Dic₇₈ #64621

order	312
isoclinism	isoclinic class of Dic₇₈
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

Dic₇₈

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₉
commutator, socle, prime radical, cosocle	C₂×C₃×C₁₃	C₂²
Fitting	C₃×C₄×C₁₃	C₂
hypercenter	C₄	D₃₉
hypocenter	C₃×C₁₃	Q₈
LCS	G ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 312

A minimal faithful permutation representation — 2 generators on 24 points:

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

Polycyclic presentation:

2^3,3,13|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^12

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 37);