Copied to
clipboard

## G = C62⋊10Dic3order 432 = 24·33

### 9th semidirect product of C62 and Dic3 acting via Dic3/C2=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×A4 — C62⋊10Dic3
 Chief series C1 — C22 — C2×C6 — C62 — C32×A4 — A4×C3×C6 — C62⋊10Dic3
 Lower central C32×A4 — C62⋊10Dic3
 Upper central C1 — C2

Generators and relations for C6210Dic3
G = < a,b,c,d | a6=b6=c6=1, d2=c3, ab=ba, cac-1=ab3, dad-1=a2b3, cbc-1=a3b4, dbd-1=a3b2, dcd-1=c-1 >

Subgroups: 1368 in 210 conjugacy classes, 69 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C32, C32, Dic3, A4, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C2×Dic3, C2×A4, C22×C6, C33, C3⋊Dic3, C3×A4, C62, C62, C6.D4, A4⋊C4, C32×C6, C2×C3⋊Dic3, C6×A4, C2×C62, C335C4, C32×A4, C625C4, C6.7S4, A4×C3×C6, C6210Dic3
Quotients: C1, C2, C4, S3, Dic3, C3⋊S3, S4, C3⋊Dic3, A4⋊C4, C33⋊C2, C3⋊S4, C335C4, C6.7S4, C324S4, C6210Dic3

Smallest permutation representation of C6210Dic3
On 108 points
Generators in S108
(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 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)(97 98 99 100 101 102)(103 104 105 106 107 108)
(1 13 7 11 18 4)(2 14 8 12 16 5)(3 15 9 10 17 6)(19 25 24 29 36 31)(20 26 22 30 34 32)(21 27 23 28 35 33)(37 96 71 40 93 68)(38 91 72 41 94 69)(39 92 67 42 95 70)(43 90 51 46 87 54)(44 85 52 47 88 49)(45 86 53 48 89 50)(55 97 108)(56 98 103)(57 99 104)(58 100 105)(59 101 106)(60 102 107)(61 78 82)(62 73 83)(63 74 84)(64 75 79)(65 76 80)(66 77 81)
(1 107 95 11 104 92)(2 105 93 12 108 96)(3 103 91 10 106 94)(4 99 42 7 102 39)(5 97 40 8 100 37)(6 101 38 9 98 41)(13 57 70 18 60 67)(14 55 68 16 58 71)(15 59 72 17 56 69)(19 79 43 29 82 46)(20 83 47 30 80 44)(21 81 45 28 84 48)(22 73 49 32 76 52)(23 77 53 33 74 50)(24 75 51 31 78 54)(25 61 90 36 64 87)(26 65 88 34 62 85)(27 63 86 35 66 89)
(1 80 11 83)(2 82 12 79)(3 84 10 81)(4 62 7 65)(5 64 8 61)(6 66 9 63)(13 73 18 76)(14 75 16 78)(15 77 17 74)(19 105 29 108)(20 107 30 104)(21 103 28 106)(22 60 32 57)(23 56 33 59)(24 58 31 55)(25 97 36 100)(26 99 34 102)(27 101 35 98)(37 87 40 90)(38 86 41 89)(39 85 42 88)(43 96 46 93)(44 95 47 92)(45 94 48 91)(49 70 52 67)(50 69 53 72)(51 68 54 71)

G:=sub<Sym(108)| (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)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108), (1,13,7,11,18,4)(2,14,8,12,16,5)(3,15,9,10,17,6)(19,25,24,29,36,31)(20,26,22,30,34,32)(21,27,23,28,35,33)(37,96,71,40,93,68)(38,91,72,41,94,69)(39,92,67,42,95,70)(43,90,51,46,87,54)(44,85,52,47,88,49)(45,86,53,48,89,50)(55,97,108)(56,98,103)(57,99,104)(58,100,105)(59,101,106)(60,102,107)(61,78,82)(62,73,83)(63,74,84)(64,75,79)(65,76,80)(66,77,81), (1,107,95,11,104,92)(2,105,93,12,108,96)(3,103,91,10,106,94)(4,99,42,7,102,39)(5,97,40,8,100,37)(6,101,38,9,98,41)(13,57,70,18,60,67)(14,55,68,16,58,71)(15,59,72,17,56,69)(19,79,43,29,82,46)(20,83,47,30,80,44)(21,81,45,28,84,48)(22,73,49,32,76,52)(23,77,53,33,74,50)(24,75,51,31,78,54)(25,61,90,36,64,87)(26,65,88,34,62,85)(27,63,86,35,66,89), (1,80,11,83)(2,82,12,79)(3,84,10,81)(4,62,7,65)(5,64,8,61)(6,66,9,63)(13,73,18,76)(14,75,16,78)(15,77,17,74)(19,105,29,108)(20,107,30,104)(21,103,28,106)(22,60,32,57)(23,56,33,59)(24,58,31,55)(25,97,36,100)(26,99,34,102)(27,101,35,98)(37,87,40,90)(38,86,41,89)(39,85,42,88)(43,96,46,93)(44,95,47,92)(45,94,48,91)(49,70,52,67)(50,69,53,72)(51,68,54,71)>;

