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G = C61⋊C6order 366 = 2·3·61

The semidirect product of C61 and C6 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C61⋊C6, D61⋊C3, C61⋊C3⋊C2, SmallGroup(366,1)

Series: Derived Chief Lower central Upper central

C1C61 — C61⋊C6
C1C61C61⋊C3 — C61⋊C6
C61 — C61⋊C6
C1

Generators and relations for C61⋊C6
 G = < a,b | a61=b6=1, bab-1=a14 >

61C2
61C3
61C6

Character table of C61⋊C6

 class 123A3B6A6B61A61B61C61D61E61F61G61H61I61J
 size 161616161616666666666
ρ11111111111111111    trivial
ρ21-111-1-11111111111    linear of order 2
ρ31-1ζ3ζ32ζ65ζ61111111111    linear of order 6
ρ411ζ3ζ32ζ3ζ321111111111    linear of order 3
ρ511ζ32ζ3ζ32ζ31111111111    linear of order 3
ρ61-1ζ32ζ3ζ6ζ651111111111    linear of order 6
ρ7600000ζ61536151614361186110618ζ61556144613861236117616ζ61546137613161306124617ζ615761566152619615614ζ614961466134612761156112ζ614561416136612561206116ζ615061406132612961216111ζ6160614861476114611361ζ61586142613961226119613ζ61596135613361286126612    orthogonal faithful
ρ8600000ζ615761566152619615614ζ61586142613961226119613ζ614961466134612761156112ζ61596135613361286126612ζ61556144613861236117616ζ61536151614361186110618ζ614561416136612561206116ζ61546137613161306124617ζ615061406132612961216111ζ6160614861476114611361    orthogonal faithful
ρ9600000ζ61546137613161306124617ζ61536151614361186110618ζ615061406132612961216111ζ614961466134612761156112ζ614561416136612561206116ζ6160614861476114611361ζ61596135613361286126612ζ61586142613961226119613ζ615761566152619615614ζ61556144613861236117616    orthogonal faithful
ρ10600000ζ61586142613961226119613ζ6160614861476114611361ζ615761566152619615614ζ615061406132612961216111ζ61596135613361286126612ζ61556144613861236117616ζ614961466134612761156112ζ61536151614361186110618ζ61546137613161306124617ζ614561416136612561206116    orthogonal faithful
ρ11600000ζ6160614861476114611361ζ614561416136612561206116ζ61586142613961226119613ζ61546137613161306124617ζ615061406132612961216111ζ61596135613361286126612ζ615761566152619615614ζ61556144613861236117616ζ61536151614361186110618ζ614961466134612761156112    orthogonal faithful
ρ12600000ζ61556144613861236117616ζ61596135613361286126612ζ61536151614361186110618ζ61586142613961226119613ζ615761566152619615614ζ614961466134612761156112ζ61546137613161306124617ζ614561416136612561206116ζ6160614861476114611361ζ615061406132612961216111    orthogonal faithful
ρ13600000ζ615061406132612961216111ζ61546137613161306124617ζ61596135613361286126612ζ614561416136612561206116ζ6160614861476114611361ζ61586142613961226119613ζ61556144613861236117616ζ615761566152619615614ζ614961466134612761156112ζ61536151614361186110618    orthogonal faithful
ρ14600000ζ614561416136612561206116ζ614961466134612761156112ζ6160614861476114611361ζ61536151614361186110618ζ61546137613161306124617ζ615061406132612961216111ζ61586142613961226119613ζ61596135613361286126612ζ61556144613861236117616ζ615761566152619615614    orthogonal faithful
ρ15600000ζ614961466134612761156112ζ615761566152619615614ζ614561416136612561206116ζ61556144613861236117616ζ61536151614361186110618ζ61546137613161306124617ζ6160614861476114611361ζ615061406132612961216111ζ61596135613361286126612ζ61586142613961226119613    orthogonal faithful
ρ16600000ζ61596135613361286126612ζ615061406132612961216111ζ61556144613861236117616ζ6160614861476114611361ζ61586142613961226119613ζ615761566152619615614ζ61536151614361186110618ζ614961466134612761156112ζ614561416136612561206116ζ61546137613161306124617    orthogonal faithful

Smallest permutation representation of C61⋊C6
On 61 points: primitive
Generators in S61
(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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61)
(2 49 48 61 14 15)(3 36 34 60 27 29)(4 23 20 59 40 43)(5 10 6 58 53 57)(7 45 39 56 18 24)(8 32 25 55 31 38)(9 19 11 54 44 52)(12 41 30 51 22 33)(13 28 16 50 35 47)(17 37 21 46 26 42)

G:=sub<Sym(61)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61), (2,49,48,61,14,15)(3,36,34,60,27,29)(4,23,20,59,40,43)(5,10,6,58,53,57)(7,45,39,56,18,24)(8,32,25,55,31,38)(9,19,11,54,44,52)(12,41,30,51,22,33)(13,28,16,50,35,47)(17,37,21,46,26,42)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61), (2,49,48,61,14,15)(3,36,34,60,27,29)(4,23,20,59,40,43)(5,10,6,58,53,57)(7,45,39,56,18,24)(8,32,25,55,31,38)(9,19,11,54,44,52)(12,41,30,51,22,33)(13,28,16,50,35,47)(17,37,21,46,26,42) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61)], [(2,49,48,61,14,15),(3,36,34,60,27,29),(4,23,20,59,40,43),(5,10,6,58,53,57),(7,45,39,56,18,24),(8,32,25,55,31,38),(9,19,11,54,44,52),(12,41,30,51,22,33),(13,28,16,50,35,47),(17,37,21,46,26,42)])

Matrix representation of C61⋊C6 in GL6(𝔽367)

010000
001000
000100
000010
000001
366279214291214279
,
100000
1076126812811653
10133731319815192
113357227272250154
25930119454355268
259154113416399

G:=sub<GL(6,GF(367))| [0,0,0,0,0,366,1,0,0,0,0,279,0,1,0,0,0,214,0,0,1,0,0,291,0,0,0,1,0,214,0,0,0,0,1,279],[1,107,101,113,259,259,0,61,337,357,301,154,0,268,313,227,194,11,0,128,198,272,54,34,0,116,151,250,355,163,0,53,92,154,268,99] >;

C61⋊C6 in GAP, Magma, Sage, TeX

C_{61}\rtimes C_6
% in TeX

G:=Group("C61:C6");
// GroupNames label

G:=SmallGroup(366,1);
// by ID

G=gap.SmallGroup(366,1);
# by ID

G:=PCGroup([3,-2,-3,-61,3242,1274]);
// Polycyclic

G:=Group<a,b|a^61=b^6=1,b*a*b^-1=a^14>;
// generators/relations

Export

Subgroup lattice of C61⋊C6 in TeX
Character table of C61⋊C6 in TeX

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