C₅⋊D₅×C₆₇ #16809778

order	3350
isoclinism	isoclinic class of C₅⋊D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₆₇ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅⋊D₅×C₆₇
unambiguous semidirect product	C₅⋊(D₅×C₆₇)

Splittings

Fitting, socle, Hall	C₅²×C₆₇ ⋊ C₂
derived, Sylow	C₅² ⋊ C₂×C₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₆₇	C₅⋊D₅
commutator, cosocle, hypocenter	C₅²	C₂×C₆₇
Frattini	C₁	G
Fitting, socle	C₅²×C₆₇	C₂
LCS	G ▷ C₅²
UCS	C₁ ◁ C₆₇

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 1675

A minimal faithful permutation representation — 3 generators on 77 points:

(2,3)(4,5)(7,8)(9,10)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(6,7,9,10,8);(1,2,4,5,3)

Conjugacy classes & character table

938 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇×C₅⋊D₅×C₆₇, C₆₇×C₅⋊D₃₅, C₅⋊D₅×C₆₇⋊C₁₁, …
maximal normal subgroup of	C₇×C₅⋊D₅×C₆₇, C₅⋊D₅×C₆₇⋊C₁₁, C₁₁×C₅⋊D₅×C₆₇
maximal quotient of	C₇×C₅⋊D₅×C₆₇, C₆₇×C₅⋊D₃₅, C₁₁×C₅⋊D₅×C₆₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3350, 9);

C5⋊D5×C67 #16809778