direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8.D6, C24.8C23, M4(2)⋊18D6, C12.59C24, C23.58D12, Dic12⋊8C22, D12.22C23, Dic6.22C23, (C2×C4).58D12, (C2×C8).101D6, C4.49(C2×D12), C8.8(C22×S3), C24⋊C2⋊9C22, (C2×C12).204D4, C12.293(C2×D4), C6⋊1(C8.C22), (C2×M4(2))⋊4S3, (C6×M4(2))⋊4C2, C4.56(S3×C23), C6.26(C22×D4), (C2×Dic12)⋊14C2, (C2×C24).69C22, (C22×C6).119D4, C2.28(C22×D12), C22.74(C2×D12), (C22×C4).282D6, (C2×C12).512C23, (C22×Dic6)⋊18C2, (C2×Dic6)⋊63C22, C4○D12.50C22, (C2×D12).230C22, (C3×M4(2))⋊20C22, (C22×C12).267C22, (C2×C24⋊C2)⋊5C2, C3⋊1(C2×C8.C22), (C2×C6).63(C2×D4), (C2×C4○D12).23C2, (C2×C4).224(C22×S3), SmallGroup(192,1306)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 664 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], S3 [×2], C6, C6 [×2], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×7], Q8 [×13], C23, C23, Dic3 [×6], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8 [×10], C4○D4 [×6], C24 [×4], Dic6 [×6], Dic6 [×7], C4×S3 [×4], D12 [×2], D12, C2×Dic3 [×7], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×C6, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C24⋊C2 [×8], Dic12 [×8], C2×C24 [×2], C3×M4(2) [×4], C2×Dic6, C2×Dic6 [×6], C2×Dic6 [×3], S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C22×Dic3, C2×C3⋊D4, C22×C12, C2×C8.C22, C2×C24⋊C2 [×2], C2×Dic12 [×2], C8.D6 [×8], C6×M4(2), C22×Dic6, C2×C4○D12, C2×C8.D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C8.C22 [×2], C22×D4, C2×D12 [×6], S3×C23, C2×C8.C22, C8.D6 [×2], C22×D12, C2×C8.D6
Generators and relations
G = < a,b,c,d | a2=b8=1, c6=d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c5 >
►On 96 points
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 49)(12 50)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 83)(26 84)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 82)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)
(1 92 37 25 7 86 43 31)(2 87 38 32 8 93 44 26)(3 94 39 27 9 88 45 33)(4 89 40 34 10 95 46 28)(5 96 41 29 11 90 47 35)(6 91 42 36 12 85 48 30)(13 66 82 50 19 72 76 56)(14 61 83 57 20 67 77 51)(15 68 84 52 21 62 78 58)(16 63 73 59 22 69 79 53)(17 70 74 54 23 64 80 60)(18 65 75 49 24 71 81 55)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 83 19 77)(14 76 20 82)(15 81 21 75)(16 74 22 80)(17 79 23 73)(18 84 24 78)(25 85 31 91)(26 90 32 96)(27 95 33 89)(28 88 34 94)(29 93 35 87)(30 86 36 92)(37 48 43 42)(38 41 44 47)(39 46 45 40)(49 58 55 52)(50 51 56 57)(53 54 59 60)(61 72 67 66)(62 65 68 71)(63 70 69 64)
G:=sub<Sym(96)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,49)(12,50)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,83)(26,84)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72), (1,92,37,25,7,86,43,31)(2,87,38,32,8,93,44,26)(3,94,39,27,9,88,45,33)(4,89,40,34,10,95,46,28)(5,96,41,29,11,90,47,35)(6,91,42,36,12,85,48,30)(13,66,82,50,19,72,76,56)(14,61,83,57,20,67,77,51)(15,68,84,52,21,62,78,58)(16,63,73,59,22,69,79,53)(17,70,74,54,23,64,80,60)(18,65,75,49,24,71,81,55), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,83,19,77)(14,76,20,82)(15,81,21,75)(16,74,22,80)(17,79,23,73)(18,84,24,78)(25,85,31,91)(26,90,32,96)(27,95,33,89)(28,88,34,94)(29,93,35,87)(30,86,36,92)(37,48,43,42)(38,41,44,47)(39,46,45,40)(49,58,55,52)(50,51,56,57)(53,54,59,60)(61,72,67,66)(62,65,68,71)(63,70,69,64)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,49)(12,50)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