direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8×D12, C24⋊25D4, C42.256D6, C3⋊1(C8×D4), C4⋊2(S3×C8), (C4×C8)⋊3S3, C12⋊4(C2×C8), D6⋊1(C2×C8), C6.4(C4×D4), (C4×C24)⋊23C2, D6⋊C8⋊41C2, C2.1(C4×D12), D6⋊C4.18C4, C12⋊C8⋊36C2, C6.3(C8○D4), C4.74(C2×D12), (C2×C8).339D6, C6.4(C22×C8), (C4×D12).29C2, (C2×D12).18C4, C2.2(C8○D12), C12.294(C2×D4), C4⋊Dic3.23C4, C4.125(C4○D12), C12.241(C4○D4), (C4×C12).323C22, (C2×C24).342C22, (C2×C12).804C23, C2.6(S3×C2×C8), (S3×C2×C8)⋊11C2, C22.36(S3×C2×C4), (C2×C4).103(C4×S3), (C2×C12).220(C2×C4), (C2×C3⋊C8).290C22, (S3×C2×C4).266C22, (C2×C6).59(C22×C4), (C22×S3).31(C2×C4), (C2×C4).746(C22×S3), (C2×Dic3).46(C2×C4), SmallGroup(192,245)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8×D12
G = < a,b,c | a8=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 312 in 134 conjugacy classes, 63 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, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, 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, C22×C8, S3×C8, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C2×D12, C8×D4, C12⋊C8, D6⋊C8, C4×C24, C4×D12, S3×C2×C8, C8×D12
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, C23, D6, C2×C8, C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C4×D4, C22×C8, C8○D4, S3×C8, S3×C2×C4, C2×D12, C4○D12, C8×D4, C4×D12, S3×C2×C8, C8○D12, C8×D12
►On 96 points
(1 82 67 58 23 34 42 93)(2 83 68 59 24 35 43 94)(3 84 69 60 13 36 44 95)(4 73 70 49 14 25 45 96)(5 74 71 50 15 26 46 85)(6 75 72 51 16 27 47 86)(7 76 61 52 17 28 48 87)(8 77 62 53 18 29 37 88)(9 78 63 54 19 30 38 89)(10 79 64 55 20 31 39 90)(11 80 65 56 21 32 40 91)(12 81 66 57 22 33 41 92)
(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 23)(14 22)(15 21)(16 20)(17 19)(25 33)(26 32)(27 31)(28 30)(34 36)(38 48)(39 47)(40 46)(41 45)(42 44)(49 57)(50 56)(51 55)(52 54)(58 60)(61 63)(64 72)(65 71)(66 70)(67 69)(73 81)(74 80)(75 79)(76 78)(82 84)(85 91)(86 90)(87 89)(92 96)(93 95)
G:=sub<Sym(96)| (1,82,67,58,23,34,42,93)(2,83,68,59,24,35,43,94)(3,84,69,60,13,36,44,95)(4,73,70,49,14,25,45,96)(5,74,71,50,15,26,46,85)(6,75,72,51,16,27,47,86)(7,76,61,52,17,28,48,87)(8,77,62,53,18,29,37,88)(9,78,63,54,19,30,38,89)(10,79,64,55,20,31,39,90)(11,80,65,56,21,32,40,91)(12,81,66,57,22,33,41,92), (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)(13,23)(14,22)(15,21)(16,20)(17,19)(25,33)(26,32)(27,31)(28,30)(34,36)(38,48)(39,47)(40,46)(41,45)(42,44)(49,57)(50,56)(51,55)(52,54)(58,60)(61,63)(64,72)(65,71)(66,70)(67,69)(73,81)(74,80)(75,79)(76,78)(82,84)(85,91)(86,90)(87,89)(92,96)(93,95)>;
G:=Group( (1,82,67,58,23,34,42,93)(2,83,68,59,24,35,43,94)(3,84,69,60,13,36,44,95)(4,73,70,49,14,25,45,96)(5,74,71,50,15,26,46,85)(6,75,72,51,16,27,47,86)(7,76,61,52,17,28,48,87)(8,77,62,53,18,29,37,88)(9,78,63,54,19,30,38,89)(10,79,64,55,20,31,39,90)(11,80,65,56,21,32,40,91)(12,81,66,57,22,33,41,92), (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)(13,23)(14,22)(15,21)(16,20)(17,19)(25,33)(26,32)(27,31)(28,30)(34,36)(38,48)(39,47)(40,46)(41,45)(42,44)(49,57)(50,56)(51,55)(52,54)(58,60)(61,63)(64,72)(65,71)(66,70)(67,69)(73,81)(74,80)(75,79)(76,78)(82,84)(85,91)(86,90)(87,89)(92,96)(93,95) );
G=PermutationGroup([[(1,82,67,58,23,34,42,93),(2,83,68,59,24,35,43,94),(3,84,69,60,13,36,44,95),(4,73,70,49,14,25,45,96),(5,74,71,50,15,26,46,85),(6,75,72,51,16,27,47,86),(7,76,61,52,17,28,48,87),(8,77,62,53,18,29,37,88),(9,78,63,54,19,30,38,89),(10,79,64,55,20,31,39,90),(11,80,65,56,21,32,40,91),(12,81,66,57,22,33,41,92)], [(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),(13,23),(14,22),(15,21),(16,20),(17,19),(25,33),(26,32),(27,31),(28,30),(34,36),(38,48),(39,47),(40,46),(41,45),(42,44),(49,57),(50,56),(51,55),(52,54),(58,60),(61,63),(64,72),(65,71),(66,70),(67,69),(73,81),(74,80),(75,79),(76,78),(82,84),(85,91),(86,90),(87,89),(92,96),(93,95)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | ··· | 8T | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 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 | 8 | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | S3 | D4 | D6 | D6 | C4○D4 | D12 | C4×S3 | C8○D4 | S3×C8 | C4○D12 | C8○D12 |
| kernel | C8×D12 | C12⋊C8 | D6⋊C8 | C4×C24 | C4×D12 | S3×C2×C8 | C4⋊Dic3 | D6⋊C4 | C2×D12 | D12 | C4×C8 | C24 | C42 | C2×C8 | C12 | C8 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 16 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 |
Matrix representation of C8×D12 ►in GL3(𝔽73) generated by
| 10 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 63 |
| 72 | 0 | 0 |
| 0 | 59 | 7 |
| 0 | 66 | 66 |
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 1 | 72 |
G:=sub<GL(3,GF(73))| [10,0,0,0,63,0,0,0,63],[72,0,0,0,59,66,0,7,66],[72,0,0,0,1,1,0,0,72] >;
C8×D12 in GAP, Magma, Sage, TeX
C_8\times D_{12} % in TeX
G:=Group("C8xD12"); // GroupNames label
G:=SmallGroup(192,245);
// by ID
G=gap.SmallGroup(192,245);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,58,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^8=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations