metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6:C8.6C2, C4:C4.154D6, C8:Dic3:15C2, (C2xC8).125D6, (C2xQ8).45D6, Q8:C4:13S3, C6.50(C4oD8), C6.Q16:12C2, D6:3Q8.5C2, C4.58(C4oD12), Q8:2Dic3:10C2, (C22xS3).21D4, C22.205(S3xD4), C12.164(C4oD4), (C6xQ8).38C22, C4.89(D4:2S3), (C2xC24).136C22, (C2xC12).255C23, (C2xDic3).157D4, C2.17(Q16:S3), C6.63(C8.C22), C3:3(C23.20D4), C4:Dic3.99C22, C2.19(C23.9D6), C2.19(Q8.7D6), C6.27(C22.D4), C4:C4:7S3.3C2, (C2xC6).268(C2xD4), (C2xC3:C8).45C22, (S3xC2xC4).27C22, (C3xQ8:C4):13C2, (C3xC4:C4).56C22, (C2xC4).362(C22xS3), SmallGroup(192,374)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8:Dic3:C2
G = < a,b,c,d | a8=b6=d2=1, c2=b3, ab=ba, cac-1=a3, dad=ab3, cbc-1=dbd=b-1, dcd=a4b3c >
Subgroups: 264 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2xC4, C2xC4, Q8, C23, Dic3, C12, C12, D6, C2xC6, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, C22xC4, C2xQ8, C3:C8, C24, C4xS3, C2xDic3, C2xDic3, C2xC12, C2xC12, C3xQ8, C22xS3, C22:C8, Q8:C4, Q8:C4, C4.Q8, C2.D8, C42:C2, C22:Q8, C2xC3:C8, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C3xC4:C4, C2xC24, S3xC2xC4, C6xQ8, C23.20D4, C6.Q16, C8:Dic3, D6:C8, Q8:2Dic3, C3xQ8:C4, C4:C4:7S3, D6:3Q8, C8:Dic3:C2
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C22xS3, C22.D4, C4oD8, C8.C22, C4oD12, S3xD4, D4:2S3, C23.20D4, C23.9D6, Q8.7D6, Q16:S3, C8:Dic3:C2
►On 96 points
Generators in S96
(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)
(1 32 21 59 74 81)(2 25 22 60 75 82)(3 26 23 61 76 83)(4 27 24 62 77 84)(5 28 17 63 78 85)(6 29 18 64 79 86)(7 30 19 57 80 87)(8 31 20 58 73 88)(9 44 39 51 71 91)(10 45 40 52 72 92)(11 46 33 53 65 93)(12 47 34 54 66 94)(13 48 35 55 67 95)(14 41 36 56 68 96)(15 42 37 49 69 89)(16 43 38 50 70 90)
(1 9 59 51)(2 12 60 54)(3 15 61 49)(4 10 62 52)(5 13 63 55)(6 16 64 50)(7 11 57 53)(8 14 58 56)(17 67 85 48)(18 70 86 43)(19 65 87 46)(20 68 88 41)(21 71 81 44)(22 66 82 47)(23 69 83 42)(24 72 84 45)(25 94 75 34)(26 89 76 37)(27 92 77 40)(28 95 78 35)(29 90 79 38)(30 93 80 33)(31 96 73 36)(32 91 74 39)
(2 60)(4 62)(6 64)(8 58)(9 55)(10 14)(11 49)(12 16)(13 51)(15 53)(17 78)(18 29)(19 80)(20 31)(21 74)(22 25)(23 76)(24 27)(26 83)(28 85)(30 87)(32 81)(33 42)(34 70)(35 44)(36 72)(37 46)(38 66)(39 48)(40 68)(41 92)(43 94)(45 96)(47 90)(50 54)(52 56)(65 89)(67 91)(69 93)(71 95)(73 88)(75 82)(77 84)(79 86)
G:=sub<Sym(96)| (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), (1,32,21,59,74,81)(2,25,22,60,75,82)(3,26,23,61,76,83)(4,27,24,62,77,84)(5,28,17,63,78,85)(6,29,18,64,79,86)(7,30,19,57,80,87)(8,31,20,58,73,88)(9,44,39,51,71,91)(10,45,40,52,72,92)(11,46,33,53,65,93)(12,47,34,54,66,94)(13,48,35,55,67,95)(14,41,36,56,68,96)(15,42,37,49,69,89)(16,43,38,50,70,90), (1,9,59,51)(2,12,60,54)(3,15,61,49)(4,10,62,52)(5,13,63,55)(6,16,64,50)(7,11,57,53)(8,14,58,56)(17,67,85,48)(18,70,86,43)(19,65,87,46)(20,68,88,41)(21,71,81,44)(22,66,82,47)(23,69,83,42)(24,72,84,45)(25,94,75,34)(26,89,76,37)(27,92,77,40)(28,95,78,35)(29,90,79,38)(30,93,80,33)(31,96,73,36)(32,91,74,39), (2,60)(4,62)(6,64)(8,58)(9,55)(10,14)(11,49)(12,16)(13,51)(15,53)(17,78)(18,29)(19,80)(20,31)(21,74)(22,25)(23,76)(24,27)(26,83)(28,85)(30,87)(32,81)(33,42)(34,70)(35,44)(36,72)(37,46)(38,66)(39,48)(40,68)(41,92)(43,94)(45,96)(47,90)(50,54)(52,56)(65,89)(67,91)(69,93)(71,95)(73,88)(75,82)(77,84)(79,86)>;
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), (1,32,21,59,74,81)(2,25,22,60,75,82)(3,26,23,61,76,83)(4,27,24,62,77,84)(5,28,17,63,78,85)(6,29,18,64,79,86)(7,30,19,57,80,87)(8,31,20,58,73,88)(9,44,39,51,71,91)(10,45,40,52,72,92)(11,46,33,53,65,93)(12,47,34,54,66,94)(13,48,35,55,67,95)(14,41,36,56,68,96)(15,42,37,49,69,89)(16,43,38,50,70,90), (1,9,59,51)(2,12,60,54)(3,15,61,49)(4,10,62,52)(5,13,63,55)(6,16,64,50)(7,11,57,53)(8,14,58,56)(17,67,85,48)(18,70,86,43)(19,65,87,46)(20,68,88,41)(21,71,81,44)(22,66,82,47)(23,69,83,42)(24,72,84,45)(25,94,75,34)(26,89,76,37)(27,92,77,40)(28,95,78,35)(29,90,79,38)(30,93,80,33)(31,96,73,36)(32,91,74,39), (2,60)(4,62)(6,64)(8,58)(9,55)(10,14)(11,49)(12,16)(13,51)(15,53)(17,78)(18,29)(19,80)(20,31)(21,74)(22,25)(23,76)(24,27)(26,83)(28,85)(30,87)(32,81)(33,42)(34,70)(35,44)(36,72)(37,46)(38,66)(39,48)(40,68)(41,92)(43,94)(45,96)(47,90)(50,54)(52,56)(65,89)(67,91)(69,93)(71,95)(73,88)(75,82)(77,84)(79,86) );
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)], [(1,32,21,59,74,81),(2,25,22,60,75,82),(3,26,23,61,76,83),(4,27,24,62,77,84),(5,28,17,63,78,85),(6,29,18,64,79,86),(7,30,19,57,80,87),(8,31,20,58,73,88),(9,44,39,51,71,91),(10,45,40,52,72,92),(11,46,33,53,65,93),(12,47,34,54,66,94),(13,48,35,55,67,95),(14,41,36,56,68,96),(15,42,37,49,69,89),(16,43,38,50,70,90)], [(1,9,59,51),(2,12,60,54),(3,15,61,49),(4,10,62,52),(5,13,63,55),(6,16,64,50),(7,11,57,53),(8,14,58,56),(17,67,85,48),(18,70,86,43),(19,65,87,46),(20,68,88,41),(21,71,81,44),(22,66,82,47),(23,69,83,42),(24,72,84,45),(25,94,75,34),(26,89,76,37),(27,92,77,40),(28,95,78,35),(29,90,79,38),(30,93,80,33),(31,96,73,36),(32,91,74,39)], [(2,60),(4,62),(6,64),(8,58),(9,55),(10,14),(11,49),(12,16),(13,51),(15,53),(17,78),(18,29),(19,80),(20,31),(21,74),(22,25),(23,76),(24,27),(26,83),(28,85),(30,87),(32,81),(33,42),(34,70),(35,44),(36,72),(37,46),(38,66),(39,48),(40,68),(41,92),(43,94),(45,96),(47,90),(50,54),(52,56),(65,89),(67,91),(69,93),(71,95),(73,88),(75,82),(77,84),(79,86)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 8 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4oD4 | C4oD8 | C4oD12 | C8.C22 | D4:2S3 | S3xD4 | Q8.7D6 | Q16:S3 |
| kernel | C8:Dic3:C2 | C6.Q16 | C8:Dic3 | D6:C8 | Q8:2Dic3 | C3xQ8:C4 | C4:C4:7S3 | D6:3Q8 | Q8:C4 | C2xDic3 | C22xS3 | C4:C4 | C2xC8 | C2xQ8 | C12 | C6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C8:Dic3:C2 ►in GL6(F73)
| 22 | 0 | 0 | 0 | 0 | 0 |
| 49 | 63 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 35 | 71 | 0 | 0 | 0 | 0 |
| 28 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 35 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [22,49,0,0,0,0,0,63,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,27,0,0,0,0,27,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[35,28,0,0,0,0,71,38,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[1,35,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;
C8:Dic3:C2 in GAP, Magma, Sage, TeX
C_8\rtimes {\rm Dic}_3\rtimes C_2 % in TeX
G:=Group("C8:Dic3:C2"); // GroupNames label
G:=SmallGroup(192,374);
// by ID
G=gap.SmallGroup(192,374);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,64,926,219,184,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^6=d^2=1,c^2=b^3,a*b=b*a,c*a*c^-1=a^3,d*a*d=a*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=a^4*b^3*c>;
// generators/relations