metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3:C8:23D4, C3:4(C8:8D4), C4:C4.61D6, (C2xC6):4SD16, (C2xD4).41D6, C4:D4.6S3, C4.172(S3xD4), C6.98(C4oD8), (C2xC12).264D4, C12.150(C2xD4), C6.56(C2xSD16), (C22xC6).87D4, D4:Dic3:17C2, C6.SD16:35C2, C6.95(C4:D4), C22:2(D4.S3), C12.Q8:36C2, (C6xD4).57C22, (C22xC4).357D6, C12.185(C4oD4), C12.48D4:24C2, C4.61(D4:2S3), (C2xC12).360C23, C23.47(C3:D4), C2.17(Q8.13D6), C4:Dic3.144C22, C2.16(C23.14D6), (C22xC12).164C22, (C2xDic6).103C22, (C22xC3:C8):4C2, (C2xD4.S3):10C2, (C3xC4:D4).5C2, (C2xC6).491(C2xD4), C2.10(C2xD4.S3), (C2xC3:C8).249C22, (C2xC4).106(C3:D4), (C3xC4:C4).108C22, (C2xC4).460(C22xS3), C22.166(C2xC3:D4), SmallGroup(192,600)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3:C8:23D4
G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=b3, bd=db, dcd=c-1 >
Subgroups: 320 in 124 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, C12, C12, C2xC6, C2xC6, C2xC6, C22:C4, C4:C4, C4:C4, C2xC8, SD16, C22xC4, C2xD4, C2xD4, C2xQ8, C3:C8, C3:C8, Dic6, C2xDic3, C2xC12, C2xC12, C3xD4, C22xC6, C22xC6, D4:C4, Q8:C4, C4.Q8, C4:D4, C22:Q8, C22xC8, C2xSD16, C2xC3:C8, C2xC3:C8, Dic3:C4, C4:Dic3, D4.S3, C6.D4, C3xC22:C4, C3xC4:C4, C2xDic6, C22xC12, C6xD4, C6xD4, C8:8D4, C12.Q8, C6.SD16, D4:Dic3, C22xC3:C8, C12.48D4, C2xD4.S3, C3xC4:D4, C3:C8:23D4
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2xD4, C4oD4, C3:D4, C22xS3, C4:D4, C2xSD16, C4oD8, D4.S3, S3xD4, D4:2S3, C2xC3:D4, C8:8D4, C2xD4.S3, C23.14D6, Q8.13D6, C3:C8:23D4
►On 96 points
(1 46 61)(2 62 47)(3 48 63)(4 64 41)(5 42 57)(6 58 43)(7 44 59)(8 60 45)(9 86 34)(10 35 87)(11 88 36)(12 37 81)(13 82 38)(14 39 83)(15 84 40)(16 33 85)(17 28 93)(18 94 29)(19 30 95)(20 96 31)(21 32 89)(22 90 25)(23 26 91)(24 92 27)(49 75 71)(50 72 76)(51 77 65)(52 66 78)(53 79 67)(54 68 80)(55 73 69)(56 70 74)
(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 74 91 16)(2 77 92 11)(3 80 93 14)(4 75 94 9)(5 78 95 12)(6 73 96 15)(7 76 89 10)(8 79 90 13)(17 39 48 54)(18 34 41 49)(19 37 42 52)(20 40 43 55)(21 35 44 50)(22 38 45 53)(23 33 46 56)(24 36 47 51)(25 82 60 67)(26 85 61 70)(27 88 62 65)(28 83 63 68)(29 86 64 71)(30 81 57 66)(31 84 58 69)(32 87 59 72)
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 49)(25 69)(26 70)(27 71)(28 72)(29 65)(30 66)(31 67)(32 68)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 89)
G:=sub<Sym(96)| (1,46,61)(2,62,47)(3,48,63)(4,64,41)(5,42,57)(6,58,43)(7,44,59)(8,60,45)(9,86,34)(10,35,87)(11,88,36)(12,37,81)(13,82,38)(14,39,83)(15,84,40)(16,33,85)(17,28,93)(18,94,29)(19,30,95)(20,96,31)(21,32,89)(22,90,25)(23,26,91)(24,92,27)(49,75,71)(50,72,76)(51,77,65)(52,66,78)(53,79,67)(54,68,80)(55,73,69)(56,70,74), (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), (1,74,91,16)(2,77,92,11)(3,80,93,14)(4,75,94,9)(5,78,95,12)(6,73,96,15)(7,76,89,10)(8,79,90,13)(17,39,48,54)(18,34,41,49)(19,37,42,52)(20,40,43,55)(21,35,44,50)(22,38,45,53)(23,33,46,56)(24,36,47,51)(25,82,60,67)(26,85,61,70)(27,88,62,65)(28,83,63,68)(29,86,64,71)(30,81,57,66)(31,84,58,69)(32,87,59,72), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,49)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,89)>;
G:=Group( (1,46,61)(2,62,47)(3,48,63)(4,64,41)(5,42,57)(6,58,43)(7,44,59)(8,60,45)(9,86,34)(10,35,87)(11,88,36)(12,37,81)(13,82,38)(14,39,83)(15,84,40)(16,33,85)(17,28,93)(18,94,29)(19,30,95)(20,96,31)(21,32,89)(22,90,25)(23,26,91)(24,92,27)(49,75,71)(50,72,76)(51,77,65)(52,66,78)(53,79,67)(54,68,80)(55,73,69)(56,70,74), (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), (1,74,91,16)(2,77,92,11)(3,80,93,14)(4,75,94,9)(5,78,95,12)(6,73,96,15)(7,76,89,10)(8,79,90,13)(17,39,48,54)(18,34,41,49)(19,37,42,52)(20,40,43,55)(21,35,44,50)(22,38,45,53)(23,33,46,56)(24,36,47,51)(25,82,60,67)(26,85,61,70)(27,88,62,65)(28,83,63,68)(29,86,64,71)(30,81,57,66)(31,84,58,69)(32,87,59,72), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,49)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,89) );
G=PermutationGroup([[(1,46,61),(2,62,47),(3,48,63),(4,64,41),(5,42,57),(6,58,43),(7,44,59),(8,60,45),(9,86,34),(10,35,87),(11,88,36),(12,37,81),(13,82,38),(14,39,83),(15,84,40),(16,33,85),(17,28,93),(18,94,29),(19,30,95),(20,96,31),(21,32,89),(22,90,25),(23,26,91),(24,92,27),(49,75,71),(50,72,76),(51,77,65),(52,66,78),(53,79,67),(54,68,80),(55,73,69),(56,70,74)], [(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)], [(1,74,91,16),(2,77,92,11),(3,80,93,14),(4,75,94,9),(5,78,95,12),(6,73,96,15),(7,76,89,10),(8,79,90,13),(17,39,48,54),(18,34,41,49),(19,37,42,52),(20,40,43,55),(21,35,44,50),(22,38,45,53),(23,33,46,56),(24,36,47,51),(25,82,60,67),(26,85,61,70),(27,88,62,65),(28,83,63,68),(29,86,64,71),(30,81,57,66),(31,84,58,69),(32,87,59,72)], [(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,49),(25,69),(26,70),(27,71),(28,72),(29,65),(30,66),(31,67),(32,68),(33,46),(34,47),(35,48),(36,41),(37,42),(38,43),(39,44),(40,45),(57,81),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,88),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,89)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 8 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4oD4 | SD16 | C3:D4 | C3:D4 | C4oD8 | S3xD4 | D4:2S3 | D4.S3 | Q8.13D6 |
| kernel | C3:C8:23D4 | C12.Q8 | C6.SD16 | D4:Dic3 | C22xC3:C8 | C12.48D4 | C2xD4.S3 | C3xC4:D4 | C4:D4 | C3:C8 | C2xC12 | C22xC6 | C4:C4 | C22xC4 | C2xD4 | C12 | C2xC6 | C2xC4 | C23 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C3:C8:23D4 ►in GL6(F73)
| 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 | 64 | 0 |
| 0 | 0 | 0 | 0 | 20 | 8 |
| 67 | 6 | 0 | 0 | 0 | 0 |
| 67 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 36 | 0 | 0 |
| 0 | 0 | 4 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 4 |
| 0 | 0 | 0 | 0 | 16 | 64 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 23 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 41 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 23 | 0 | 0 |
| 0 | 0 | 35 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,20,0,0,0,0,0,8],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,1,4,0,0,0,0,36,72,0,0,0,0,0,0,9,16,0,0,0,0,4,64],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,23,46,0,0,0,0,0,0,72,41,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,35,0,0,0,0,23,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
C3:C8:23D4 in GAP, Magma, Sage, TeX
C_3\rtimes C_8\rtimes_{23}D_4 % in TeX
G:=Group("C3:C8:23D4"); // GroupNames label
G:=SmallGroup(192,600);
// by ID
G=gap.SmallGroup(192,600);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,219,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=b^3,b*d=d*b,d*c*d=c^-1>;
// generators/relations