metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.2D56, C23.31D28, C14.9C4≀C2, (C2×D28)⋊2C4, C4⋊Dic7⋊4C4, C22⋊C8⋊2D7, (C2×C14).1D8, C28⋊7D4.1C2, (C2×C28).440D4, (C2×C14).2SD16, C14.6(C23⋊C4), (C22×C14).40D4, (C22×C4).55D14, C7⋊2(C22.SD16), C2.3(C2.D56), C2.7(D28⋊4C4), C22.2(C56⋊C2), C14.11(D4⋊C4), C14.C42⋊26C2, C22.59(D14⋊C4), (C22×C28).41C22, C2.7(C23.1D14), (C7×C22⋊C8)⋊2C2, (C2×C4).13(C4×D7), (C2×C28).25(C2×C4), (C2×C4).211(C7⋊D4), (C2×C14).42(C22⋊C4), SmallGroup(448,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.2D56
G = < a,b,c,d | a2=b2=c56=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >
Subgroups: 636 in 90 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, Dic7, C28, D14, C2×C14, C2×C14, C2.C42, C22⋊C8, C4⋊D4, C56, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C22.SD16, C4⋊Dic7, D14⋊C4, C2×C56, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, C14.C42, C7×C22⋊C8, C28⋊7D4, C22.2D56
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, D14, C23⋊C4, D4⋊C4, C4≀C2, C4×D7, D28, C7⋊D4, C22.SD16, C56⋊C2, D56, D14⋊C4, C23.1D14, C2.D56, D28⋊4C4, C22.2D56
►On 112 points
Generators in S112
(1 70)(3 72)(5 74)(7 76)(9 78)(11 80)(13 82)(15 84)(17 86)(19 88)(21 90)(23 92)(25 94)(27 96)(29 98)(31 100)(33 102)(35 104)(37 106)(39 108)(41 110)(43 112)(45 58)(47 60)(49 62)(51 64)(53 66)(55 68)
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 90)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 110)(42 111)(43 112)(44 57)(45 58)(46 59)(47 60)(48 61)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)
(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)
(1 104 70 35)(2 103)(3 33 72 102)(4 32)(5 100 74 31)(6 99)(7 29 76 98)(8 28)(9 96 78 27)(10 95)(11 25 80 94)(12 24)(13 92 82 23)(14 91)(15 21 84 90)(16 20)(17 88 86 19)(18 87)(22 83)(26 79)(30 75)(34 71)(36 56)(37 68 106 55)(38 67)(39 53 108 66)(40 52)(41 64 110 51)(42 63)(43 49 112 62)(44 48)(45 60 58 47)(46 59)(50 111)(54 107)(57 61)(65 109)(69 105)(73 101)(77 97)(81 93)(85 89)
G:=sub<Sym(112)| (1,70)(3,72)(5,74)(7,76)(9,78)(11,80)(13,82)(15,84)(17,86)(19,88)(21,90)(23,92)(25,94)(27,96)(29,98)(31,100)(33,102)(35,104)(37,106)(39,108)(41,110)(43,112)(45,58)(47,60)(49,62)(51,64)(53,66)(55,68), (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,112)(44,57)(45,58)(46,59)(47,60)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69), (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), (1,104,70,35)(2,103)(3,33,72,102)(4,32)(5,100,74,31)(6,99)(7,29,76,98)(8,28)(9,96,78,27)(10,95)(11,25,80,94)(12,24)(13,92,82,23)(14,91)(15,21,84,90)(16,20)(17,88,86,19)(18,87)(22,83)(26,79)(30,75)(34,71)(36,56)(37,68,106,55)(38,67)(39,53,108,66)(40,52)(41,64,110,51)(42,63)(43,49,112,62)(44,48)(45,60,58,47)(46,59)(50,111)(54,107)(57,61)(65,109)(69,105)(73,101)(77,97)(81,93)(85,89)>;
