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(C₅⋊D₅)⋊C₃ #3875

order	150
isoclinism	isoclinic class of (C₅⋊D₅)⋊C₃
automorphism	Aut(G) = AΓL(1,𝔽₂₅) (order 1200) · Inn(G) = G · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₅

Aliases

unambiguous semidirect product

(C₅⋊D₅)⋊C₃

Splittings

Fitting, socle, derived, Sylow	C₅² ⋊ C₂×C₃
Hall	C₅²⋊C₃ ⋊ C₂
Hall	C₅⋊D₅ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅²	C₂×C₃
LCS	G ▷ C₅²
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2,3,5^2|c^a=c^4,d^a=d^4,c^b=c^3d^2,d^b=cd

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	6	6	6	6	25	25	25	25	25
order	1	5	5	5	5	2	3	3	6	6
χ₁	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	1	1	1	1	1	-1	1	1	-1	-1
χ₅	1	1	1	1	1	-1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₆	1	1	1	1	1	-1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₇	6	$1 - \sqrt{5}$	$1 + \sqrt{5}$	$\frac{- 3 - \sqrt{5}}{2}$	$\frac{- 3 + \sqrt{5}}{2}$	0	0	0	0	0
χ₈	6	$1 + \sqrt{5}$	$1 - \sqrt{5}$	$\frac{- 3 + \sqrt{5}}{2}$	$\frac{- 3 - \sqrt{5}}{2}$	0	0	0	0	0
χ₉	6	$\frac{- 3 - \sqrt{5}}{2}$	$\frac{- 3 + \sqrt{5}}{2}$	$1 + \sqrt{5}$	$1 - \sqrt{5}$	0	0	0	0	0
χ₁₀	6	$\frac{- 3 + \sqrt{5}}{2}$	$\frac{- 3 - \sqrt{5}}{2}$	$1 - \sqrt{5}$	$1 + \sqrt{5}$	0	0	0	0	0

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

Relationships

derived subgroup of	AΓL(1,𝔽₂₅), C₂²×AΓL(1,𝔽₂₅), C₃×AΓL(1,𝔽₂₅), …
maximal subgroup of	#64480, #1043⋊C₃, #3874⋊C₂, …
maximal normal subgroup of	#64480, #1043⋊C₃, #3874⋊C₂, …
maximal quotient of	#1039⋊C₃, C₂×(C₅⋊D₅)⋊C₃, (C₅⋊D₅)⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(150, 6);