D₇×C₁₇×C₅₃ #27162195

order	12614
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₇×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₇×C₁₇×C₅₃
unambiguous semidirect product	C₇⋊(C₂×C₁₇×C₅₃)

Splittings

Fitting, socle, Hall	C₇×C₁₇×C₅₃ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂×C₁₇×C₅₃
Hall	C₇×C₅₃ ⋊ C₂×C₁₇
Hall	C₇×C₁₇ ⋊ C₂×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₇×C₅₃	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₁₇×C₅₃
Frattini	C₁	G
Fitting, socle	C₇×C₁₇×C₅₃	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₁₇×C₅₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6307

A minimal faithful permutation representation — 2 generators on 77 points:

(2,3)(4,5)(6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(1,2,4,6,7,5,3)

Conjugacy classes & character table

4505 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇×C₁₇×D₅₃, D₇×D₁₇×C₅₃, D₇×D₉₀₁, …
maximal normal subgroup of	D₇×C₁₇×D₅₃, D₇×D₁₇×C₅₃, D₇×D₉₀₁, …
maximal quotient of	C₁₇×Dic₇×C₅₃, C₂×D₇×C₁₇×C₅₃, C₃×D₇×C₁₇×C₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12614, 5);

D7×C17×C53 #27162195