,83)(26,84)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72), (1,92,37,25,7,86,43,31)(2,87,38,32,8,93,44,26)(3,94,39,27,9,88,45,33)(4,89,40,34,10,95,46,28)(5,96,41,29,11,90,47,35)(6,91,42,36,12,85,48,30)(13,66,82,50,19,72,76,56)(14,61,83,57,20,67,77,51)(15,68,84,52,21,62,78,58)(16,63,73,59,22,69,79,53)(17,70,74,54,23,64,80,60)(18,65,75,49,24,71,81,55), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,83,19,77)(14,76,20,82)(15,81,21,75)(16,74,22,80)(17,79,23,73)(18,84,24,78)(25,85,31,91)(26,90,32,96)(27,95,33,89)(28,88,34,94)(29,93,35,87)(30,86,36,92)(37,48,43,42)(38,41,44,47)(39,46,45,40)(49,58,55,52)(50,51,56,57)(53,54,59,60)(61,72,67,66)(62,65,68,71)(63,70,69,64) );
G=PermutationGroup([(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,49),(12,50),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,85),(20,86),(21,87),(22,88),(23,89),(24,90),(25,83),(26,84),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,81),(36,82),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72)], [(1,92,37,25,7,86,43,31),(2,87,38,32,8,93,44,26),(3,94,39,27,9,88,45,33),(4,89,40,34,10,95,46,28),(5,96,41,29,11,90,47,35),(6,91,42,36,12,85,48,30),(13,66,82,50,19,72,76,56),(14,61,83,57,20,67,77,51),(15,68,84,52,21,62,78,58),(16,63,73,59,22,69,79,53),(17,70,74,54,23,64,80,60),(18,65,75,49,24,71,81,55)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,83,19,77),(14,76,20,82),(15,81,21,75),(16,74,22,80),(17,79,23,73),(18,84,24,78),(25,85,31,91),(26,90,32,96),(27,95,33,89),(28,88,34,94),(29,93,35,87),(30,86,36,92),(37,48,43,42),(38,41,44,47),(39,46,45,40),(49,58,55,52),(50,51,56,57),(53,54,59,60),(61,72,67,66),(62,65,68,71),(63,70,69,64)])
Matrix representation ►G ⊆ GL8(𝔽73)
| 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 | 0 | 0 | 72 | 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 | 72 |
| 72 | 71 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 66 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 62 | 1 |
| 0 | 0 | 0 | 0 | 17 | 71 | 72 | 0 |
| 0 | 0 | 0 | 0 | 66 | 72 | 62 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 8 | 65 |
| 0 | 0 | 0 | 0 | 9 | 6 | 45 | 5 |
| 0 | 0 | 0 | 0 | 39 | 27 | 40 | 27 |
| 0 | 0 | 0 | 0 | 55 | 27 | 68 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 8 | 65 |
| 0 | 0 | 0 | 0 | 14 | 67 | 45 | 51 |
| 0 | 0 | 0 | 0 | 40 | 46 | 40 | 39 |
| 0 | 0 | 0 | 0 | 56 | 46 | 68 | 11 |
G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[72,1,0,0,0,0,0,0,71,1,0,0,0,0,0,0,0,0,7,14,0,0,0,0,0,0,59,66,0,0,0,0,0,0,0,0,1,0,17,66,0,0,0,0,0,0,71,72,0,0,0,0,3,62,72,62,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,28,9,39,55,0,0,0,0,0,6,27,27,0,0,0,0,8,45,40,68,0,0,0,0,65,5,27,72],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,28,14,40,56,0,0,0,0,0,67,46,46,0,0,0,0,8,45,40,68,0,0,0,0,65,51,39,11] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D12 | D12 | C8.C22 | C8.D6 |
| kernel | C2×C8.D6 | C2×C24⋊C2 | C2×Dic12 | C8.D6 | C6×M4(2) | C22×Dic6 | C2×C4○D12 | C2×M4(2) | C2×C12 | C22×C6 | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 3 | 1 | 2 | 4 | 1 | 6 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2\times C_8.D_6
% in TeX
G:=Group("C2xC8.D6"); // GroupNames label
G:=SmallGroup(192,1306);
// by ID
G=gap.SmallGroup(192,1306);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,80,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^6=d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations