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#1044

order	100
isoclinism	isoclinic class of #1044
automorphism	Aut(G) = AGL(1,𝔽₅)²⋊C₂ (order 800) · Inn(G) = G · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₅

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow

C₅² ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²	C₄
prime radical, cosocle	C₅⋊D₅	C₂
LCS	G ▷ C₅²
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 10 points:

(1,2)(3,6,4,8)(5,9,10,7);(2,8,10,5,6)

Polycyclic presentation:

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	4	4	4	4	4	4	25	25	25
order	1	5	5	5	5	5	5	2	4	4
χ₁	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	1	-1	-1
χ₃	1	1	1	1	1	1	1	-1	$i$	$- i$
χ₄	1	1	1	1	1	1	1	-1	$- i$	$i$
χ₅	4	4	-1	-1	-1	-1	-1	0	0	0
χ₆	4	-1	4	-1	-1	-1	-1	0	0	0
χ₇	4	-1	-1	$- 1 + \sqrt{5}$	$\frac{3 - \sqrt{5}}{2}$	$\frac{3 + \sqrt{5}}{2}$	$- 1 - \sqrt{5}$	0	0	0
χ₈	4	-1	-1	$\frac{3 - \sqrt{5}}{2}$	$- 1 - \sqrt{5}$	$- 1 + \sqrt{5}$	$\frac{3 + \sqrt{5}}{2}$	0	0	0
χ₉	4	-1	-1	$\frac{3 + \sqrt{5}}{2}$	$- 1 + \sqrt{5}$	$- 1 - \sqrt{5}$	$\frac{3 - \sqrt{5}}{2}$	0	0	0
χ₁₀	4	-1	-1	$- 1 - \sqrt{5}$	$\frac{3 + \sqrt{5}}{2}$	$\frac{3 - \sqrt{5}}{2}$	$- 1 + \sqrt{5}$	0	0	0

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

Relationships

derived subgroup of	AGL(1,𝔽₅)²⋊C₂, #13107437, #13107438, …
maximal subgroup of	D₅⋊AGL(1,𝔽₅), D₅²⋊C₂, #6057, …
maximal normal subgroup of	D₅⋊AGL(1,𝔽₅), D₅²⋊C₂, #6057, …
maximal quotient of	#6034, C₂×#1044, C₃×#1044, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(100, 12);