Finite Group Databasebeta

← Prev (#72209673) Next (#72209675) →

#72209674

order	2904
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, Sylow	C₁₁² ⋊ SL(2,𝔽₃)
derived, Hall	#11757263 ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	#11757263	C₃
Fitting, socle	C₁₁²	SL(2,𝔽₃)
LCS	G ▷ #11757263
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

3,2^3,11^2|b^2=d,c^2=d,b^a=c,c^a=bcd,c^b=cd,e^a=e^5f^9,e^b=e^8f^4,e^c=e^6f^5,e^d=e^10,f^a=e^10f^5,f^b=e^3f^3,f^c=e^8f^5,f^d=f^10

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	24	24	24	24	24	121	484	484	484	484	726
order	1	11	11	11	11	11	2	3	3	6	6	4
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1
χ₃	1	1	1	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1
χ₄	2	2	2	2	2	2	-2	-1	-1	1	1	0
χ₅	2	2	2	2	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	2	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	3	3	3	3	0	0	0	0	-1
χ₈	24	2ζ²−2ζ⁴−2ζ⁷−ζ⁸+2ζ¹³−ζ¹⁴−ζ¹⁹+2ζ²⁰−ζ²⁵−2ζ²⁶−2ζ²⁹+2ζ³¹	2ζ⁴−ζ⁵+2ζ⁷−2ζ⁸−2ζ¹⁴−ζ¹⁶−ζ¹⁷−2ζ¹⁹−2ζ²⁵+2ζ²⁶−ζ²⁸+2ζ²⁹	2ζ−2ζ²−ζ⁴−ζ⁷+2ζ¹⁰−2ζ¹³−2ζ²⁰+2ζ²³−ζ²⁶−ζ²⁹−2ζ³¹+2ζ³²	−2ζ−ζ²+2ζ⁵−2ζ¹⁰−ζ¹³+2ζ¹⁶+2ζ¹⁷−ζ²⁰−2ζ²³+2ζ²⁸−ζ³¹−2ζ³²	−ζ−2ζ⁵+2ζ⁸−ζ¹⁰+2ζ¹⁴−2ζ¹⁶−2ζ¹⁷+2ζ¹⁹−ζ²³+2ζ²⁵−2ζ²⁸−ζ³²	0	0	0	0	0	0
χ₉	24	2ζ⁴−ζ⁵+2ζ⁷−2ζ⁸−2ζ¹⁴−ζ¹⁶−ζ¹⁷−2ζ¹⁹−2ζ²⁵+2ζ²⁶−ζ²⁸+2ζ²⁹	−ζ−2ζ⁵+2ζ⁸−ζ¹⁰+2ζ¹⁴−2ζ¹⁶−2ζ¹⁷+2ζ¹⁹−ζ²³+2ζ²⁵−2ζ²⁸−ζ³²	2ζ²−2ζ⁴−2ζ⁷−ζ⁸+2ζ¹³−ζ¹⁴−ζ¹⁹+2ζ²⁰−ζ²⁵−2ζ²⁶−2ζ²⁹+2ζ³¹	2ζ−2ζ²−ζ⁴−ζ⁷+2ζ¹⁰−2ζ¹³−2ζ²⁰+2ζ²³−ζ²⁶−ζ²⁹−2ζ³¹+2ζ³²	−2ζ−ζ²+2ζ⁵−2ζ¹⁰−ζ¹³+2ζ¹⁶+2ζ¹⁷−ζ²⁰−2ζ²³+2ζ²⁸−ζ³¹−2ζ³²	0	0	0	0	0	0
