direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C28.D4, C24.2Dic7, (D4×C14).10C4, (C2×C28).190D4, C28.204(C2×D4), (C23×C14).4C4, (C2×D4).7Dic7, (C22×D4).3D7, (C2×D4).197D14, C14⋊2(C4.D4), C4.9(C23.D7), C28.32(C22⋊C4), (C2×C28).471C23, (C22×C4).148D14, C4.Dic7⋊22C22, C23.32(C2×Dic7), (D4×C14).239C22, C22.5(C22×Dic7), (C22×C28).196C22, C22.34(C23.D7), (D4×C2×C14).3C2, C7⋊3(C2×C4.D4), C4.90(C2×C7⋊D4), (C2×C28).117(C2×C4), C2.9(C2×C23.D7), C14.73(C2×C22⋊C4), (C2×C4.Dic7)⋊18C2, (C2×C4).24(C2×Dic7), (C2×C4).197(C7⋊D4), (C22×C14).15(C2×C4), (C2×C4).129(C22×D7), (C2×C14).194(C22×C4), (C2×C14).112(C22⋊C4), SmallGroup(448,750)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C28.D4
G = < a,b,c,d | a2=b28=1, c4=b14, d2=b21, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b13, dcd-1=b7c3 >
Subgroups: 532 in 186 conjugacy classes, 71 normal (21 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, C28, C2×C14, C2×C14, C4.D4, C2×M4(2), C22×D4, C7⋊C8, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C2×C4.D4, C2×C7⋊C8, C4.Dic7, C4.Dic7, C22×C28, D4×C14, D4×C14, C23×C14, C28.D4, C2×C4.Dic7, D4×C2×C14, C2×C28.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C4.D4, C2×C22⋊C4, C2×Dic7, C7⋊D4, C22×D7, C2×C4.D4, C23.D7, C22×Dic7, C2×C7⋊D4, C28.D4, C2×C23.D7, C2×C28.D4
►On 112 points
Generators in S112
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)
(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 92 70 50 15 106 84 36)(2 91 71 49 16 105 57 35)(3 90 72 48 17 104 58 34)(4 89 73 47 18 103 59 33)(5 88 74 46 19 102 60 32)(6 87 75 45 20 101 61 31)(7 86 76 44 21 100 62 30)(8 85 77 43 22 99 63 29)(9 112 78 42 23 98 64 56)(10 111 79 41 24 97 65 55)(11 110 80 40 25 96 66 54)(12 109 81 39 26 95 67 53)(13 108 82 38 27 94 68 52)(14 107 83 37 28 93 69 51)
(1 99 22 92 15 85 8 106)(2 112 23 105 16 98 9 91)(3 97 24 90 17 111 10 104)(4 110 25 103 18 96 11 89)(5 95 26 88 19 109 12 102)(6 108 27 101 20 94 13 87)(7 93 28 86 21 107 14 100)(29 77 50 70 43 63 36 84)(30 62 51 83 44 76 37 69)(31 75 52 68 45 61 38 82)(32 60 53 81 46 74 39 67)(33 73 54 66 47 59 40 80)(34 58 55 79 48 72 41 65)(35 71 56 64 49 57 42 78)
G:=sub<Sym(112)| (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,106)(30,107)(31,108)(32,109)(33,110)(34,111)(35,112)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105), (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), (1,92,70,50,15,106,84,36)(2,91,71,49,16,105,57,35)(3,90,72,48,17,104,58,34)(4,89,73,47,18,103,59,33)(5,88,74,46,19,102,60,32)(6,87,75,45,20,101,61,31)(7,86,76,44,21,100,62,30)(8,85,77,43,22,99,63,29)(9,112,78,42,23,98,64,56)(10,111,79,41,24,97,65,55)(11,110,80,40,25,96,66,54)(12,109,81,39,26,95,67,53)(13,108,82,38,27,94,68,52)(14,107,83,37,28,93,69,51), (1,99,22,92,15,85,8,106)(2,112,23,105,16,98,9,91)(3,97,24,90,17,111,10,104)(4,110,25,103,18,96,11,89)(5,95,26,88,19,109,12,102)(6,108,27,101,20,94,13,87)(7,93,28,86,21,107,14,100)(29,77,50,70,43,63,36,84)(30,62,51,83,44,76,37,69)(31,75,52,68,45,61,38,82)(32,60,53,81,46,74,39,67)(33,73,54,66,47,59,40,80)(34,58,55,79,48,72,41,65)(35,71,56,64,49,57,42,78)>;
