direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2xC8.Dic3, C23.16Dic6, C24.74(C2xC4), (C2xC24).19C4, C4.84(C2xD12), (C2xC8).314D6, C12.48(C4:C4), (C2xC12).62Q8, C6:1(C8.C4), C12.304(C2xD4), (C2xC12).402D4, (C2xC4).170D12, (C22xC8).14S3, (C2xC4).51Dic6, (C2xC8).11Dic3, C8.19(C2xDic3), (C22xC6).23Q8, (C22xC24).18C2, C4.24(C4:Dic3), (C22xC4).442D6, C22.8(C2xDic6), (C2xC24).386C22, (C2xC12).795C23, C12.172(C22xC4), C4.26(C22xDic3), C22.14(C4:Dic3), C4.Dic3.35C22, (C22xC12).540C22, C6.47(C2xC4:C4), C3:2(C2xC8.C4), (C2xC6).40(C2xQ8), (C2xC6).52(C4:C4), C2.13(C2xC4:Dic3), (C2xC12).307(C2xC4), (C2xC4).84(C2xDic3), (C2xC4.Dic3).5C2, (C2xC4).713(C22xS3), SmallGroup(192,666)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xC8.Dic3
G = < a,b,c,d | a2=b8=1, c6=b4, d2=b4c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >
Subgroups: 184 in 106 conjugacy classes, 71 normal (41 characteristic)
C1, C2, C2 [x2], C2 [x2], C3, C4 [x4], C22 [x3], C22 [x2], C6, C6 [x2], C6 [x2], C8 [x4], C8 [x4], C2xC4 [x6], C23, C12 [x4], C2xC6 [x3], C2xC6 [x2], C2xC8 [x2], C2xC8 [x4], C2xC8 [x2], M4(2) [x6], C22xC4, C3:C8 [x4], C24 [x4], C2xC12 [x6], C22xC6, C8.C4 [x4], C22xC8, C2xM4(2) [x2], C2xC3:C8 [x2], C4.Dic3 [x4], C4.Dic3 [x2], C2xC24 [x2], C2xC24 [x4], C22xC12, C2xC8.C4, C8.Dic3 [x4], C2xC4.Dic3 [x2], C22xC24, C2xC8.Dic3
Quotients: C1, C2 [x7], C4 [x4], C22 [x7], S3, C2xC4 [x6], D4 [x2], Q8 [x2], C23, Dic3 [x4], D6 [x3], C4:C4 [x4], C22xC4, C2xD4, C2xQ8, Dic6 [x2], D12 [x2], C2xDic3 [x6], C22xS3, C8.C4 [x2], C2xC4:C4, C4:Dic3 [x4], C2xDic6, C2xD12, C22xDic3, C2xC8.C4, C8.Dic3 [x2], C2xC4:Dic3, C2xC8.Dic3
►On 96 points
(1 33)(2 34)(3 35)(4 36)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 49)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 73)(46 74)(47 75)(48 76)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 85)(70 86)(71 87)(72 88)
(1 94 36 69 7 88 30 63)(2 95 25 70 8 89 31 64)(3 96 26 71 9 90 32 65)(4 85 27 72 10 91 33 66)(5 86 28 61 11 92 34 67)(6 87 29 62 12 93 35 68)(13 83 59 40 19 77 53 46)(14 84 60 41 20 78 54 47)(15 73 49 42 21 79 55 48)(16 74 50 43 22 80 56 37)(17 75 51 44 23 81 57 38)(18 76 52 45 24 82 58 39)
(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 16 10 13 7 22 4 19)(2 21 11 18 8 15 5 24)(3 14 12 23 9 20 6 17)(25 49 34 58 31 55 28 52)(26 54 35 51 32 60 29 57)(27 59 36 56 33 53 30 50)(37 91 46 88 43 85 40 94)(38 96 47 93 44 90 41 87)(39 89 48 86 45 95 42 92)(61 76 70 73 67 82 64 79)(62 81 71 78 68 75 65 84)(63 74 72 83 69 80 66 77)
G:=sub<Sym(96)| (1,33)(2,34)(3,35)(4,36)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,49)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,73)(46,74)(47,75)(48,76)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,85)(70,86)(71,87)(72,88), (1,94,36,69,7,88,30,63)(2,95,25,70,8,89,31,64)(3,96,26,71,9,90,32,65)(4,85,27,72,10,91,33,66)(5,86,28,61,11,92,34,67)(6,87,29,62,12,93,35,68)(13,83,59,40,19,77,53,46)(14,84,60,41,20,78,54,47)(15,73,49,42,21,79,55,48)(16,74,50,43,22,80,56,37)(17,75,51,44,23,81,57,38)(18,76,52,45,24,82,58,39), (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), (1,16,10,13,7,22,4,19)(2,21,11,18,8,15,5,24)(3,14,12,23,9,20,6,17)(25,49,34,58,31,55,28,52)(26,54,35,51,32,60,29,57)(27,59,36,56,33,53,30,50)(37,91,46,88,43,85,40,94)(38,96,47,93,44,90,41,87)(39,89,48,86,45,95,42,92)(61,76,70,73,67,82,64,79)(62,81,71,78,68,75,65,84)(63,74,72,83,69,80,66,77)>;