G:=Group( (1,70)(3,72)(5,74)(7,76)(9,78)(11,80)(13,82)(15,84)(17,86)(19,88)(21,90)(23,92)(25,94)(27,96)(29,98)(31,100)(33,102)(35,104)(37,106)(39,108)(41,110)(43,112)(45,58)(47,60)(49,62)(51,64)(53,66)(55,68), (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,112)(44,57)(45,58)(46,59)(47,60)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69), (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), (1,104,70,35)(2,103)(3,33,72,102)(4,32)(5,100,74,31)(6,99)(7,29,76,98)(8,28)(9,96,78,27)(10,95)(11,25,80,94)(12,24)(13,92,82,23)(14,91)(15,21,84,90)(16,20)(17,88,86,19)(18,87)(22,83)(26,79)(30,75)(34,71)(36,56)(37,68,106,55)(38,67)(39,53,108,66)(40,52)(41,64,110,51)(42,63)(43,49,112,62)(44,48)(45,60,58,47)(46,59)(50,111)(54,107)(57,61)(65,109)(69,105)(73,101)(77,97)(81,93)(85,89) );
G=PermutationGroup([[(1,70),(3,72),(5,74),(7,76),(9,78),(11,80),(13,82),(15,84),(17,86),(19,88),(21,90),(23,92),(25,94),(27,96),(29,98),(31,100),(33,102),(35,104),(37,106),(39,108),(41,110),(43,112),(45,58),(47,60),(49,62),(51,64),(53,66),(55,68)], [(1,70),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,80),(12,81),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,88),(20,89),(21,90),(22,91),(23,92),(24,93),(25,94),(26,95),(27,96),(28,97),(29,98),(30,99),(31,100),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(41,110),(42,111),(43,112),(44,57),(45,58),(46,59),(47,60),(48,61),(49,62),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69)], [(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)], [(1,104,70,35),(2,103),(3,33,72,102),(4,32),(5,100,74,31),(6,99),(7,29,76,98),(8,28),(9,96,78,27),(10,95),(11,25,80,94),(12,24),(13,92,82,23),(14,91),(15,21,84,90),(16,20),(17,88,86,19),(18,87),(22,83),(26,79),(30,75),(34,71),(36,56),(37,68,106,55),(38,67),(39,53,108,66),(40,52),(41,64,110,51),(42,63),(43,49,112,62),(44,48),(45,60,58,47),(46,59),(50,111),(54,107),(57,61),(65,109),(69,105),(73,101),(77,97),(81,93),(85,89)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 56 | 2 | 2 | 4 | 28 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D7 | D8 | SD16 | D14 | C4≀C2 | C4×D7 | C7⋊D4 | D28 | C56⋊C2 | D56 | C23⋊C4 | C23.1D14 | D28⋊4C4 |
| kernel | C22.2D56 | C14.C42 | C7×C22⋊C8 | C28⋊7D4 | C4⋊Dic7 | C2×D28 | C2×C28 | C22×C14 | C22⋊C8 | C2×C14 | C2×C14 | C22×C4 | C14 | C2×C4 | C2×C4 | C23 | C22 | C22 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 12 | 12 | 1 | 6 | 6 |
Matrix representation of C22.2D56 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 2 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 12 | 29 | 0 | 0 |
| 84 | 43 | 0 | 0 |
| 0 | 0 | 111 | 111 |
| 0 | 0 | 66 | 2 |
| 13 | 9 | 0 | 0 |
| 19 | 100 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 97 | 112 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,112,2,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[12,84,0,0,29,43,0,0,0,0,111,66,0,0,111,2],[13,19,0,0,9,100,0,0,0,0,15,97,0,0,0,112] >;
C22.2D56 in GAP, Magma, Sage, TeX
C_2^2._2D_{56} % in TeX
G:=Group("C2^2.2D56"); // GroupNames label
G:=SmallGroup(448,27);
// by ID
G=gap.SmallGroup(448,27);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,85,92,422,387,100,1123,570,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^56=1,d^2=a,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations