metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊9D12, C24⋊19D4, D6⋊1M4(2), C42.15D6, C8⋊C4⋊8S3, D6⋊C8⋊36C2, C3⋊1(C8⋊9D4), D6⋊C4.4C4, C12⋊C8⋊3C2, C6.10(C4×D4), (C4×D12).4C2, (C2×D12).7C4, C2.13(C4×D12), (C2×C8).157D6, C4.77(C2×D12), C4⋊Dic3.9C4, C6.21(C8○D4), C12.297(C2×D4), C2.7(D12.C4), (C4×C12).13C22, C6.17(C2×M4(2)), C2.10(S3×M4(2)), C4.130(C4○D12), C12.246(C4○D4), (C2×C24).269C22, (C2×C12).812C23, (S3×C2×C8)⋊26C2, (C2×C4).29(C4×S3), (C3×C8⋊C4)⋊12C2, (C2×C8⋊S3)⋊24C2, C22.99(S3×C2×C4), (C2×C12).37(C2×C4), (C2×C3⋊C8).295C22, (S3×C2×C4).177C22, (C2×C6).67(C22×C4), (C22×S3).33(C2×C4), (C2×C4).754(C22×S3), (C2×Dic3).14(C2×C4), SmallGroup(192,265)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊9D12
G = < a,b,c | a8=b12=c2=1, bab-1=cac=a5, cbc=b-1 >
Subgroups: 312 in 124 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C3⋊C8, C24, C24, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), S3×C8, C8⋊S3, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C2×D12, C8⋊9D4, C12⋊C8, D6⋊C8, C3×C8⋊C4, C4×D12, S3×C2×C8, C2×C8⋊S3, C8⋊9D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, M4(2), C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C4×D4, C2×M4(2), C8○D4, S3×C2×C4, C2×D12, C4○D12, C8⋊9D4, C4×D12, S3×M4(2), D12.C4, C8⋊9D12
►On 96 points
(1 16 56 81 71 34 43 86)(2 35 57 87 72 17 44 82)(3 18 58 83 61 36 45 88)(4 25 59 89 62 19 46 84)(5 20 60 73 63 26 47 90)(6 27 49 91 64 21 48 74)(7 22 50 75 65 28 37 92)(8 29 51 93 66 23 38 76)(9 24 52 77 67 30 39 94)(10 31 53 95 68 13 40 78)(11 14 54 79 69 32 41 96)(12 33 55 85 70 15 42 80)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(37 39)(40 48)(41 47)(42 46)(43 45)(49 53)(50 52)(54 60)(55 59)(56 58)(61 71)(62 70)(63 69)(64 68)(65 67)(73 96)(74 95)(75 94)(76 93)(77 92)(78 91)(79 90)(80 89)(81 88)(82 87)(83 86)(84 85)
G:=sub<Sym(96)| (1,16,56,81,71,34,43,86)(2,35,57,87,72,17,44,82)(3,18,58,83,61,36,45,88)(4,25,59,89,62,19,46,84)(5,20,60,73,63,26,47,90)(6,27,49,91,64,21,48,74)(7,22,50,75,65,28,37,92)(8,29,51,93,66,23,38,76)(9,24,52,77,67,30,39,94)(10,31,53,95,68,13,40,78)(11,14,54,79,69,32,41,96)(12,33,55,85,70,15,42,80), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(37,39)(40,48)(41,47)(42,46)(43,45)(49,53)(50,52)(54,60)(55,59)(56,58)(61,71)(62,70)(63,69)(64,68)(65,67)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)>;
G:=Group( (1,16,56,81,71,34,43,86)(2,35,57,87,72,17,44,82)(3,18,58,83,61,36,45,88)(4,25,59,89,62,19,46,84)(5,20,60,73,63,26,47,90)(6,27,49,91,64,21,48,74)(7,22,50,75,65,28,37,92)(8,29,51,93,66,23,38,76)(9,24,52,77,67,30,39,94)(10,31,53,95,68,13,40,78)(11,14,54,79,69,32,41,96)(12,33,55,85,70,15,42,80), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(37,39)(40,48)(41,47)(42,46)(43,45)(49,53)(50,52)(54,60)(55,59)(56,58)(61,71)(62,70)(63,69)(64,68)(65,67)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85) );
G=PermutationGroup([[(1,16,56,81,71,34,43,86),(2,35,57,87,72,17,44,82),(3,18,58,83,61,36,45,88),(4,25,59,89,62,19,46,84),(5,20,60,73,63,26,47,90),(6,27,49,91,64,21,48,74),(7,22,50,75,65,28,37,92),(8,29,51,93,66,23,38,76),(9,24,52,77,67,30,39,94),(10,31,53,95,68,13,40,78),(11,14,54,79,69,32,41,96),(12,33,55,85,70,15,42,80)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,27),(14,26),(15,25),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,28),(37,39),(40,48),(41,47),(42,46),(43,45),(49,53),(50,52),(54,60),(55,59),(56,58),(61,71),(62,70),(63,69),(64,68),(65,67),(73,96),(74,95),(75,94),(76,93),(77,92),(78,91),(79,90),(80,89),(81,88),(82,87),(83,86),(84,85)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | C4○D4 | M4(2) | D12 | C4×S3 | C8○D4 | C4○D12 | S3×M4(2) | D12.C4 |
| kernel | C8⋊9D12 | C12⋊C8 | D6⋊C8 | C3×C8⋊C4 | C4×D12 | S3×C2×C8 | C2×C8⋊S3 | C4⋊Dic3 | D6⋊C4 | C2×D12 | C8⋊C4 | C24 | C42 | C2×C8 | C12 | D6 | C8 | C2×C4 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of C8⋊9D12 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 71 |
| 0 | 0 | 0 | 0 | 23 | 47 |
| 60 | 67 | 0 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 26 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 53 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 26 | 72 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,23,0,0,0,0,71,47],[60,4,0,0,0,0,67,13,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,26,0,0,0,0,0,72],[72,53,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,1,26,0,0,0,0,0,72] >;
C8⋊9D12 in GAP, Magma, Sage, TeX
C_8\rtimes_9D_{12} % in TeX
G:=Group("C8:9D12"); // GroupNames label
G:=SmallGroup(192,265);
// by ID
G=gap.SmallGroup(192,265);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,387,58,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^8=b^12=c^2=1,b*a*b^-1=c*a*c=a^5,c*b*c=b^-1>;
// generators/relations