G:=Group( (1,33)(2,34)(3,35)(4,36)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,49)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,73)(46,74)(47,75)(48,76)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,85)(70,86)(71,87)(72,88), (1,94,36,69,7,88,30,63)(2,95,25,70,8,89,31,64)(3,96,26,71,9,90,32,65)(4,85,27,72,10,91,33,66)(5,86,28,61,11,92,34,67)(6,87,29,62,12,93,35,68)(13,83,59,40,19,77,53,46)(14,84,60,41,20,78,54,47)(15,73,49,42,21,79,55,48)(16,74,50,43,22,80,56,37)(17,75,51,44,23,81,57,38)(18,76,52,45,24,82,58,39), (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), (1,16,10,13,7,22,4,19)(2,21,11,18,8,15,5,24)(3,14,12,23,9,20,6,17)(25,49,34,58,31,55,28,52)(26,54,35,51,32,60,29,57)(27,59,36,56,33,53,30,50)(37,91,46,88,43,85,40,94)(38,96,47,93,44,90,41,87)(39,89,48,86,45,95,42,92)(61,76,70,73,67,82,64,79)(62,81,71,78,68,75,65,84)(63,74,72,83,69,80,66,77) );
G=PermutationGroup([(1,33),(2,34),(3,35),(4,36),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,49),(37,77),(38,78),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,73),(46,74),(47,75),(48,76),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,85),(70,86),(71,87),(72,88)], [(1,94,36,69,7,88,30,63),(2,95,25,70,8,89,31,64),(3,96,26,71,9,90,32,65),(4,85,27,72,10,91,33,66),(5,86,28,61,11,92,34,67),(6,87,29,62,12,93,35,68),(13,83,59,40,19,77,53,46),(14,84,60,41,20,78,54,47),(15,73,49,42,21,79,55,48),(16,74,50,43,22,80,56,37),(17,75,51,44,23,81,57,38),(18,76,52,45,24,82,58,39)], [(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)], [(1,16,10,13,7,22,4,19),(2,21,11,18,8,15,5,24),(3,14,12,23,9,20,6,17),(25,49,34,58,31,55,28,52),(26,54,35,51,32,60,29,57),(27,59,36,56,33,53,30,50),(37,91,46,88,43,85,40,94),(38,96,47,93,44,90,41,87),(39,89,48,86,45,95,42,92),(61,76,70,73,67,82,64,79),(62,81,71,78,68,75,65,84),(63,74,72,83,69,80,66,77)])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6G | 8A | ··· | 8H | 8I | ··· | 8P | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | - | - | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | Q8 | Dic3 | D6 | D6 | Dic6 | D12 | Dic6 | C8.C4 | C8.Dic3 |
| kernel | C2xC8.Dic3 | C8.Dic3 | C2xC4.Dic3 | C22xC24 | C2xC24 | C22xC8 | C2xC12 | C2xC12 | C22xC6 | C2xC8 | C2xC8 | C22xC4 | C2xC4 | C2xC4 | C23 | C6 | C2 |
| # reps | 1 | 4 | 2 | 1 | 8 | 1 | 2 | 1 | 1 | 4 | 2 | 1 | 2 | 4 | 2 | 8 | 16 |
Matrix representation of C2xC8.Dic3 ►in GL4(F73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 63 | 0 | 0 | 0 |
| 0 | 51 | 0 | 0 |
| 0 | 0 | 66 | 14 |
| 0 | 0 | 59 | 7 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 72 |
| 0 | 22 | 0 | 0 |
| 22 | 0 | 0 | 0 |
| 0 | 0 | 10 | 41 |
| 0 | 0 | 51 | 63 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[63,0,0,0,0,51,0,0,0,0,66,59,0,0,14,7],[46,0,0,0,0,46,0,0,0,0,0,1,0,0,72,72],[0,22,0,0,22,0,0,0,0,0,10,51,0,0,41,63] >;
C2xC8.Dic3 in GAP, Magma, Sage, TeX
C_2\times C_8.{\rm Dic}_3 % in TeX
G:=Group("C2xC8.Dic3"); // GroupNames label
G:=SmallGroup(192,666);
// by ID
G=gap.SmallGroup(192,666);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,100,136,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^6=b^4,d^2=b^4*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations