direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D8.S3, D8.7D6, C6⋊2SD32, C24.21D4, C12.22D8, C24.25C23, Dic12⋊12C22, C3⋊3(C2×SD32), C3⋊C16⋊8C22, (C2×D8).2S3, (C6×D8).3C2, C6.64(C2×D8), (C2×C6).43D8, C4.9(D4⋊S3), (C2×C8).234D6, C12.161(C2×D4), (C2×C12).181D4, C8.14(C3⋊D4), C8.31(C22×S3), (C3×D8).7C22, (C2×Dic12)⋊17C2, (C2×C24).86C22, C22.22(D4⋊S3), (C2×C3⋊C16)⋊7C2, C4.3(C2×C3⋊D4), C2.19(C2×D4⋊S3), (C2×C4).143(C3⋊D4), SmallGroup(192,707)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D8.S3
G = < a,b,c,d,e | a2=b8=c2=d3=1, e2=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=b5c, ede-1=d-1 >
Subgroups: 280 in 90 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C16, C2×C8, D8, D8, Q16, C2×D4, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C3×D4, C22×C6, C2×C16, SD32, C2×D8, C2×Q16, C3⋊C16, Dic12, Dic12, C2×C24, C3×D8, C3×D8, C2×Dic6, C6×D4, C2×SD32, C2×C3⋊C16, D8.S3, C2×Dic12, C6×D8, C2×D8.S3
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, SD32, C2×D8, D4⋊S3, C2×C3⋊D4, C2×SD32, D8.S3, C2×D4⋊S3, C2×D8.S3
►On 96 points
Generators in S96
(1 62)(2 63)(3 64)(4 57)(5 58)(6 59)(7 60)(8 61)(9 82)(10 83)(11 84)(12 85)(13 86)(14 87)(15 88)(16 81)(17 76)(18 77)(19 78)(20 79)(21 80)(22 73)(23 74)(24 75)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 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 61)(2 60)(3 59)(4 58)(5 57)(6 64)(7 63)(8 62)(9 84)(10 83)(11 82)(12 81)(13 88)(14 87)(15 86)(16 85)(17 80)(18 79)(19 78)(20 77)(21 76)(22 75)(23 74)(24 73)(25 44)(26 43)(27 42)(28 41)(29 48)(30 47)(31 46)(32 45)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 56)(40 55)(65 89)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)
(1 39 32)(2 40 25)(3 33 26)(4 34 27)(5 35 28)(6 36 29)(7 37 30)(8 38 31)(9 96 22)(10 89 23)(11 90 24)(12 91 17)(13 92 18)(14 93 19)(15 94 20)(16 95 21)(41 57 52)(42 58 53)(43 59 54)(44 60 55)(45 61 56)(46 62 49)(47 63 50)(48 64 51)(65 74 83)(66 75 84)(67 76 85)(68 77 86)(69 78 87)(70 79 88)(71 80 81)(72 73 82)
(1 88 5 84)(2 87 6 83)(3 86 7 82)(4 85 8 81)(9 64 13 60)(10 63 14 59)(11 62 15 58)(12 61 16 57)(17 56 21 52)(18 55 22 51)(19 54 23 50)(20 53 24 49)(25 69 29 65)(26 68 30 72)(27 67 31 71)(28 66 32 70)(33 77 37 73)(34 76 38 80)(35 75 39 79)(36 74 40 78)(41 91 45 95)(42 90 46 94)(43 89 47 93)(44 96 48 92)
G:=sub<Sym(96)| (1,62)(2,63)(3,64)(4,57)(5,58)(6,59)(7,60)(8,61)(9,82)(10,83)(11,84)(12,85)(13,86)(14,87)(15,88)(16,81)(17,76)(18,77)(19,78)(20,79)(21,80)(22,73)(23,74)(24,75)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,64)(7,63)(8,62)(9,84)(10,83)(11,82)(12,81)(13,88)(14,87)(15,86)(16,85)(17,80)(18,79)(19,78)(20,77)(21,76)(22,75)(23,74)(24,73)(25,44)(26,43)(27,42)(28,41)(29,48)(30,47)(31,46)(32,45)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,56)(40,55)(65,89)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90), (1,39,32)(2,40,25)(3,33,26)(4,34,27)(5,35,28)(6,36,29)(7,37,30)(8,38,31)(9,96,22)(10,89,23)(11,90,24)(12,91,17)(13,92,18)(14,93,19)(15,94,20)(16,95,21)(41,57,52)(42,58,53)(43,59,54)(44,60,55)(45,61,56)(46,62,49)(47,63,50)(48,64,51)(65,74,83)(66,75,84)(67,76,85)(68,77,86)(69,78,87)(70,79,88)(71,80,81)(72,73,82), (1,88,5,84)(2,87,6,83)(3,86,7,82)(4,85,8,81)(9,64,13,60)(10,63,14,59)(11,62,15,58)(12,61,16,57)(17,56,21,52)(18,55,22,51)(19,54,23,50)(20,53,24,49)(25,69,29,65)(26,68,30,72)(27,67,31,71)(28,66,32,70)(33,77,37,73)(34,76,38,80)(35,75,39,79)(36,74,40,78)(41,91,45,95)(42,90,46,94)(43,89,47,93)(44,96,48,92)>;