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)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108), (1,13,7,11,18,4)(2,14,8,12,16,5)(3,15,9,10,17,6)(19,25,24,29,36,31)(20,26,22,30,34,32)(21,27,23,28,35,33)(37,96,71,40,93,68)(38,91,72,41,94,69)(39,92,67,42,95,70)(43,90,51,46,87,54)(44,85,52,47,88,49)(45,86,53,48,89,50)(55,97,108)(56,98,103)(57,99,104)(58,100,105)(59,101,106)(60,102,107)(61,78,82)(62,73,83)(63,74,84)(64,75,79)(65,76,80)(66,77,81), (1,107,95,11,104,92)(2,105,93,12,108,96)(3,103,91,10,106,94)(4,99,42,7,102,39)(5,97,40,8,100,37)(6,101,38,9,98,41)(13,57,70,18,60,67)(14,55,68,16,58,71)(15,59,72,17,56,69)(19,79,43,29,82,46)(20,83,47,30,80,44)(21,81,45,28,84,48)(22,73,49,32,76,52)(23,77,53,33,74,50)(24,75,51,31,78,54)(25,61,90,36,64,87)(26,65,88,34,62,85)(27,63,86,35,66,89), (1,80,11,83)(2,82,12,79)(3,84,10,81)(4,62,7,65)(5,64,8,61)(6,66,9,63)(13,73,18,76)(14,75,16,78)(15,77,17,74)(19,105,29,108)(20,107,30,104)(21,103,28,106)(22,60,32,57)(23,56,33,59)(24,58,31,55)(25,97,36,100)(26,99,34,102)(27,101,35,98)(37,87,40,90)(38,86,41,89)(39,85,42,88)(43,96,46,93)(44,95,47,92)(45,94,48,91)(49,70,52,67)(50,69,53,72)(51,68,54,71) );

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),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96),(97,98,99,100,101,102),(103,104,105,106,107,108)], [(1,13,7,11,18,4),(2,14,8,12,16,5),(3,15,9,10,17,6),(19,25,24,29,36,31),(20,26,22,30,34,32),(21,27,23,28,35,33),(37,96,71,40,93,68),(38,91,72,41,94,69),(39,92,67,42,95,70),(43,90,51,46,87,54),(44,85,52,47,88,49),(45,86,53,48,89,50),(55,97,108),(56,98,103),(57,99,104),(58,100,105),(59,101,106),(60,102,107),(61,78,82),(62,73,83),(63,74,84),(64,75,79),(65,76,80),(66,77,81)], [(1,107,95,11,104,92),(2,105,93,12,108,96),(3,103,91,10,106,94),(4,99,42,7,102,39),(5,97,40,8,100,37),(6,101,38,9,98,41),(13,57,70,18,60,67),(14,55,68,16,58,71),(15,59,72,17,56,69),(19,79,43,29,82,46),(20,83,47,30,80,44),(21,81,45,28,84,48),(22,73,49,32,76,52),(23,77,53,33,74,50),(24,75,51,31,78,54),(25,61,90,36,64,87),(26,65,88,34,62,85),(27,63,86,35,66,89)], [(1,80,11,83),(2,82,12,79),(3,84,10,81),(4,62,7,65),(5,64,8,61),(6,66,9,63),(13,73,18,76),(14,75,16,78),(15,77,17,74),(19,105,29,108),(20,107,30,104),(21,103,28,106),(22,60,32,57),(23,56,33,59),(24,58,31,55),(25,97,36,100),(26,99,34,102),(27,101,35,98),(37,87,40,90),(38,86,41,89),(39,85,42,88),(43,96,46,93),(44,95,47,92),(45,94,48,91),(49,70,52,67),(50,69,53,72),(51,68,54,71)]])

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E ··· 3M 4A 4B 4C 4D 6A 6B 6C 6D 6E ··· 6L 6M ··· 6U order 1 2 2 2 3 3 3 3 3 ··· 3 4 4 4 4 6 6 6 6 6 ··· 6 6 ··· 6 size 1 1 3 3 2 2 2 2 8 ··· 8 54 54 54 54 2 2 2 2 6 ··· 6 8 ··· 8

42 irreducible representations

 dim 1 1 1 2 2 2 2 3 3 6 6 type + + + + - - + + - image C1 C2 C4 S3 S3 Dic3 Dic3 S4 A4⋊C4 C3⋊S4 C6.7S4 kernel C62⋊10Dic3 A4×C3×C6 C32×A4 C6×A4 C2×C62 C3×A4 C62 C3×C6 C32 C6 C3 # reps 1 1 2 12 1 12 1 2 2 4 4

Matrix representation of C6210Dic3 in GL7(𝔽13)

 12 12 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 11 0 0 0 0 0 0 3 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 1 2 1
,
 12 12 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 12 0 12
,
 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 0 4 2 0 0 0 0 0 0 10 0 0 0 0 0 0 0 12 11 11 0 0 0 0 12 0 0 0 0 0 0 8 8 1
,
 11 2 0 0 0 0 0 4 2 0 0 0 0 0 0 0 8 0 0 0 0 0 0 11 5 0 0 0 0 0 0 0 12 11 11 0 0 0 0 0 1 0 0 0 0 0 0 0 1

G:=sub<GL(7,GF(13))| [12,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,11,3,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,12,2,0,0,0,0,0,0,1],[12,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[0,1,0,0,0,0,0,12,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,2,10,0,0,0,0,0,0,0,12,12,8,0,0,0,0,11,0,8,0,0,0,0,11,0,1],[11,4,0,0,0,0,0,2,2,0,0,0,0,0,0,0,8,11,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,11,1,0,0,0,0,0,11,0,1] >;

C6210Dic3 in GAP, Magma, Sage, TeX

C_6^2\rtimes_{10}{\rm Dic}_3
% in TeX

G:=Group("C6^2:10Dic3");
// GroupNames label

G:=SmallGroup(432,621);
// by ID

G=gap.SmallGroup(432,621);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,14,170,675,2524,9077,2287,5298,3989]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^6=1,d^2=c^3,a*b=b*a,c*a*c^-1=a*b^3,d*a*d^-1=a^2*b^3,c*b*c^-1=a^3*b^4,d*b*d^-1=a^3*b^2,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