G:=Group( (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,106)(30,107)(31,108)(32,109)(33,110)(34,111)(35,112)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105), (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), (1,92,70,50,15,106,84,36)(2,91,71,49,16,105,57,35)(3,90,72,48,17,104,58,34)(4,89,73,47,18,103,59,33)(5,88,74,46,19,102,60,32)(6,87,75,45,20,101,61,31)(7,86,76,44,21,100,62,30)(8,85,77,43,22,99,63,29)(9,112,78,42,23,98,64,56)(10,111,79,41,24,97,65,55)(11,110,80,40,25,96,66,54)(12,109,81,39,26,95,67,53)(13,108,82,38,27,94,68,52)(14,107,83,37,28,93,69,51), (1,99,22,92,15,85,8,106)(2,112,23,105,16,98,9,91)(3,97,24,90,17,111,10,104)(4,110,25,103,18,96,11,89)(5,95,26,88,19,109,12,102)(6,108,27,101,20,94,13,87)(7,93,28,86,21,107,14,100)(29,77,50,70,43,63,36,84)(30,62,51,83,44,76,37,69)(31,75,52,68,45,61,38,82)(32,60,53,81,46,74,39,67)(33,73,54,66,47,59,40,80)(34,58,55,79,48,72,41,65)(35,71,56,64,49,57,42,78) );
G=PermutationGroup([[(1,77),(2,78),(3,79),(4,80),(5,81),(6,82),(7,83),(8,84),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,106),(30,107),(31,108),(32,109),(33,110),(34,111),(35,112),(36,85),(37,86),(38,87),(39,88),(40,89),(41,90),(42,91),(43,92),(44,93),(45,94),(46,95),(47,96),(48,97),(49,98),(50,99),(51,100),(52,101),(53,102),(54,103),(55,104),(56,105)], [(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)], [(1,92,70,50,15,106,84,36),(2,91,71,49,16,105,57,35),(3,90,72,48,17,104,58,34),(4,89,73,47,18,103,59,33),(5,88,74,46,19,102,60,32),(6,87,75,45,20,101,61,31),(7,86,76,44,21,100,62,30),(8,85,77,43,22,99,63,29),(9,112,78,42,23,98,64,56),(10,111,79,41,24,97,65,55),(11,110,80,40,25,96,66,54),(12,109,81,39,26,95,67,53),(13,108,82,38,27,94,68,52),(14,107,83,37,28,93,69,51)], [(1,99,22,92,15,85,8,106),(2,112,23,105,16,98,9,91),(3,97,24,90,17,111,10,104),(4,110,25,103,18,96,11,89),(5,95,26,88,19,109,12,102),(6,108,27,101,20,94,13,87),(7,93,28,86,21,107,14,100),(29,77,50,70,43,63,36,84),(30,62,51,83,44,76,37,69),(31,75,52,68,45,61,38,82),(32,60,53,81,46,74,39,67),(33,73,54,66,47,59,40,80),(34,58,55,79,48,72,41,65),(35,71,56,64,49,57,42,78)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D7 | D14 | Dic7 | D14 | Dic7 | C7⋊D4 | C4.D4 | C28.D4 |
| kernel | C2×C28.D4 | C28.D4 | C2×C4.Dic7 | D4×C2×C14 | D4×C14 | C23×C14 | C2×C28 | C22×D4 | C22×C4 | C2×D4 | C2×D4 | C24 | C2×C4 | C14 | C2 |
| # reps | 1 | 4 | 2 | 1 | 4 | 4 | 4 | 3 | 3 | 6 | 6 | 6 | 24 | 2 | 12 |
Matrix representation of C2×C28.D4 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 85 | 0 | 0 | 0 | 0 | 0 |
| 100 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 83 | 14 | 0 | 0 |
| 0 | 0 | 49 | 30 | 0 | 0 |
| 0 | 0 | 24 | 83 | 0 | 49 |
| 0 | 0 | 40 | 83 | 64 | 0 |
| 3 | 10 | 0 | 0 | 0 | 0 |
| 112 | 110 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 1 | 112 |
| 0 | 0 | 0 | 112 | 112 | 0 |
| 0 | 0 | 28 | 0 | 112 | 0 |
| 3 | 10 | 0 | 0 | 0 | 0 |
| 112 | 110 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 112 | 1 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 28 | 1 | 0 | 112 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[85,100,0,0,0,0,0,4,0,0,0,0,0,0,83,49,24,40,0,0,14,30,83,83,0,0,0,0,0,64,0,0,0,0,49,0],[3,112,0,0,0,0,10,110,0,0,0,0,0,0,1,0,0,28,0,0,0,0,112,0,0,0,8,1,112,112,0,0,0,112,0,0],[3,112,0,0,0,0,10,110,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,1,0,0,0,112,0,0,0,0,8,1,112,112] >;
C2×C28.D4 in GAP, Magma, Sage, TeX
C_2\times C_{28}.D_4 % in TeX
G:=Group("C2xC28.D4"); // GroupNames label
G:=SmallGroup(448,750);
// by ID
G=gap.SmallGroup(448,750);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,297,136,1684,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=1,c^4=b^14,d^2=b^21,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^13,d*c*d^-1=b^7*c^3>;
// generators/relations