G:=Group( (1,62)(2,63)(3,64)(4,57)(5,58)(6,59)(7,60)(8,61)(9,82)(10,83)(11,84)(12,85)(13,86)(14,87)(15,88)(16,81)(17,76)(18,77)(19,78)(20,79)(21,80)(22,73)(23,74)(24,75)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,64)(7,63)(8,62)(9,84)(10,83)(11,82)(12,81)(13,88)(14,87)(15,86)(16,85)(17,80)(18,79)(19,78)(20,77)(21,76)(22,75)(23,74)(24,73)(25,44)(26,43)(27,42)(28,41)(29,48)(30,47)(31,46)(32,45)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,56)(40,55)(65,89)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90), (1,39,32)(2,40,25)(3,33,26)(4,34,27)(5,35,28)(6,36,29)(7,37,30)(8,38,31)(9,96,22)(10,89,23)(11,90,24)(12,91,17)(13,92,18)(14,93,19)(15,94,20)(16,95,21)(41,57,52)(42,58,53)(43,59,54)(44,60,55)(45,61,56)(46,62,49)(47,63,50)(48,64,51)(65,74,83)(66,75,84)(67,76,85)(68,77,86)(69,78,87)(70,79,88)(71,80,81)(72,73,82), (1,88,5,84)(2,87,6,83)(3,86,7,82)(4,85,8,81)(9,64,13,60)(10,63,14,59)(11,62,15,58)(12,61,16,57)(17,56,21,52)(18,55,22,51)(19,54,23,50)(20,53,24,49)(25,69,29,65)(26,68,30,72)(27,67,31,71)(28,66,32,70)(33,77,37,73)(34,76,38,80)(35,75,39,79)(36,74,40,78)(41,91,45,95)(42,90,46,94)(43,89,47,93)(44,96,48,92) );
G=PermutationGroup([[(1,62),(2,63),(3,64),(4,57),(5,58),(6,59),(7,60),(8,61),(9,82),(10,83),(11,84),(12,85),(13,86),(14,87),(15,88),(16,81),(17,76),(18,77),(19,78),(20,79),(21,80),(22,73),(23,74),(24,75),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,49),(40,50),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96)], [(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)], [(1,61),(2,60),(3,59),(4,58),(5,57),(6,64),(7,63),(8,62),(9,84),(10,83),(11,82),(12,81),(13,88),(14,87),(15,86),(16,85),(17,80),(18,79),(19,78),(20,77),(21,76),(22,75),(23,74),(24,73),(25,44),(26,43),(27,42),(28,41),(29,48),(30,47),(31,46),(32,45),(33,54),(34,53),(35,52),(36,51),(37,50),(38,49),(39,56),(40,55),(65,89),(66,96),(67,95),(68,94),(69,93),(70,92),(71,91),(72,90)], [(1,39,32),(2,40,25),(3,33,26),(4,34,27),(5,35,28),(6,36,29),(7,37,30),(8,38,31),(9,96,22),(10,89,23),(11,90,24),(12,91,17),(13,92,18),(14,93,19),(15,94,20),(16,95,21),(41,57,52),(42,58,53),(43,59,54),(44,60,55),(45,61,56),(46,62,49),(47,63,50),(48,64,51),(65,74,83),(66,75,84),(67,76,85),(68,77,86),(69,78,87),(70,79,88),(71,80,81),(72,73,82)], [(1,88,5,84),(2,87,6,83),(3,86,7,82),(4,85,8,81),(9,64,13,60),(10,63,14,59),(11,62,15,58),(12,61,16,57),(17,56,21,52),(18,55,22,51),(19,54,23,50),(20,53,24,49),(25,69,29,65),(26,68,30,72),(27,67,31,71),(28,66,32,70),(33,77,37,73),(34,76,38,80),(35,75,39,79),(36,74,40,78),(41,91,45,95),(42,90,46,94),(43,89,47,93),(44,96,48,92)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 16A | ··· | 16H | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 16 | ··· | 16 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 24 | 24 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | D8 | C3⋊D4 | C3⋊D4 | SD32 | D4⋊S3 | D4⋊S3 | D8.S3 |
| kernel | C2×D8.S3 | C2×C3⋊C16 | D8.S3 | C2×Dic12 | C6×D8 | C2×D8 | C24 | C2×C12 | C2×C8 | D8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 4 |
Matrix representation of C2×D8.S3 ►in GL5(𝔽97)
| 96 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 |
| 0 | 0 | 0 | 0 | 96 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 65 |
| 0 | 0 | 0 | 94 | 83 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 94 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 61 | 0 | 0 | 0 |
| 0 | 0 | 35 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 54 | 76 |
| 0 | 0 | 0 | 65 | 43 |
G:=sub<GL(5,GF(97))| [96,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,96,0,0,0,0,0,96],[1,0,0,0,0,0,96,0,0,0,0,0,96,0,0,0,0,0,0,94,0,0,0,65,83],[1,0,0,0,0,0,96,0,0,0,0,0,1,0,0,0,0,0,0,94,0,0,0,32,0],[1,0,0,0,0,0,61,0,0,0,0,0,35,0,0,0,0,0,1,0,0,0,0,0,1],[96,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,54,65,0,0,0,76,43] >;
C2×D8.S3 in GAP, Magma, Sage, TeX
C_2\times D_8.S_3
% in TeX
G:=Group("C2xD8.S3"); // GroupNames label
G:=SmallGroup(192,707);
// by ID
G=gap.SmallGroup(192,707);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,675,185,192,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^3=1,e^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^5*c,e*d*e^-1=d^-1>;
// generators/relations