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C₂×A₅ #1225

order	120
isoclinism	isoclinic class of A₅
automorphism	Aut(G) = S₅ (order 120) · Inn(G) = A₅ · Out(G) = C₂
holomorph	G ⋊ Aut(G) = C₂×#72219548
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

Aliases

C₂×A₅

Subgroups

	subgroup	G/H
center, Fitting, hypercenter	C₂	A₅
commutator, hypocenter	A₅	C₂
Frattini	C₁	G
LCS	G ▷ A₅
UCS	C₁ ◁ C₂

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 7
transitive degree	μ_t(G) = 10

A minimal faithful permutation representation — 2 generators on 7 points:

(3,7)(4,5);(1,2)(3,6,5)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	12	12	15	20	1	12	12	15	20
order	1	5	5	2	3	2	10	10	2	6
χ₁	1	1	1	1	1	1	1	1	1	1
χ₂	3	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$	-1	0	3	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$	-1	0
χ₃	3	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$	-1	0	3	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$	-1	0
χ₄	4	-1	-1	0	1	4	-1	-1	0	1
χ₅	5	0	0	1	-1	5	0	0	1	-1
χ₆	1	1	1	1	1	-1	-1	-1	-1	-1
χ₇	3	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$	-1	0	-3	$\frac{- 1 + \sqrt{5}}{2}$	$\frac{- 1 - \sqrt{5}}{2}$	1	0
χ₈	3	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$	-1	0	-3	$\frac{- 1 - \sqrt{5}}{2}$	$\frac{- 1 + \sqrt{5}}{2}$	1	0
χ₉	4	-1	-1	0	1	-4	1	1	0	-1
χ₁₀	5	0	0	1	-1	-5	0	0	-1	1

Relationships

derived subgroup of	C₄⋊S₅, A₅⋊Q₈, C₂²⋊S₅, …
socle of	A₅⋊C₄, C₄×A₅, C₂×S₅, …
maximal subgroup of	A₅⋊C₄, C₄×A₅, C₂×S₅, …
maximal normal subgroup of	A₅⋊C₄, C₄×A₅, C₂×S₅, …
maximal quotient of	C₄×A₅, #6732, C₂×SL(2,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 35);