metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊5(C4×S3), C8⋊S3⋊2C4, C24⋊10(C2×C4), (C2×C8).65D6, C2.D8⋊10S3, (C4×S3).2Q8, C4.31(S3×Q8), C4⋊C4.171D6, D6.6(C4⋊C4), C8⋊Dic3⋊19C2, C12.22(C2×Q8), C6.Q16⋊20C2, C2.5(D8⋊S3), C22.91(S3×D4), C6.40(C8⋊C22), Dic3.7(C4⋊C4), C12.50(C22×C4), C12.Q8⋊20C2, C2.5(Q16⋊S3), (C22×S3).84D4, C3⋊2(M4(2)⋊C4), (C2×C24).143C22, (C2×C12).297C23, (C2×Dic3).167D4, C6.69(C8.C22), C4⋊Dic3.123C22, C3⋊C8⋊5(C2×C4), C4.81(S3×C2×C4), C6.15(C2×C4⋊C4), (S3×C4⋊C4).8C2, C2.16(S3×C4⋊C4), (C3×C2.D8)⋊7C2, (C4×S3).7(C2×C4), C4⋊C4⋊7S3.8C2, (C2×C8⋊S3).4C2, (C2×C6).302(C2×D4), (C2×C3⋊C8).68C22, (S3×C2×C4).37C22, (C3×C4⋊C4).90C22, (C2×C4).400(C22×S3), SmallGroup(192,440)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊S3⋊C4
G = < a,b,c,d | a8=b3=c2=d4=1, ab=ba, cac=a5, dad-1=a-1, cbc=b-1, bd=db, cd=dc >
Subgroups: 304 in 118 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C8⋊S3, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, M4(2)⋊C4, C6.Q16, C12.Q8, C8⋊Dic3, C3×C2.D8, S3×C4⋊C4, C4⋊C4⋊7S3, C2×C8⋊S3, C8⋊S3⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4×S3, C22×S3, C2×C4⋊C4, C8⋊C22, C8.C22, S3×C2×C4, S3×D4, S3×Q8, M4(2)⋊C4, S3×C4⋊C4, D8⋊S3, Q16⋊S3, C8⋊S3⋊C4
►On 96 points
(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 71 82)(2 72 83)(3 65 84)(4 66 85)(5 67 86)(6 68 87)(7 69 88)(8 70 81)(9 40 77)(10 33 78)(11 34 79)(12 35 80)(13 36 73)(14 37 74)(15 38 75)(16 39 76)(17 49 58)(18 50 59)(19 51 60)(20 52 61)(21 53 62)(22 54 63)(23 55 64)(24 56 57)(25 47 89)(26 48 90)(27 41 91)(28 42 92)(29 43 93)(30 44 94)(31 45 95)(32 46 96)
(2 6)(4 8)(9 77)(10 74)(11 79)(12 76)(13 73)(14 78)(15 75)(16 80)(17 62)(18 59)(19 64)(20 61)(21 58)(22 63)(23 60)(24 57)(25 89)(26 94)(27 91)(28 96)(29 93)(30 90)(31 95)(32 92)(33 37)(35 39)(42 46)(44 48)(49 53)(51 55)(65 84)(66 81)(67 86)(68 83)(69 88)(70 85)(71 82)(72 87)
(1 52 45 40)(2 51 46 39)(3 50 47 38)(4 49 48 37)(5 56 41 36)(6 55 42 35)(7 54 43 34)(8 53 44 33)(9 82 20 31)(10 81 21 30)(11 88 22 29)(12 87 23 28)(13 86 24 27)(14 85 17 26)(15 84 18 25)(16 83 19 32)(57 91 73 67)(58 90 74 66)(59 89 75 65)(60 96 76 72)(61 95 77 71)(62 94 78 70)(63 93 79 69)(64 92 80 68)
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), (1,71,82)(2,72,83)(3,65,84)(4,66,85)(5,67,86)(6,68,87)(7,69,88)(8,70,81)(9,40,77)(10,33,78)(11,34,79)(12,35,80)(13,36,73)(14,37,74)(15,38,75)(16,39,76)(17,49,58)(18,50,59)(19,51,60)(20,52,61)(21,53,62)(22,54,63)(23,55,64)(24,56,57)(25,47,89)(26,48,90)(27,41,91)(28,42,92)(29,43,93)(30,44,94)(31,45,95)(32,46,96), (2,6)(4,8)(9,77)(10,74)(11,79)(12,76)(13,73)(14,78)(15,75)(16,80)(17,62)(18,59)(19,64)(20,61)(21,58)(22,63)(23,60)(24,57)(25,89)(26,94)(27,91)(28,96)(29,93)(30,90)(31,95)(32,92)(33,37)(35,39)(42,46)(44,48)(49,53)(51,55)(65,84)(66,81)(67,86)(68,83)(69,88)(70,85)(71,82)(72,87), (1,52,45,40)(2,51,46,39)(3,50,47,38)(4,49,48,37)(5,56,41,36)(6,55,42,35)(7,54,43,34)(8,53,44,33)(9,82,20,31)(10,81,21,30)(11,88,22,29)(12,87,23,28)(13,86,24,27)(14,85,17,26)(15,84,18,25)(16,83,19,32)(57,91,73,67)(58,90,74,66)(59,89,75,65)(60,96,76,72)(61,95,77,71)(62,94,78,70)(63,93,79,69)(64,92,80,68)>;
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), (1,71,82)(2,72,83)(3,65,84)(4,66,85)(5,67,86)(6,68,87)(7,69,88)(8,70,81)(9,40,77)(10,33,78)(11,34,79)(12,35,80)(13,36,73)(14,37,74)(15,38,75)(16,39,76)(17,49,58)(18,50,59)(19,51,60)(20,52,61)(21,53,62)(22,54,63)(23,55,64)(24,56,57)(25,47,89)(26,48,90)(27,41,91)(28,42,92)(29,43,93)(30,44,94)(31,45,95)(32,46,96), (2,6)(4,8)(9,77)(10,74)(11,79)(12,76)(13,73)(14,78)(15,75)(16,80)(17,62)(18,59)(19,64)(20,61)(21,58)(22,63)(23,60)(24,57)(25,89)(26,94)(27,91)(28,96)(29,93)(30,90)(31,95)(32,92)(33,37)(35,39)(42,46)(44,48)(49,53)(51,55)(65,84)(66,81)(67,86)(68,83)(69,88)(70,85)(71,82)(72,87), (1,52,45,40)(2,51,46,39)(3,50,47,38)(4,49,48,37)(5,56,41,36)(6,55,42,35)(7,54,43,34)(8,53,44,33)(9,82,20,31)(10,81,21,30)(11,88,22,29)(12,87,23,28)(13,86,24,27)(14,85,17,26)(15,84,18,25)(16,83,19,32)(57,91,73,67)(58,90,74,66)(59,89,75,65)(60,96,76,72)(61,95,77,71)(62,94,78,70)(63,93,79,69)(64,92,80,68) );
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)], [(1,71,82),(2,72,83),(3,65,84),(4,66,85),(5,67,86),(6,68,87),(7,69,88),(8,70,81),(9,40,77),(10,33,78),(11,34,79),(12,35,80),(13,36,73),(14,37,74),(15,38,75),(16,39,76),(17,49,58),(18,50,59),(19,51,60),(20,52,61),(21,53,62),(22,54,63),(23,55,64),(24,56,57),(25,47,89),(26,48,90),(27,41,91),(28,42,92),(29,43,93),(30,44,94),(31,45,95),(32,46,96)], [(2,6),(4,8),(9,77),(10,74),(11,79),(12,76),(13,73),(14,78),(15,75),(16,80),(17,62),(18,59),(19,64),(20,61),(21,58),(22,63),(23,60),(24,57),(25,89),(26,94),(27,91),(28,96),(29,93),(30,90),(31,95),(32,92),(33,37),(35,39),(42,46),(44,48),(49,53),(51,55),(65,84),(66,81),(67,86),(68,83),(69,88),(70,85),(71,82),(72,87)], [(1,52,45,40),(2,51,46,39),(3,50,47,38),(4,49,48,37),(5,56,41,36),(6,55,42,35),(7,54,43,34),(8,53,44,33),(9,82,20,31),(10,81,21,30),(11,88,22,29),(12,87,23,28),(13,86,24,27),(14,85,17,26),(15,84,18,25),(16,83,19,32),(57,91,73,67),(58,90,74,66),(59,89,75,65),(60,96,76,72),(61,95,77,71),(62,94,78,70),(63,93,79,69),(64,92,80,68)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 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 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | D4 | D6 | D6 | C4×S3 | C8⋊C22 | C8.C22 | S3×Q8 | S3×D4 | D8⋊S3 | Q16⋊S3 |
| kernel | C8⋊S3⋊C4 | C6.Q16 | C12.Q8 | C8⋊Dic3 | C3×C2.D8 | S3×C4⋊C4 | C4⋊C4⋊7S3 | C2×C8⋊S3 | C8⋊S3 | C2.D8 | C4×S3 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C8 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C8⋊S3⋊C4 ►in GL8(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 45 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 55 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 70 | 3 | 0 |
| 0 | 0 | 0 | 0 | 72 | 70 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 12 | 34 | 3 |
| 0 | 0 | 0 | 0 | 36 | 69 | 1 | 3 |
| 72 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 2 | 72 | 0 |
| 0 | 0 | 0 | 0 | 25 | 2 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 68 | 48 | 7 | 13 |
| 0 | 0 | 0 | 0 | 28 | 58 | 13 | 66 |
G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,45,55,0,0,0,0,0,0,72,28,0,0,0,0,0,0,0,0,39,72,3,36,0,0,0,0,70,70,12,69,0,0,0,0,3,0,34,1,0,0,0,0,0,3,3,3],[72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,47,25,0,0,0,0,0,1,2,2,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,41,16,0,0,0,0,0,0,50,32,0,0,0,0,0,0,0,0,60,7,68,28,0,0,0,0,7,13,48,58,0,0,0,0,0,0,7,13,0,0,0,0,0,0,13,66] >;
C8⋊S3⋊C4 in GAP, Magma, Sage, TeX
C_8\rtimes S_3\rtimes C_4
% in TeX
G:=Group("C8:S3:C4"); // GroupNames label
G:=SmallGroup(192,440);
// by ID
G=gap.SmallGroup(192,440);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,219,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^3=c^2=d^4=1,a*b=b*a,c*a*c=a^5,d*a*d^-1=a^-1,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations