metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.8C42, C42⋊6Dic7, C14.7C4≀C2, (C4×C28)⋊12C4, C4⋊Dic7⋊5C4, C4.Dic7⋊1C4, C28.27(C4⋊C4), (C2×C28).58Q8, C4.8(C4×Dic7), (C2×C42).7D7, C7⋊1(C42⋊6C4), (C2×C28).478D4, (C2×C4).162D28, C4.44(D14⋊C4), C4.20(C4⋊Dic7), (C2×C4).42Dic14, C2.3(Dic14⋊C4), C28.59(C22⋊C4), C4.22(Dic7⋊C4), (C22×C14).176D4, (C22×C4).412D14, C23.72(C7⋊D4), C22.36(D14⋊C4), C22.8(C23.D7), C14.1(C2.C42), C2.3(C14.C42), C22.12(Dic7⋊C4), (C22×C28).532C22, C23.21D14.1C2, (C2×C4×C28).15C2, (C2×C4).98(C4×D7), (C2×C14).33(C4⋊C4), (C2×C28).217(C2×C4), (C2×C4).71(C2×Dic7), (C2×C4.Dic7).1C2, (C2×C4).231(C7⋊D4), (C2×C14).49(C22⋊C4), SmallGroup(448,80)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.8C42
G = < a,b,c | a28=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=a7b >
Subgroups: 356 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2×C42, C42⋊C2, C2×M4(2), C7⋊C8, C2×Dic7, C2×C28, C2×C28, C22×C14, C42⋊6C4, C2×C7⋊C8, C4.Dic7, C4.Dic7, C4×Dic7, C4⋊Dic7, C23.D7, C4×C28, C4×C28, C22×C28, C22×C28, C2×C4.Dic7, C23.21D14, C2×C4×C28, C28.8C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, C4≀C2, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C42⋊6C4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, Dic14⋊C4, C14.C42, C28.8C42
►On 112 points
(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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 103 34 68)(2 102 35 67)(3 101 36 66)(4 100 37 65)(5 99 38 64)(6 98 39 63)(7 97 40 62)(8 96 41 61)(9 95 42 60)(10 94 43 59)(11 93 44 58)(12 92 45 57)(13 91 46 84)(14 90 47 83)(15 89 48 82)(16 88 49 81)(17 87 50 80)(18 86 51 79)(19 85 52 78)(20 112 53 77)(21 111 54 76)(22 110 55 75)(23 109 56 74)(24 108 29 73)(25 107 30 72)(26 106 31 71)(27 105 32 70)(28 104 33 69)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(57 85 71 99)(58 86 72 100)(59 87 73 101)(60 88 74 102)(61 89 75 103)(62 90 76 104)(63 91 77 105)(64 92 78 106)(65 93 79 107)(66 94 80 108)(67 95 81 109)(68 96 82 110)(69 97 83 111)(70 98 84 112)
G:=sub<Sym(112)| (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,103,34,68)(2,102,35,67)(3,101,36,66)(4,100,37,65)(5,99,38,64)(6,98,39,63)(7,97,40,62)(8,96,41,61)(9,95,42,60)(10,94,43,59)(11,93,44,58)(12,92,45,57)(13,91,46,84)(14,90,47,83)(15,89,48,82)(16,88,49,81)(17,87,50,80)(18,86,51,79)(19,85,52,78)(20,112,53,77)(21,111,54,76)(22,110,55,75)(23,109,56,74)(24,108,29,73)(25,107,30,72)(26,106,31,71)(27,105,32,70)(28,104,33,69), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(57,85,71,99)(58,86,72,100)(59,87,73,101)(60,88,74,102)(61,89,75,103)(62,90,76,104)(63,91,77,105)(64,92,78,106)(65,93,79,107)(66,94,80,108)(67,95,81,109)(68,96,82,110)(69,97,83,111)(70,98,84,112)>;
G:=Group( (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,103,34,68)(2,102,35,67)(3,101,36,66)(4,100,37,65)(5,99,38,64)(6,98,39,63)(7,97,40,62)(8,96,41,61)(9,95,42,60)(10,94,43,59)(11,93,44,58)(12,92,45,57)(13,91,46,84)(14,90,47,83)(15,89,48,82)(16,88,49,81)(17,87,50,80)(18,86,51,79)(19,85,52,78)(20,112,53,77)(21,111,54,76)(22,110,55,75)(23,109,56,74)(24,108,29,73)(25,107,30,72)(26,106,31,71)(27,105,32,70)(28,104,33,69), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(57,85,71,99)(58,86,72,100)(59,87,73,101)(60,88,74,102)(61,89,75,103)(62,90,76,104)(63,91,77,105)(64,92,78,106)(65,93,79,107)(66,94,80,108)(67,95,81,109)(68,96,82,110)(69,97,83,111)(70,98,84,112) );
G=PermutationGroup([[(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,103,34,68),(2,102,35,67),(3,101,36,66),(4,100,37,65),(5,99,38,64),(6,98,39,63),(7,97,40,62),(8,96,41,61),(9,95,42,60),(10,94,43,59),(11,93,44,58),(12,92,45,57),(13,91,46,84),(14,90,47,83),(15,89,48,82),(16,88,49,81),(17,87,50,80),(18,86,51,79),(19,85,52,78),(20,112,53,77),(21,111,54,76),(22,110,55,75),(23,109,56,74),(24,108,29,73),(25,107,30,72),(26,106,31,71),(27,105,32,70),(28,104,33,69)], [(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,40),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47),(57,85,71,99),(58,86,72,100),(59,87,73,101),(60,88,74,102),(61,89,75,103),(62,90,76,104),(63,91,77,105),(64,92,78,106),(65,93,79,107),(66,94,80,108),(67,95,81,109),(68,96,82,110),(69,97,83,111),(70,98,84,112)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14U | 28A | ··· | 28BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | D7 | Dic7 | D14 | C4≀C2 | Dic14 | C4×D7 | D28 | C7⋊D4 | C7⋊D4 | Dic14⋊C4 |
| kernel | C28.8C42 | C2×C4.Dic7 | C23.21D14 | C2×C4×C28 | C4.Dic7 | C4⋊Dic7 | C4×C28 | C2×C28 | C2×C28 | C22×C14 | C2×C42 | C42 | C22×C4 | C14 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 3 | 6 | 3 | 8 | 6 | 12 | 6 | 6 | 6 | 48 |
Matrix representation of C28.8C42 ►in GL4(𝔽113) generated by
| 98 | 0 | 0 | 0 |
| 66 | 15 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 83 |
| 99 | 45 | 0 | 0 |
| 81 | 14 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 |
| 1 | 0 | 0 | 0 |
| 10 | 98 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(113))| [98,66,0,0,0,15,0,0,0,0,64,0,0,0,0,83],[99,81,0,0,45,14,0,0,0,0,0,112,0,0,1,0],[1,10,0,0,0,98,0,0,0,0,112,0,0,0,0,1] >;
C28.8C42 in GAP, Magma, Sage, TeX
C_{28}._8C_4^2 % in TeX
G:=Group("C28.8C4^2"); // GroupNames label
G:=SmallGroup(448,80);
// by ID
G=gap.SmallGroup(448,80);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,1123,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=c^4=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;
// generators/relations