direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12.C8, C6⋊2M5(2), C24.77C23, (C2×C12).8C8, C3⋊3(C2×M5(2)), C3⋊C16⋊12C22, C24.84(C2×C4), C12.45(C2×C8), (C2×C24).28C4, (C2×C8).326D6, C23.4(C3⋊C8), (C22×C6).7C8, C6.27(C22×C8), C8.63(C22×S3), (C22×C8).16S3, (C2×C8).15Dic3, C8.26(C2×Dic3), (C22×C24).27C2, (C22×C12).27C4, C12.176(C22×C4), (C2×C24).431C22, C4.30(C22×Dic3), (C22×C4).20Dic3, C4.9(C2×C3⋊C8), (C2×C3⋊C16)⋊12C2, (C2×C4).6(C3⋊C8), C2.7(C22×C3⋊C8), C22.6(C2×C3⋊C8), (C2×C6).37(C2×C8), (C2×C12).305(C2×C4), (C2×C4).101(C2×Dic3), SmallGroup(192,656)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C12.C8
G = < a,b,c | a2=b24=1, c4=b18, ab=ba, ac=ca, cbc-1=b5 >
Subgroups: 120 in 90 conjugacy classes, 71 normal (29 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C16, C2×C8, C2×C8, C22×C4, C24, C24, C2×C12, C2×C12, C22×C6, C2×C16, M5(2), C22×C8, C3⋊C16, C2×C24, C2×C24, C22×C12, C2×M5(2), C2×C3⋊C16, C12.C8, C22×C24, C2×C12.C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, Dic3, D6, C2×C8, C22×C4, C3⋊C8, C2×Dic3, C22×S3, M5(2), C22×C8, C2×C3⋊C8, C22×Dic3, C2×M5(2), C12.C8, C22×C3⋊C8, C2×C12.C8
►On 96 points
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(49 94)(50 95)(51 96)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)
(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 49 10 70 19 67 4 64 13 61 22 58 7 55 16 52)(2 54 11 51 20 72 5 69 14 66 23 63 8 60 17 57)(3 59 12 56 21 53 6 50 15 71 24 68 9 65 18 62)(25 96 34 93 43 90 28 87 37 84 46 81 31 78 40 75)(26 77 35 74 44 95 29 92 38 89 47 86 32 83 41 80)(27 82 36 79 45 76 30 73 39 94 48 91 33 88 42 85)
G:=sub<Sym(96)| (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(49,94)(50,95)(51,96)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93), (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), (1,49,10,70,19,67,4,64,13,61,22,58,7,55,16,52)(2,54,11,51,20,72,5,69,14,66,23,63,8,60,17,57)(3,59,12,56,21,53,6,50,15,71,24,68,9,65,18,62)(25,96,34,93,43,90,28,87,37,84,46,81,31,78,40,75)(26,77,35,74,44,95,29,92,38,89,47,86,32,83,41,80)(27,82,36,79,45,76,30,73,39,94,48,91,33,88,42,85)>;
G:=Group( (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(49,94)(50,95)(51,96)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93), (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), (1,49,10,70,19,67,4,64,13,61,22,58,7,55,16,52)(2,54,11,51,20,72,5,69,14,66,23,63,8,60,17,57)(3,59,12,56,21,53,6,50,15,71,24,68,9,65,18,62)(25,96,34,93,43,90,28,87,37,84,46,81,31,78,40,75)(26,77,35,74,44,95,29,92,38,89,47,86,32,83,41,80)(27,82,36,79,45,76,30,73,39,94,48,91,33,88,42,85) );
G=PermutationGroup([[(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(49,94),(50,95),(51,96),(52,73),(53,74),(54,75),(55,76),(56,77),(57,78),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93)], [(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)], [(1,49,10,70,19,67,4,64,13,61,22,58,7,55,16,52),(2,54,11,51,20,72,5,69,14,66,23,63,8,60,17,57),(3,59,12,56,21,53,6,50,15,71,24,68,9,65,18,62),(25,96,34,93,43,90,28,87,37,84,46,81,31,78,40,75),(26,77,35,74,44,95,29,92,38,89,47,86,32,83,41,80),(27,82,36,79,45,76,30,73,39,94,48,91,33,88,42,85)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6G | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12H | 16A | ··· | 16P | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | Dic3 | D6 | Dic3 | C3⋊C8 | C3⋊C8 | M5(2) | C12.C8 |
| kernel | C2×C12.C8 | C2×C3⋊C16 | C12.C8 | C22×C24 | C2×C24 | C22×C12 | C2×C12 | C22×C6 | C22×C8 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 12 | 4 | 1 | 3 | 3 | 1 | 6 | 2 | 8 | 16 |
Matrix representation of C2×C12.C8 ►in GL3(𝔽97) generated by
| 96 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 73 | 0 |
| 0 | 59 | 9 |
| 22 | 0 | 0 |
| 0 | 17 | 95 |
| 0 | 31 | 80 |
G:=sub<GL(3,GF(97))| [96,0,0,0,1,0,0,0,1],[1,0,0,0,73,59,0,0,9],[22,0,0,0,17,31,0,95,80] >;
C2×C12.C8 in GAP, Magma, Sage, TeX
C_2\times C_{12}.C_8 % in TeX
G:=Group("C2xC12.C8"); // GroupNames label
G:=SmallGroup(192,656);
// by ID
G=gap.SmallGroup(192,656);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,758,80,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^2=b^24=1,c^4=b^18,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations