C₇⋊SL(2,ℤ/4) #66651

order	336
isoclinism	isoclinic class of C₇⋊S₄
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	C₇⋊SL(2,ℤ/4)
unambiguous semidirect product	A₄⋊Dic₇

derived	C₇×A₄ ⋊ C₄
Sylow	C₇ ⋊ SL(2,ℤ/4)

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₇⋊S₄
commutator, hypocenter	C₇×A₄	C₄
Fitting, socle	C₂³×C₇	D₃
prime radical, cosocle	C₂×C₇×A₄	C₂
LCS	G ▷ C₇×A₄
UCS	C₁ ◁ C₂

38 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 2 generators on 15 points:

(1,2,3,4)(5,6,8,7)(10,11)(12,13)(14,15);(6,7,8)(9,10,12,14,15,13,11)

Polycyclic presentation:

2^2,3,7,2^2|a^2=b,c^a=c^2,d^a=d^6,e^a=f,f^a=e,e^c=f,f^c=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 120);