D₅₇⋊C₄ #90694

order	456
isoclinism	isoclinic class of D₃×D₁₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

D₅₇⋊C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×D₁₉
commutator, hypocenter	C₃×C₁₉	C₂×C₄
Fitting, socle, cosocle	C₂×C₃×C₁₉	C₂²
prime radical	C₂×D₅₇	C₂
LCS	G ▷ C₃×C₁₉
UCS	C₁ ◁ C₂

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 228

A minimal faithful permutation representation — 2 generators on 26 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 14);