χ₁₀	24	2ζ−2ζ²−ζ⁴−ζ⁷+2ζ¹⁰−2ζ¹³−2ζ²⁰+2ζ²³−ζ²⁶−ζ²⁹−2ζ³¹+2ζ³²	2ζ²−2ζ⁴−2ζ⁷−ζ⁸+2ζ¹³−ζ¹⁴−ζ¹⁹+2ζ²⁰−ζ²⁵−2ζ²⁶−2ζ²⁹+2ζ³¹	−2ζ−ζ²+2ζ⁵−2ζ¹⁰−ζ¹³+2ζ¹⁶+2ζ¹⁷−ζ²⁰−2ζ²³+2ζ²⁸−ζ³¹−2ζ³²	−ζ−2ζ⁵+2ζ⁸−ζ¹⁰+2ζ¹⁴−2ζ¹⁶−2ζ¹⁷+2ζ¹⁹−ζ²³+2ζ²⁵−2ζ²⁸−ζ³²	2ζ⁴−ζ⁵+2ζ⁷−2ζ⁸−2ζ¹⁴−ζ¹⁶−ζ¹⁷−2ζ¹⁹−2ζ²⁵+2ζ²⁶−ζ²⁸+2ζ²⁹	0	0	0	0	0	0
χ₁₁	24	−2ζ−ζ²+2ζ⁵−2ζ¹⁰−ζ¹³+2ζ¹⁶+2ζ¹⁷−ζ²⁰−2ζ²³+2ζ²⁸−ζ³¹−2ζ³²	2ζ−2ζ²−ζ⁴−ζ⁷+2ζ¹⁰−2ζ¹³−2ζ²⁰+2ζ²³−ζ²⁶−ζ²⁹−2ζ³¹+2ζ³²	−ζ−2ζ⁵+2ζ⁸−ζ¹⁰+2ζ¹⁴−2ζ¹⁶−2ζ¹⁷+2ζ¹⁹−ζ²³+2ζ²⁵−2ζ²⁸−ζ³²	2ζ⁴−ζ⁵+2ζ⁷−2ζ⁸−2ζ¹⁴−ζ¹⁶−ζ¹⁷−2ζ¹⁹−2ζ²⁵+2ζ²⁶−ζ²⁸+2ζ²⁹	2ζ²−2ζ⁴−2ζ⁷−ζ⁸+2ζ¹³−ζ¹⁴−ζ¹⁹+2ζ²⁰−ζ²⁵−2ζ²⁶−2ζ²⁹+2ζ³¹	0	0	0	0	0	0
χ₁₂	24	−ζ−2ζ⁵+2ζ⁸−ζ¹⁰+2ζ¹⁴−2ζ¹⁶−2ζ¹⁷+2ζ¹⁹−ζ²³+2ζ²⁵−2ζ²⁸−ζ³²	−2ζ−ζ²+2ζ⁵−2ζ¹⁰−ζ¹³+2ζ¹⁶+2ζ¹⁷−ζ²⁰−2ζ²³+2ζ²⁸−ζ³¹−2ζ³²	2ζ⁴−ζ⁵+2ζ⁷−2ζ⁸−2ζ¹⁴−ζ¹⁶−ζ¹⁷−2ζ¹⁹−2ζ²⁵+2ζ²⁶−ζ²⁸+2ζ²⁹	2ζ²−2ζ⁴−2ζ⁷−ζ⁸+2ζ¹³−ζ¹⁴−ζ¹⁹+2ζ²⁰−ζ²⁵−2ζ²⁶−2ζ²⁹+2ζ³¹	2ζ−2ζ²−ζ⁴−ζ⁷+2ζ¹⁰−2ζ¹³−2ζ²⁰+2ζ²³−ζ²⁶−ζ²⁹−2ζ³¹+2ζ³²	0	0	0	0	0	0

Here ζ is a primitive root of unity of order 33 (ζ³³ = 1); i = ζ⁸.

Relationships

maximal subgroup of

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [3, 2, 2, 2, 11, 11]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.3);;
gap> SetConjugate(coll, 3, 1, F.2*F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^5*F.6^9);;
gap> SetConjugate(coll, 5, 2, F.5^8*F.6^4);;
gap> SetConjugate(coll, 5, 3, F.5^6*F.6^5);;
gap> SetConjugate(coll, 5, 4, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.5^10*F.6^5);;
gap> SetConjugate(coll, 6, 2, F.5^3*F.6^3);;
gap> SetConjugate(coll, 6, 3, F.5^8*F.6^5);;
gap> SetConjugate(coll, 6, 4, F.6^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);