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G = C41⋊C8order 328 = 23·41

The semidirect product of C41 and C8 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C41⋊C8, D41.C4, C41⋊C4.C2, SmallGroup(328,12)

Series: Derived Chief Lower central Upper central

C1C41 — C41⋊C8
C1C41D41C41⋊C4 — C41⋊C8
C41 — C41⋊C8
C1

Generators and relations for C41⋊C8
 G = < a,b | a41=b8=1, bab-1=a38 >

41C2
41C4
41C8

Character table of C41⋊C8

 class 124A4B8A8B8C8D41A41B41C41D41E
 size 14141414141414188888
ρ11111111111111    trivial
ρ21111-1-1-1-111111    linear of order 2
ρ311-1-1i-i-ii11111    linear of order 4
ρ411-1-1-iii-i11111    linear of order 4
ρ51-1i-iζ85ζ83ζ87ζ811111    linear of order 8
ρ61-1-iiζ87ζ8ζ85ζ8311111    linear of order 8
ρ71-1-iiζ83ζ85ζ8ζ8711111    linear of order 8
ρ81-1i-iζ8ζ87ζ83ζ8511111    linear of order 8
ρ980000000ζ4134412541224121412041194116417ζ4133413141304124411741114110418ζ4140413841324127411441941341ζ413741364129412641154112415414ζ413941354128412341184113416412    orthogonal faithful
ρ1080000000ζ413941354128412341184113416412ζ4140413841324127411441941341ζ413741364129412641154112415414ζ4134412541224121412041194116417ζ4133413141304124411741114110418    orthogonal faithful
ρ1180000000ζ4140413841324127411441941341ζ4134412541224121412041194116417ζ413941354128412341184113416412ζ4133413141304124411741114110418ζ413741364129412641154112415414    orthogonal faithful
ρ1280000000ζ413741364129412641154112415414ζ413941354128412341184113416412ζ4133413141304124411741114110418ζ4140413841324127411441941341ζ4134412541224121412041194116417    orthogonal faithful
ρ1380000000ζ4133413141304124411741114110418ζ413741364129412641154112415414ζ4134412541224121412041194116417ζ413941354128412341184113416412ζ4140413841324127411441941341    orthogonal faithful

Smallest permutation representation of C41⋊C8
On 41 points: primitive
Generators in S41
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41)
(2 28 33 4 41 15 10 39)(3 14 24 7 40 29 19 36)(5 27 6 13 38 16 37 30)(8 26 20 22 35 17 23 21)(9 12 11 25 34 31 32 18)

G:=sub<Sym(41)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41), (2,28,33,4,41,15,10,39)(3,14,24,7,40,29,19,36)(5,27,6,13,38,16,37,30)(8,26,20,22,35,17,23,21)(9,12,11,25,34,31,32,18)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41), (2,28,33,4,41,15,10,39)(3,14,24,7,40,29,19,36)(5,27,6,13,38,16,37,30)(8,26,20,22,35,17,23,21)(9,12,11,25,34,31,32,18) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)], [(2,28,33,4,41,15,10,39),(3,14,24,7,40,29,19,36),(5,27,6,13,38,16,37,30),(8,26,20,22,35,17,23,21),(9,12,11,25,34,31,32,18)])

Matrix representation of C41⋊C8 in GL8(𝔽2297)

01000000
00100000
00010000
00001000
00000100
00000010
00000001
22962270171222421711224217122270
,
10000000
3671648956150595714777332153
8477211269515208784272177
148821882702361912612002591
192722586581693701831273730
101417171805288295180619151035
12831221135111641014194312411903
3701532805150897213701190763

G:=sub<GL(8,GF(2297))| [0,0,0,0,0,0,0,2296,1,0,0,0,0,0,0,2270,0,1,0,0,0,0,0,1712,0,0,1,0,0,0,0,2242,0,0,0,1,0,0,0,1711,0,0,0,0,1,0,0,2242,0,0,0,0,0,1,0,1712,0,0,0,0,0,0,1,2270],[1,367,847,1488,1927,1014,1283,370,0,1648,72,2188,2258,1717,1221,1532,0,956,1126,270,658,1805,1351,805,0,1505,95,236,1693,288,1164,1508,0,957,1520,191,70,295,1014,972,0,1477,878,261,1831,1806,1943,1370,0,733,427,2002,273,1915,1241,1190,0,2153,2177,591,730,1035,1903,763] >;

C41⋊C8 in GAP, Magma, Sage, TeX

C_{41}\rtimes C_8
% in TeX

G:=Group("C41:C8");
// GroupNames label

G:=SmallGroup(328,12);
// by ID

G=gap.SmallGroup(328,12);
# by ID

G:=PCGroup([4,-2,-2,-2,-41,8,21,3459,2055,1291]);
// Polycyclic

G:=Group<a,b|a^41=b^8=1,b*a*b^-1=a^38>;
// generators/relations

Export

Subgroup lattice of C41⋊C8 in TeX
Character table of C41⋊C8 in TeX

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