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#6057⋊C₃ #10597169

order	600
isoclinism	isoclinic class of #6057⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₅

Aliases

unambiguous semidirect product

Splittings

Fitting, socle, Sylow	C₅² ⋊ SL(2,𝔽₃)
derived, Hall	#6057 ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	#6057	C₃
Fitting, socle	C₅²	SL(2,𝔽₃)
LCS	G ▷ #6057
UCS	C₁

17 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 25

Polycyclic presentation:

3,2^3,5^2|b^2=d,c^2=d,b^a=bc,c^a=b,e^a=ef^3,f^a=e^4f^3,c^b=cd,e^b=e^3f^3,f^b=f^2,e^c=e^3,f^c=e^4f^2,e^d=e^4,f^d=f^4

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈
size	1	24	25	100	100	100	100	150
order	1	5	2	3	3	6	6	4
χ₁	1	1	1	1	1	1	1	1
χ₂	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1
χ₃	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1
χ₄	2	2	-2	-1	-1	1	1	0
χ₅	2	2	-2	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	0
χ₆	2	2	-2	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	0
χ₇	3	3	3	0	0	0	0	-1
χ₈	24	-1	0	0	0	0	0	0

Here i is the imaginary unit (i² = −1).

Relationships

derived subgroup of	C₂²×#72210663, C₄×#72210663
maximal subgroup of	((D₅⋊AGL(1,𝔽₅))⋊C₂)⋊C₃, C₂×#6057⋊C₃, C₃×#6057⋊C₃, …
maximal normal subgroup of	((D₅⋊AGL(1,𝔽₅))⋊C₂)⋊C₃, C₂×#6057⋊C₃, C₃×#6057⋊C₃
maximal quotient of	C₂×#6057⋊C₃, #6057⋊C₉, C₃×#6057⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(600, 150);