C₁₉⋊C₈ #3884

order	152
isoclinism	isoclinic class of D₁₉
automorphism	Aut(G) = C₂²×AGL(1,𝔽₁₉) (order 1368) · Inn(G) = D₁₉ · Out(G) = C₂²×C₉
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

C₁₉⋊C₈

C₁₉ ⋊ C₈

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₁₉
commutator, hypocenter	C₁₉	C₈
Fitting, cosocle	C₄×C₁₉	C₂
socle, prime radical	C₂×C₁₉	C₄
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₄

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 152

A minimal faithful permutation representation — 2 generators on 27 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21,22,24,23,25,26,27);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(152, 1);