metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊6D12, C24⋊26D4, C12⋊4M4(2), C42.257D6, (C4×C8)⋊14S3, D6⋊C8⋊1C2, C6.5(C4×D4), (C4×C24)⋊25C2, C3⋊1(C8⋊6D4), D6⋊C4.1C4, C12⋊C8⋊2C2, C2.8(C4×D12), C4⋊2(C8⋊S3), (C4×D12).2C2, (C2×D12).6C4, C6.4(C8○D4), C4.75(C2×D12), (C2×C8).283D6, C4⋊Dic3.7C4, C2.7(C8○D12), C12.295(C2×D4), C6.4(C2×M4(2)), C4.126(C4○D12), C12.242(C4○D4), (C4×C12).324C22, (C2×C12).806C23, (C2×C24).343C22, C2.7(C2×C8⋊S3), (C2×C8⋊S3)⋊10C2, C22.95(S3×C2×C4), (C2×C4).104(C4×S3), (C2×C12).221(C2×C4), (C2×C3⋊C8).187C22, (C22×S3).9(C2×C4), (S3×C2×C4).175C22, (C2×C6).61(C22×C4), (C2×C4).748(C22×S3), (C2×Dic3).11(C2×C4), SmallGroup(192,247)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊6D12
G = < a,b,c | a8=b12=c2=1, ab=ba, cac=a5, cbc=b-1 >
Subgroups: 312 in 122 conjugacy classes, 55 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, C12, 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, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C8⋊S3, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C2×D12, C8⋊6D4, C12⋊C8, D6⋊C8, C4×C24, C4×D12, C2×C8⋊S3, C8⋊6D12
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, C8⋊S3, S3×C2×C4, C2×D12, C4○D12, C8⋊6D4, C4×D12, C2×C8⋊S3, C8○D12, C8⋊6D12
►On 96 points
(1 62 39 90 20 36 77 60)(2 63 40 91 21 25 78 49)(3 64 41 92 22 26 79 50)(4 65 42 93 23 27 80 51)(5 66 43 94 24 28 81 52)(6 67 44 95 13 29 82 53)(7 68 45 96 14 30 83 54)(8 69 46 85 15 31 84 55)(9 70 47 86 16 32 73 56)(10 71 48 87 17 33 74 57)(11 72 37 88 18 34 75 58)(12 61 38 89 19 35 76 59)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 72)(26 71)(27 70)(28 69)(29 68)(30 67)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 88)(50 87)(51 86)(52 85)(53 96)(54 95)(55 94)(56 93)(57 92)(58 91)(59 90)(60 89)(73 80)(74 79)(75 78)(76 77)(81 84)(82 83)
G:=sub<Sym(96)| (1,62,39,90,20,36,77,60)(2,63,40,91,21,25,78,49)(3,64,41,92,22,26,79,50)(4,65,42,93,23,27,80,51)(5,66,43,94,24,28,81,52)(6,67,44,95,13,29,82,53)(7,68,45,96,14,30,83,54)(8,69,46,85,15,31,84,55)(9,70,47,86,16,32,73,56)(10,71,48,87,17,33,74,57)(11,72,37,88,18,34,75,58)(12,61,38,89,19,35,76,59), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,88)(50,87)(51,86)(52,85)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(73,80)(74,79)(75,78)(76,77)(81,84)(82,83)>;
G:=Group( (1,62,39,90,20,36,77,60)(2,63,40,91,21,25,78,49)(3,64,41,92,22,26,79,50)(4,65,42,93,23,27,80,51)(5,66,43,94,24,28,81,52)(6,67,44,95,13,29,82,53)(7,68,45,96,14,30,83,54)(8,69,46,85,15,31,84,55)(9,70,47,86,16,32,73,56)(10,71,48,87,17,33,74,57)(11,72,37,88,18,34,75,58)(12,61,38,89,19,35,76,59), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,88)(50,87)(51,86)(52,85)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(73,80)(74,79)(75,78)(76,77)(81,84)(82,83) );
G=PermutationGroup([[(1,62,39,90,20,36,77,60),(2,63,40,91,21,25,78,49),(3,64,41,92,22,26,79,50),(4,65,42,93,23,27,80,51),(5,66,43,94,24,28,81,52),(6,67,44,95,13,29,82,53),(7,68,45,96,14,30,83,54),(8,69,46,85,15,31,84,55),(9,70,47,86,16,32,73,56),(10,71,48,87,17,33,74,57),(11,72,37,88,18,34,75,58),(12,61,38,89,19,35,76,59)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14),(15,24),(16,23),(17,22),(18,21),(19,20),(25,72),(26,71),(27,70),(28,69),(29,68),(30,67),(31,66),(32,65),(33,64),(34,63),(35,62),(36,61),(37,40),(38,39),(41,48),(42,47),(43,46),(44,45),(49,88),(50,87),(51,86),(52,85),(53,96),(54,95),(55,94),(56,93),(57,92),(58,91),(59,90),(60,89),(73,80),(74,79),(75,78),(76,77),(81,84),(82,83)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | M4(2) | C4○D4 | D12 | C4×S3 | C8○D4 | C8⋊S3 | C4○D12 | C8○D12 |
| kernel | C8⋊6D12 | C12⋊C8 | D6⋊C8 | C4×C24 | C4×D12 | C2×C8⋊S3 | C4⋊Dic3 | D6⋊C4 | C2×D12 | C4×C8 | C24 | C42 | C2×C8 | C12 | C12 | C8 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 4 | 8 |
Matrix representation of C8⋊6D12 ►in GL4(𝔽73) generated by
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 70 | 6 |
| 0 | 0 | 67 | 3 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 14 | 66 |
| 0 | 0 | 7 | 7 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 59 |
| 0 | 0 | 66 | 66 |
G:=sub<GL(4,GF(73))| [27,0,0,0,0,27,0,0,0,0,70,67,0,0,6,3],[0,72,0,0,1,0,0,0,0,0,14,7,0,0,66,7],[0,1,0,0,1,0,0,0,0,0,7,66,0,0,59,66] >;
C8⋊6D12 in GAP, Magma, Sage, TeX
C_8\rtimes_6D_{12} % in TeX
G:=Group("C8:6D12"); // GroupNames label
G:=SmallGroup(192,247);
// by ID
G=gap.SmallGroup(192,247);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,758,58,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^8=b^12=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations