C₂×C₇×C₃⋊D₂₀₁ #30683252

order	16884
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₂×C₇×C₃⋊D₂₀₁
unambiguous semidirect product	(C₂×C₃)⋊(C₇×D₂₀₁)
unambiguous semidirect product	(C₂×C₆₇)⋊(C₇×C₃⋊D₃)

Splittings

Fitting, socle	C₂×C₃²×C₇×C₆₇ ⋊ C₂
derived, Hall	C₃²×C₆₇ ⋊ C₂²×C₇
Hall	C₃²×C₇×C₆₇ ⋊ C₂²
Hall	C₇×C₆₇ ⋊ C₂×C₃⋊D₃
Sylow	C₆₇ ⋊ C₂×C₇×C₃⋊D₃
Hall	C₃²×C₇ ⋊ C₂×D₆₇
Sylow	C₃² ⋊ C₂×C₇×D₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇	C₃⋊D₂₀₁
commutator, cosocle, hypocenter	C₃²×C₆₇	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₇×C₆₇	C₂
LCS	G ▷ C₃²×C₆₇
UCS	C₁ ◁ C₂×C₇

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 82
transitive degree	μ_t(G) = 8442

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

Conjugacy classes & character table

4242 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16884, 221);

C2×C7×C3⋊D201 #30683252