C₇⋊C₆₄ #89239

order	448
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₂×C₁₆×AGL(1,𝔽₇) (order 1344) · Inn(G) = D₇ · Out(G) = C₂×C₃×C₁₆
homology	Abelianization (H₁) = C₆₄ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₇⋊C₆₄

Splittings

derived, Sylow

C₇ ⋊ C₆₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃₂	D₇
commutator, hypocenter	C₇	C₆₄
Fitting, cosocle	C₇×C₃₂	C₂
socle, prime radical	C₂×C₇	C₃₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₃₂

14 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 2 generators on 71 points:

(2,3)(4,5)(6,7)(8,9,10,15,11,16,20,30,12,17,21,31,24,34,40,50,13,18,22,32,25,35,41,51,27,37,43,53,46,56,60,65,14,19,23,33,26,36,42,52,28,38,44,54,47,57,61,66,29,39,45,55,48,58,62,67,49,59,63,68,64,69,70,71);(1,2,4,6,7,5,3)

Polycyclic presentation:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×C₆₄, #11725238, C₂×C₇⋊C₆₄, …
maximal normal subgroup of	D₇×C₆₄, #11725238, C₂×C₇⋊C₆₄, …
maximal quotient of	C₇⋊C₁₂₈, C₂×C₇⋊C₆₄, C₃×C₇⋊C₆₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1);

C7⋊C64 #89239