metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C56)⋊1C4, (C2×C8)⋊1Dic7, (C2×C4).14D28, C4.Dic7⋊6C4, C28.11(C4⋊C4), (C2×C28).11Q8, C23.9(C4×D7), (C2×C28).109D4, (C2×C4).5Dic14, C4.35(D14⋊C4), C23.D7.1C4, (C2×C14).20C42, C4.13(C4⋊Dic7), (C22×C4).62D14, (C2×M4(2)).7D7, C28.11(C22⋊C4), C7⋊2(M4(2)⋊4C4), C4.11(Dic7⋊C4), C4.28(C23.D7), C22.10(C4×Dic7), C22.19(D14⋊C4), (C14×M4(2)).11C2, C22.6(Dic7⋊C4), (C22×C28).125C22, C23.21D14.9C2, C14.15(C2.C42), C2.15(C14.C42), (C2×C4).21(C4×D7), (C2×C14).8(C4⋊C4), (C2×C28).304(C2×C4), (C2×C4).76(C2×Dic7), (C2×C4).180(C7⋊D4), (C22×C14).32(C2×C4), (C2×C4.Dic7).11C2, (C2×C14).10(C22⋊C4), SmallGroup(448,113)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C56)⋊C4
G = < a,b,c | a2=b56=c4=1, ab=ba, cac-1=ab28, cbc-1=ab13 >
Subgroups: 324 in 90 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic7, C28, C2×C14, C2×C14, C42⋊C2, C2×M4(2), C2×M4(2), C7⋊C8, C56, C2×Dic7, C2×C28, C22×C14, M4(2)⋊4C4, C2×C7⋊C8, C4.Dic7, C4.Dic7, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×M4(2), C22×C28, C2×C4.Dic7, C23.21D14, C14×M4(2), (C2×C56)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, M4(2)⋊4C4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C14.C42, (C2×C56)⋊C4
►On 112 points
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 105)(48 106)(49 107)(50 108)(51 109)(52 110)(53 111)(54 112)(55 57)(56 58)
(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 43 29 15)(2 86)(3 13 31 41)(4 112)(5 39 33 11)(6 82)(7 9 35 37)(8 108)(10 78)(12 104)(14 74)(16 100)(17 27 45 55)(18 70)(19 53 47 25)(20 96)(21 23 49 51)(22 66)(24 92)(26 62)(28 88)(30 58)(32 84)(34 110)(36 80)(38 106)(40 76)(42 102)(44 72)(46 98)(48 68)(50 94)(52 64)(54 90)(56 60)(57 103 85 75)(59 73 87 101)(61 99 89 71)(63 69 91 97)(65 95 93 67)(77 83 105 111)(79 109 107 81)
G:=sub<Sym(112)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,105)(48,106)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,57)(56,58), (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), (1,43,29,15)(2,86)(3,13,31,41)(4,112)(5,39,33,11)(6,82)(7,9,35,37)(8,108)(10,78)(12,104)(14,74)(16,100)(17,27,45,55)(18,70)(19,53,47,25)(20,96)(21,23,49,51)(22,66)(24,92)(26,62)(28,88)(30,58)(32,84)(34,110)(36,80)(38,106)(40,76)(42,102)(44,72)(46,98)(48,68)(50,94)(52,64)(54,90)(56,60)(57,103,85,75)(59,73,87,101)(61,99,89,71)(63,69,91,97)(65,95,93,67)(77,83,105,111)(79,109,107,81)>;
G:=Group( (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,105)(48,106)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,57)(56,58), (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), (1,43,29,15)(2,86)(3,13,31,41)(4,112)(5,39,33,11)(6,82)(7,9,35,37)(8,108)(10,78)(12,104)(14,74)(16,100)(17,27,45,55)(18,70)(19,53,47,25)(20,96)(21,23,49,51)(22,66)(24,92)(26,62)(28,88)(30,58)(32,84)(34,110)(36,80)(38,106)(40,76)(42,102)(44,72)(46,98)(48,68)(50,94)(52,64)(54,90)(56,60)(57,103,85,75)(59,73,87,101)(61,99,89,71)(63,69,91,97)(65,95,93,67)(77,83,105,111)(79,109,107,81) );
G=PermutationGroup([[(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,75),(18,76),(19,77),(20,78),(21,79),(22,80),(23,81),(24,82),(25,83),(26,84),(27,85),(28,86),(29,87),(30,88),(31,89),(32,90),(33,91),(34,92),(35,93),(36,94),(37,95),(38,96),(39,97),(40,98),(41,99),(42,100),(43,101),(44,102),(45,103),(46,104),(47,105),(48,106),(49,107),(50,108),(51,109),(52,110),(53,111),(54,112),(55,57),(56,58)], [(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)], [(1,43,29,15),(2,86),(3,13,31,41),(4,112),(5,39,33,11),(6,82),(7,9,35,37),(8,108),(10,78),(12,104),(14,74),(16,100),(17,27,45,55),(18,70),(19,53,47,25),(20,96),(21,23,49,51),(22,66),(24,92),(26,62),(28,88),(30,58),(32,84),(34,110),(36,80),(38,106),(40,76),(42,102),(44,72),(46,98),(48,68),(50,94),(52,64),(54,90),(56,60),(57,103,85,75),(59,73,87,101),(61,99,89,71),(63,69,91,97),(65,95,93,67),(77,83,105,111),(79,109,107,81)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D7 | Dic7 | D14 | Dic14 | C4×D7 | D28 | C7⋊D4 | C4×D7 | M4(2)⋊4C4 | (C2×C56)⋊C4 |
| kernel | (C2×C56)⋊C4 | C2×C4.Dic7 | C23.21D14 | C14×M4(2) | C4.Dic7 | C23.D7 | C2×C56 | C2×C28 | C2×C28 | C2×M4(2) | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 3 | 1 | 3 | 6 | 3 | 6 | 6 | 6 | 12 | 6 | 2 | 12 |
Matrix representation of (C2×C56)⋊C4 ►in GL4(𝔽113) generated by
| 91 | 12 | 0 | 0 |
| 101 | 22 | 0 | 0 |
| 0 | 0 | 22 | 12 |
| 0 | 0 | 101 | 91 |
| 79 | 100 | 51 | 23 |
| 13 | 69 | 51 | 51 |
| 111 | 73 | 44 | 13 |
| 85 | 111 | 100 | 34 |
| 98 | 0 | 0 | 0 |
| 58 | 15 | 0 | 0 |
| 88 | 72 | 94 | 100 |
| 72 | 100 | 19 | 19 |
G:=sub<GL(4,GF(113))| [91,101,0,0,12,22,0,0,0,0,22,101,0,0,12,91],[79,13,111,85,100,69,73,111,51,51,44,100,23,51,13,34],[98,58,88,72,0,15,72,100,0,0,94,19,0,0,100,19] >;
(C2×C56)⋊C4 in GAP, Magma, Sage, TeX
(C_2\times C_{56})\rtimes C_4 % in TeX
G:=Group("(C2xC56):C4"); // GroupNames label
G:=SmallGroup(448,113);
// by ID
G=gap.SmallGroup(448,113);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,387,136,1684,18822]);
// Polycyclic
G:=Group<a,b,c|a^2=b^56=c^4=1,a*b=b*a,c*a*c^-1=a*b^28,c*b*c^-1=a*b^13>;
// generators/relations