metabelian, supersoluble, monomial
Aliases: C33:12D4, C32:8D12, (C3xC12):5S3, C12:1(C3:S3), C4:(C33:C2), (C3xC6).64D6, (C32xC12):1C2, C3:1(C12:S3), (C32xC6).28C22, C6.16(C2xC3:S3), (C2xC33:C2):3C2, C2.4(C2xC33:C2), SmallGroup(216,147)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33:12D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 1372 in 224 conjugacy classes, 87 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, S3, C6, D4, C32, C12, D6, C3:S3, C3xC6, D12, C33, C3xC12, C2xC3:S3, C33:C2, C32xC6, C12:S3, C32xC12, C2xC33:C2, C33:12D4
Quotients: C1, C2, C22, S3, D4, D6, C3:S3, D12, C2xC3:S3, C33:C2, C12:S3, C2xC33:C2, C33:12D4
►On 108 points
Generators in S108
(1 54 20)(2 55 17)(3 56 18)(4 53 19)(5 51 69)(6 52 70)(7 49 71)(8 50 72)(9 27 74)(10 28 75)(11 25 76)(12 26 73)(13 30 80)(14 31 77)(15 32 78)(16 29 79)(21 91 86)(22 92 87)(23 89 88)(24 90 85)(33 106 96)(34 107 93)(35 108 94)(36 105 95)(37 57 98)(38 58 99)(39 59 100)(40 60 97)(41 63 101)(42 64 102)(43 61 103)(44 62 104)(45 65 83)(46 66 84)(47 67 81)(48 68 82)
(1 49 60)(2 50 57)(3 51 58)(4 52 59)(5 38 18)(6 39 19)(7 40 20)(8 37 17)(9 43 85)(10 44 86)(11 41 87)(12 42 88)(13 81 34)(14 82 35)(15 83 36)(16 84 33)(21 28 62)(22 25 63)(23 26 64)(24 27 61)(29 46 106)(30 47 107)(31 48 108)(32 45 105)(53 70 100)(54 71 97)(55 72 98)(56 69 99)(65 95 78)(66 96 79)(67 93 80)(68 94 77)(73 102 89)(74 103 90)(75 104 91)(76 101 92)
(1 80 43)(2 77 44)(3 78 41)(4 79 42)(5 45 92)(6 46 89)(7 47 90)(8 48 91)(9 60 93)(10 57 94)(11 58 95)(12 59 96)(13 61 54)(14 62 55)(15 63 56)(16 64 53)(17 31 104)(18 32 101)(19 29 102)(20 30 103)(21 72 82)(22 69 83)(23 70 84)(24 71 81)(25 99 36)(26 100 33)(27 97 34)(28 98 35)(37 108 75)(38 105 76)(39 106 73)(40 107 74)(49 67 85)(50 68 86)(51 65 87)(52 66 88)
(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 4)(2 3)(5 98)(6 97)(7 100)(8 99)(9 66)(10 65)(11 68)(12 67)(13 102)(14 101)(15 104)(16 103)(17 56)(18 55)(19 54)(20 53)(21 105)(22 108)(23 107)(24 106)(25 48)(26 47)(27 46)(28 45)(29 61)(30 64)(31 63)(32 62)(33 90)(34 89)(35 92)(36 91)(37 69)(38 72)(39 71)(40 70)(41 77)(42 80)(43 79)(44 78)(49 59)(50 58)(51 57)(52 60)(73 81)(74 84)(75 83)(76 82)(85 96)(86 95)(87 94)(88 93)
G:=sub<Sym(108)| (1,54,20)(2,55,17)(3,56,18)(4,53,19)(5,51,69)(6,52,70)(7,49,71)(8,50,72)(9,27,74)(10,28,75)(11,25,76)(12,26,73)(13,30,80)(14,31,77)(15,32,78)(16,29,79)(21,91,86)(22,92,87)(23,89,88)(24,90,85)(33,106,96)(34,107,93)(35,108,94)(36,105,95)(37,57,98)(38,58,99)(39,59,100)(40,60,97)(41,63,101)(42,64,102)(43,61,103)(44,62,104)(45,65,83)(46,66,84)(47,67,81)(48,68,82), (1,49,60)(2,50,57)(3,51,58)(4,52,59)(5,38,18)(6,39,19)(7,40,20)(8,37,17)(9,43,85)(10,44,86)(11,41,87)(12,42,88)(13,81,34)(14,82,35)(15,83,36)(