(C₁₉⋊C₃)⋊D₁₃₃ #29503141

order	15162
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

(C₁₉⋊C₃)⋊D₁₃₃

Splittings

Fitting, socle, derived, Hall	C₇×C₁₉² ⋊ C₂×C₃
Hall	C₇×C₁₉×C₁₉⋊C₃ ⋊ C₂
Hall	C₇⋊(C₁₉⋊D₁₉) ⋊ C₃
Hall	C₁₉×C₁₉⋊C₃ ⋊ D₇
Sylow	C₁₉² ⋊ C₃×D₇
Sylow	C₇ ⋊ (C₁₉⋊C₃)⋊D₁₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₉²	C₂×C₃
LCS	G ▷ C₇×C₁₉²
UCS	C₁

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 399

A minimal faithful permutation representation — 2 generators on 45 points:

(2,3)(4,5)(6,7)(9,13,14,10,11,15)(12,21,18,16,17,23)(19,24,26,25,22,20)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45);(1,2,4,6,7,5,3)(8,9,12,18,24,21,25,11,15,20,26,14,13,19,17,22,23,16,10)(27,28,30,32,34,36,38,40,42,44,45,43,41,39,37,35,33,31,29)

Polycyclic presentation:

2,3,7,19^2|c^a=c^6,d^a=d^18,e^a=e^18,d^b=d^7

Conjugacy classes & character table

603 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×(C₁₉⋊C₃)⋊D₁₉, D₁₉×(C₁₉⋊C₃)⋊D₇, D₁₉⋊C₃×D₁₃₃, …
maximal normal subgroup of	D₇×(C₁₉⋊C₃)⋊D₁₉, D₁₉×(C₁₉⋊C₃)⋊D₇, D₁₉⋊C₃×D₁₃₃, …
maximal quotient of	(C₁₉⋊C₃)⋊Dic₁₃₃, C₂×(C₁₉⋊C₃)⋊D₁₃₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15162, 76);

(C19⋊C3)⋊D133 #29503141