁
Fitting	C₇×C₉×C₁₇	C₂
socle	C₃×C₇×C₁₇	D₃
LCS	G ▷ C₉×C₁₇
UCS	C₁ ◁ C₇

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 1071

A minimal faithful permutation representation — 2 generators on 33 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);(9,16)(10,15)(11,14)(12,13)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×D₉×D₁₇, D₇×D₁₅₃, D₆₃⋊D₁₇, …
maximal normal subgroup of	C₇×D₉×D₁₇, D₇×D₁₅₃, D₆₃⋊D₁₇, …
maximal quotient of	C₇×Dic₁₅₃, C₂×C₇×D₁₅₃, C₄₉×D₁₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2142, 10);

C7×D153 #14523874