16,84,33)(21,28,62)(22,25,63)(23,26,64)(24,27,61)(29,46,106)(30,47,107)(31,48,108)(32,45,105)(53,70,100)(54,71,97)(55,72,98)(56,69,99)(65,95,78)(66,96,79)(67,93,80)(68,94,77)(73,102,89)(74,103,90)(75,104,91)(76,101,92), (1,80,43)(2,77,44)(3,78,41)(4,79,42)(5,45,92)(6,46,89)(7,47,90)(8,48,91)(9,60,93)(10,57,94)(11,58,95)(12,59,96)(13,61,54)(14,62,55)(15,63,56)(16,64,53)(17,31,104)(18,32,101)(19,29,102)(20,30,103)(21,72,82)(22,69,83)(23,70,84)(24,71,81)(25,99,36)(26,100,33)(27,97,34)(28,98,35)(37,108,75)(38,105,76)(39,106,73)(40,107,74)(49,67,85)(50,68,86)(51,65,87)(52,66,88), (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), (1,4)(2,3)(5,98)(6,97)(7,100)(8,99)(9,66)(10,65)(11,68)(12,67)(13,102)(14,101)(15,104)(16,103)(17,56)(18,55)(19,54)(20,53)(21,105)(22,108)(23,107)(24,106)(25,48)(26,47)(27,46)(28,45)(29,61)(30,64)(31,63)(32,62)(33,90)(34,89)(35,92)(36,91)(37,69)(38,72)(39,71)(40,70)(41,77)(42,80)(43,79)(44,78)(49,59)(50,58)(51,57)(52,60)(73,81)(74,84)(75,83)(76,82)(85,96)(86,95)(87,94)(88,93)>;
G:=Group( (1,54,20)(2,55,17)(3,56,18)(4,53,19)(5,51,69)(6,52,70)(7,49,71)(8,50,72)(9,27,74)(10,28,75)(11,25,76)(12,26,73)(13,30,80)(14,31,77)(15,32,78)(16,29,79)(21,91,86)(22,92,87)(23,89,88)(24,90,85)(33,106,96)(34,107,93)(35,108,94)(36,105,95)(37,57,98)(38,58,99)(39,59,100)(40,60,97)(41,63,101)(42,64,102)(43,61,103)(44,62,104)(45,65,83)(46,66,84)(47,67,81)(48,68,82), (1,49,60)(2,50,57)(3,51,58)(4,52,59)(5,38,18)(6,39,19)(7,40,20)(8,37,17)(9,43,85)(10,44,86)(11,41,87)(12,42,88)(13,81,34)(14,82,35)(15,83,36)(16,84,33)(21,28,62)(22,25,63)(23,26,64)(24,27,61)(29,46,106)(30,47,107)(31,48,108)(32,45,105)(53,70,100)(54,71,97)(55,72,98)(56,69,99)(65,95,78)(66,96,79)(67,93,80)(68,94,77)(73,102,89)(74,103,90)(75,104,91)(76,101,92), (1,80,43)(2,77,44)(3,78,41)(4,79,42)(5,45,92)(6,46,89)(7,47,90)(8,48,91)(9,60,93)(10,57,94)(11,58,95)(12,59,96)(13,61,54)(14,62,55)(15,63,56)(16,64,53)(17,31,104)(18,32,101)(19,29,102)(20,30,103)(21,72,82)(22,69,83)(23,70,84)(24,71,81)(25,99,36)(26,100,33)(27,97,34)(28,98,35)(37,108,75)(38,105,76)(39,106,73)(40,107,74)(49,67,85)(50,68,86)(51,65,87)(52,66,88), (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), (1,4)(2,3)(5,98)(6,97)(7,100)(8,99)(9,66)(10,65)(11,68)(12,67)(13,102)(14,101)(15,104)(16,103)(17,56)(18,55)(19,54)(20,53)(21,105)(22,108)(23,107)(24,106)(25,48)(26,47)(27,46)(28,45)(29,61)(30,64)(31,63)(32,62)(33,90)(34,89)(35,92)(36,91)(37,69)(38,72)(39,71)(40,70)(41,77)(42,80)(43,79)(44,78)(49,59)(50,58)(51,57)(52,60)(73,81)(74,84)(75,83)(76,82)(85,96)(86,95)(87,94)(88,93) );
