direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C56⋊C2, C8⋊8D14, C4.6D28, C56⋊9C22, C14⋊1SD16, C28.29D4, C28.28C23, D28.6C22, C22.12D28, Dic14⋊3C22, (C2×C8)⋊5D7, (C2×C56)⋊7C2, C7⋊1(C2×SD16), C14.9(C2×D4), (C2×D28).4C2, C2.11(C2×D28), (C2×C14).16D4, (C2×C4).79D14, (C2×Dic14)⋊5C2, C4.26(C22×D7), (C2×C28).88C22, SmallGroup(224,97)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C56⋊C2
G = < a,b,c | a2=b56=c2=1, ab=ba, ac=ca, cbc=b27 >
Subgroups: 366 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C2×C8, SD16, C2×D4, C2×Q8, Dic7, C28, D14, C2×C14, C2×SD16, C56, Dic14, Dic14, D28, D28, C2×Dic7, C2×C28, C22×D7, C56⋊C2, C2×C56, C2×Dic14, C2×D28, C2×C56⋊C2
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C2×SD16, D28, C22×D7, C56⋊C2, C2×D28, C2×C56⋊C2
►On 112 points
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(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 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 97)(50 98)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)
(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)
(1 105)(2 76)(3 103)(4 74)(5 101)(6 72)(7 99)(8 70)(9 97)(10 68)(11 95)(12 66)(13 93)(14 64)(15 91)(16 62)(17 89)(18 60)(19 87)(20 58)(21 85)(22 112)(23 83)(24 110)(25 81)(26 108)(27 79)(28 106)(29 77)(30 104)(31 75)(32 102)(33 73)(34 100)(35 71)(36 98)(37 69)(38 96)(39 67)(40 94)(41 65)(42 92)(43 63)(44 90)(45 61)(46 88)(47 59)(48 86)(49 57)(50 84)(51 111)(52 82)(53 109)(54 80)(55 107)(56 78)
G:=sub<Sym(112)| (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(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)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104), (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), (1,105)(2,76)(3,103)(4,74)(5,101)(6,72)(7,99)(8,70)(9,97)(10,68)(11,95)(12,66)(13,93)(14,64)(15,91)(16,62)(17,89)(18,60)(19,87)(20,58)(21,85)(22,112)(23,83)(24,110)(25,81)(26,108)(27,79)(28,106)(29,77)(30,104)(31,75)(32,102)(33,73)(34,100)(35,71)(36,98)(37,69)(38,96)(39,67)(40,94)(41,65)(42,92)(43,63)(44,90)(45,61)(46,88)(47,59)(48,86)(49,57)(50,84)(51,111)(52,82)(53,109)(54,80)(55,107)(56,78)>;
G:=Group( (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(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)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104), (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), (1,105)(2,76)(3,103)(4,74)(5,101)(6,72)(7,99)(8,70)(9,97)(10,68)(11,95)(12,66)(13,93)(14,64)(15,91)(16,62)(17,89)(18,60)(19,87)(20,58)(21,85)(22,112)(23,83)(24,110)(25,81)(26,108)(27,79)(28,106)(29,77)(30,104)(31,75)(32,102)(33,73)(34,100)(35,71)(36,98)(37,69)(38,96)(39,67)(40,94)(41,65)(42,92)(43,63)(44,90)(45,61)(46,88)(47,59)(48,86)(49,57)(50,84)(51,111)(52,82)(53,109)(54,80)(55,107)(56,78) );
G=PermutationGroup([[(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,111),(8,112),(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),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96),(49,97),(50,98),(51,99),(52,100),(53,101),(54,102),(55,103),(56,104)], [(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)], [(1,105),(2,76),(3,103),(4,74),(5,101),(6,72),(7,99),(8,70),(9,97),(10,68),(11,95),(12,66),(13,93),(14,64),(15,91),(16,62),(17,89),(18,60),(19,87),(20,58),(21,85),(22,112),(23,83),(24,110),(25,81),(26,108),(27,79),(28,106),(29,77),(30,104),(31,75),(32,102),(33,73),(34,100),(35,71),(36,98),(37,69),(38,96),(39,67),(40,94),(41,65),(42,92),(43,63),(44,90),(45,61),(46,88),(47,59),(48,86),(49,57),(50,84),(51,111),(52,82),(53,109),(54,80),(55,107),(56,78)]])
C2×C56⋊C2 is a maximal subgroup of
C8⋊5D28 C8.8D28 C42.16D14 C8⋊D28 C8.D28 D28.31D4 D28.32D4 D28⋊14D4 Dic14⋊14D4 Dic14⋊2D4 D4.6D28 D4⋊3D28 D28.D4 Dic14.11D4 Q8⋊2D28 Q8.D28 Dic7⋊SD16 C28⋊SD16 D28.19D4 C42.36D14 Dic14⋊8D4 Dic7⋊8SD16 C8⋊8D28 C8⋊3D28 C56⋊C2⋊C4 C8.24D28 C56⋊30D4 C56⋊2D4 D4.3D28 C56⋊11D4 C56.43D4 C56⋊15D4 C56.37D4 D4.11D28 C2×D7×SD16 D8⋊11D14
C2×C56⋊C2 is a maximal quotient of
C56⋊9Q8 C28.14Q16 C8⋊5D28 C4.5D56 C23.34D28 D28.31D4 C23.38D28 Dic14⋊14D4 C28⋊SD16 D28⋊3Q8 Dic14⋊8D4 Dic14⋊4Q8 C56⋊30D4
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | SD16 | D14 | D14 | D28 | D28 | C56⋊C2 |
| kernel | C2×C56⋊C2 | C56⋊C2 | C2×C56 | C2×Dic14 | C2×D28 | C28 | C2×C14 | C2×C8 | C14 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 6 | 3 | 6 | 6 | 24 |
Matrix representation of C2×C56⋊C2 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 13 | 9 | 0 | 0 |
| 104 | 46 | 0 | 0 |
| 0 | 0 | 89 | 8 |
| 0 | 0 | 97 | 76 |
| 112 | 0 | 0 | 0 |
| 34 | 1 | 0 | 0 |
| 0 | 0 | 15 | 24 |
| 0 | 0 | 66 | 98 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[13,104,0,0,9,46,0,0,0,0,89,97,0,0,8,76],[112,34,0,0,0,1,0,0,0,0,15,66,0,0,24,98] >;
C2×C56⋊C2 in GAP, Magma, Sage, TeX
C_2\times C_{56}\rtimes C_2 % in TeX
G:=Group("C2xC56:C2"); // GroupNames label
G:=SmallGroup(224,97);
// by ID
G=gap.SmallGroup(224,97);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,218,50,579,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^2=b^56=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^27>;
// generators/relations