C₂×C₁₉×D₁₇₇ #27212119

order	13452
isoclinism	isoclinic class of D₁₇₇
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₁₉×D₁₇₇
unambiguous semidirect product	(C₂×C₃)⋊(C₁₉×D₅₉)
unambiguous semidirect product	(C₂×C₅₉)⋊(D₃×C₁₉)

Splittings

Fitting, socle	C₂×C₃×C₁₉×C₅₉ ⋊ C₂
derived, Hall	C₃×C₅₉ ⋊ C₂²×C₁₉
Hall	C₃×C₁₉×C₅₉ ⋊ C₂²
Hall	C₁₉×C₅₉ ⋊ C₂×D₃
Sylow	C₅₉ ⋊ C₂×D₃×C₁₉
Hall	C₃×C₁₉ ⋊ C₂×D₅₉
Sylow	C₃ ⋊ C₂×C₁₉×D₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₉	D₁₇₇
commutator, cosocle, hypocenter	C₃×C₅₉	C₂²×C₁₉
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₉×C₅₉	C₂
LCS	G ▷ C₃×C₅₉
UCS	C₁ ◁ C₂×C₁₉

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 6726

A minimal faithful permutation representation — 2 generators on 83 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)(61,62)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(60,61,62)(82,83)

Conjugacy classes & character table

3420 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13452, 30);

C2×C19×D177 #27212119