C₁₇×D₃₇₁ #27162196

order	12614
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₇×D₃₇₁
unambiguous semidirect product	C₇⋊(C₁₇×D₅₃)
unambiguous semidirect product	C₅₃⋊(D₇×C₁₇)

Splittings

Fitting, socle, Hall	C₇×C₁₇×C₅₃ ⋊ C₂
derived, Hall	C₇×C₅₃ ⋊ C₂×C₁₇
Hall	C₁₇×C₅₃ ⋊ D₇
Hall	C₇×C₁₇ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₇×C₁₇
Sylow	C₇ ⋊ C₁₇×D₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₇	D₃₇₁
commutator, cosocle, hypocenter	C₇×C₅₃	C₂×C₁₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₇×C₅₃	C₂
LCS	G ▷ C₇×C₅₃
UCS	C₁ ◁ C₁₇

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

Conjugacy classes & character table

3179 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇×C₁₇×D₅₃, D₁₇×D₃₇₁, D₁₁₉⋊D₅₃, …
maximal normal subgroup of	D₇×C₁₇×D₅₃, D₁₇×D₃₇₁, D₁₁₉⋊D₅₃, …
maximal quotient of	C₁₇×Dic₃₇₁, C₂×C₁₇×D₃₇₁, C₃×C₁₇×D₃₇₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12614, 6);

C17×D371 #27162196