G=PermutationGroup([[(1,54,20),(2,55,17),(3,56,18),(4,53,19),(5,51,69),(6,52,70),(7,49,71),(8,50,72),(9,27,74),(10,28,75),(11,25,76),(12,26,73),(13,30,80),(14,31,77),(15,32,78),(16,29,79),(21,91,86),(22,92,87),(23,89,88),(24,90,85),(33,106,96),(34,107,93),(35,108,94),(36,105,95),(37,57,98),(38,58,99),(39,59,100),(40,60,97),(41,63,101),(42,64,102),(43,61,103),(44,62,104),(45,65,83),(46,66,84),(47,67,81),(48,68,82)], [(1,49,60),(2,50,57),(3,51,58),(4,52,59),(5,38,18),(6,39,19),(7,40,20),(8,37,17),(9,43,85),(10,44,86),(11,41,87),(12,42,88),(13,81,34),(14,82,35),(15,83,36),(16,84,33),(21,28,62),(22,25,63),(23,26,64),(24,27,61),(29,46,106),(30,47,107),(31,48,108),(32,45,105),(53,70,100),(54,71,97),(55,72,98),(56,69,99),(65,95,78),(66,96,79),(67,93,80),(68,94,77),(73,102,89),(74,103,90),(75,104,91),(76,101,92)], [(1,80,43),(2,77,44),(3,78,41),(4,79,42),(5,45,92),(6,46,89),(7,47,90),(8,48,91),(9,60,93),(10,57,94),(11,58,95),(12,59,96),(13,61,54),(14,62,55),(15,63,56),(16,64,53),(17,31,104),(18,32,101),(19,29,102),(20,30,103),(21,72,82),(22,69,83),(23,70,84),(24,71,81),(25,99,36),(26,100,33),(27,97,34),(28,98,35),(37,108,75),(38,105,76),(39,106,73),(40,107,74),(49,67,85),(50,68,86),(51,65,87),(52,66,88)], [(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)], [(1,4),(2,3),(5,98),(6,97),(7,100),(8,99),(9,66),(10,65),(11,68),(12,67),(13,102),(14,101),(15,104),(16,103),(17,56),(18,55),(19,54),(20,53),(21,105),(22,108),(23,107),(24,106),(25,48),(26,47),(27,46),(28,45),(29,61),(30,64),(31,63),(32,62),(33,90),(34,89),(35,92),(36,91),(37,69),(38,72),(39,71),(40,70),(41,77),(42,80),(43,79),(44,78),(49,59),(50,58),(51,57),(52,60),(73,81),(74,84),(75,83),(76,82),(85,96),(86,95),(87,94),(88,93)]])
C33:12D4 is a maximal subgroup of
C33:7D8 C33:8D8 C33:15SD16 C33:17SD16 C33:21SD16 C33:12D8 C33:15D8 C33:27SD16 C12.40S32 C12.58S32 S3xC12:S3 C3:S3xD12 C62.160D6 D4xC33:C2 (Q8xC33):C2
C33:12D4 is a maximal quotient of
C33:21SD16 C33:12D8 C33:12Q16 C62.147D6 C62.148D6
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3M | 4 | 6A | ··· | 6M | 12A | ··· | 12Z |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 54 | 54 | 2 | ··· | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
57 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D4 | D6 | D12 |
| kernel | C33:12D4 | C32xC12 | C2xC33:C2 | C3xC12 | C33 | C3xC6 | C32 |
| # reps | 1 | 1 | 2 | 13 | 1 | 13 | 26 |
Matrix representation of C33:12D4 ►in GL6(F13)
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 0 | 0 | 0 | 0 | 10 | 6 |
G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,7,10,0,0,0,0,3,6] >;
C33:12D4 in GAP, Magma, Sage, TeX
C_3^3\rtimes_{12}D_4 % in TeX
G:=Group("C3^3:12D4"); // GroupNames label
G:=SmallGroup(216,147);
// by ID
G=gap.SmallGroup(216,147);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,387,1444,5189]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations