metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊2D28, C28⋊1D4, D14⋊2D4, C4⋊C4⋊3D7, D14⋊C4⋊8C2, (C2×D28)⋊4C2, C7⋊2(C4⋊D4), C2.9(C2×D28), C2.13(D4×D7), C14.7(C2×D4), (C2×C4).12D14, (C2×C28).5C22, C14.34(C4○D4), (C2×C14).36C23, C2.6(Q8⋊2D7), (C22×D7).7C22, C22.50(C22×D7), (C2×Dic7).31C22, (C2×C4×D7)⋊1C2, (C7×C4⋊C4)⋊6C2, SmallGroup(224,90)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊D28
G = < a,b,c | a28=b4=c2=1, bab-1=a15, cac=a-1, cbc=b-1 >
Subgroups: 502 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, D7, C14, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, C28, C28, D14, D14, C2×C14, C4⋊D4, C4×D7, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, D14⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, C2×D28, C4⋊D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, D28, C22×D7, C2×D28, D4×D7, Q8⋊2D7, C4⋊D28
►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 62 94 51)(2 77 95 38)(3 64 96 53)(4 79 97 40)(5 66 98 55)(6 81 99 42)(7 68 100 29)(8 83 101 44)(9 70 102 31)(10 57 103 46)(11 72 104 33)(12 59 105 48)(13 74 106 35)(14 61 107 50)(15 76 108 37)(16 63 109 52)(17 78 110 39)(18 65 111 54)(19 80 112 41)(20 67 85 56)(21 82 86 43)(22 69 87 30)(23 84 88 45)(24 71 89 32)(25 58 90 47)(26 73 91 34)(27 60 92 49)(28 75 93 36)
(1 94)(2 93)(3 92)(4 91)(5 90)(6 89)(7 88)(8 87)(9 86)(10 85)(11 112)(12 111)(13 110)(14 109)(15 108)(16 107)(17 106)(18 105)(19 104)(20 103)(21 102)(22 101)(23 100)(24 99)(25 98)(26 97)(27 96)(28 95)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(46 56)(47 55)(48 54)(49 53)(50 52)(57 67)(58 66)(59 65)(60 64)(61 63)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)
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), (1,62,94,51)(2,77,95,38)(3,64,96,53)(4,79,97,40)(5,66,98,55)(6,81,99,42)(7,68,100,29)(8,83,101,44)(9,70,102,31)(10,57,103,46)(11,72,104,33)(12,59,105,48)(13,74,106,35)(14,61,107,50)(15,76,108,37)(16,63,109,52)(17,78,110,39)(18,65,111,54)(19,80,112,41)(20,67,85,56)(21,82,86,43)(22,69,87,30)(23,84,88,45)(24,71,89,32)(25,58,90,47)(26,73,91,34)(27,60,92,49)(28,75,93,36), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,112)(12,111)(13,110)(14,109)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,98)(26,97)(27,96)(28,95)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,56)(47,55)(48,54)(49,53)(50,52)(57,67)(58,66)(59,65)(60,64)(61,63)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)>;
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), (1,62,94,51)(2,77,95,38)(3,64,96,53)(4,79,97,40)(5,66,98,55)(6,81,99,42)(7,68,100,29)(8,83,101,44)(9,70,102,31)(10,57,103,46)(11,72,104,33)(12,59,105,48)(13,74,106,35)(14,61,107,50)(15,76,108,37)(16,63,109,52)(17,78,110,39)(18,65,111,54)(19,80,112,41)(20,67,85,56)(21,82,86,43)(22,69,87,30)(23,84,88,45)(24,71,89,32)(25,58,90,47)(26,73,91,34)(27,60,92,49)(28,75,93,36), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,112)(12,111)(13,110)(14,109)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,98)(26,97)(27,96)(28,95)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,56)(47,55)(48,54)(49,53)(50,52)(57,67)(58,66)(59,65)(60,64)(61,63)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77) );
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)], [(1,62,94,51),(2,77,95,38),(3,64,96,53),(4,79,97,40),(5,66,98,55),(6,81,99,42),(7,68,100,29),(8,83,101,44),(9,70,102,31),(10,57,103,46),(11,72,104,33),(12,59,105,48),(13,74,106,35),(14,61,107,50),(15,76,108,37),(16,63,109,52),(17,78,110,39),(18,65,111,54),(19,80,112,41),(20,67,85,56),(21,82,86,43),(22,69,87,30),(23,84,88,45),(24,71,89,32),(25,58,90,47),(26,73,91,34),(27,60,92,49),(28,75,93,36)], [(1,94),(2,93),(3,92),(4,91),(5,90),(6,89),(7,88),(8,87),(9,86),(10,85),(11,112),(12,111),(13,110),(14,109),(15,108),(16,107),(17,106),(18,105),(19,104),(20,103),(21,102),(22,101),(23,100),(24,99),(25,98),(26,97),(27,96),(28,95),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(46,56),(47,55),(48,54),(49,53),(50,52),(57,67),(58,66),(59,65),(60,64),(61,63),(68,84),(69,83),(70,82),(71,81),(72,80),(73,79),(74,78),(75,77)]])
C4⋊D28 is a maximal subgroup of
D4⋊D28 D14⋊D8 D4⋊3D28 C7⋊C8⋊D4 Q8⋊2D28 D14⋊2SD16 D28⋊4D4 C7⋊(C8⋊D4) D14.4SD16 C8⋊8D28 C56⋊7D4 C4.Q8⋊D7 D14.5D8 C8⋊7D28 C2.D8⋊D7 C8⋊3D28 C14.2- 1+4 C14.2+ 1+4 C14.112+ 1+4 C42⋊8D14 C42⋊9D14 C42.95D14 C42.97D14 C42.228D14 D4×D28 D4⋊5D28 C42.116D14 Q8⋊5D28 Q8⋊6D28 C42.131D14 C42.133D14 Dic14⋊20D4 D7×C4⋊D4 C14.382+ 1+4 D28⋊19D4 C4⋊C4⋊26D14 C14.172- 1+4 D28⋊21D4 Dic14⋊22D4 C14.562+ 1+4 C14.262- 1+4 C14.1202+ 1+4 C14.1212+ 1+4 C14.662+ 1+4 C14.682+ 1+4 C42.237D14 C42.150D14 C42.153D14 C42.156D14 C42.158D14 C42⋊23D14 C42.163D14 C42⋊25D14 C42.240D14 D28⋊12D4 C42.178D14 C42.179D14
C4⋊D28 is a maximal quotient of
C14.(C4⋊Q8) (C2×C4)⋊9D28 C2.(C4×D28) (C2×C28)⋊5D4 (C2×Dic7)⋊3D4 (C2×C4).21D28 C28⋊SD16 C4⋊D56 D28.19D4 C42.36D14 Dic14⋊8D4 C4⋊Dic28 C8⋊8D28 C56⋊7D4 C8.2D28 C8⋊7D28 C8⋊3D28 D14⋊2Q16 C8.20D28 C8.21D28 C8.24D28 C4⋊(C4⋊Dic7) C4⋊(D14⋊C4) (C2×D28)⋊10C4 (C2×C4)⋊3D28 (C2×C4).45D28
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | D14 | D28 | D4×D7 | Q8⋊2D7 |
| kernel | C4⋊D28 | D14⋊C4 | C7×C4⋊C4 | C2×C4×D7 | C2×D28 | C28 | D14 | C4⋊C4 | C14 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 2 | 2 | 3 | 2 | 9 | 12 | 3 | 3 |
Matrix representation of C4⋊D28 ►in GL4(𝔽29) generated by
| 19 | 7 | 0 | 0 |
| 22 | 28 | 0 | 0 |
| 0 | 0 | 21 | 13 |
| 0 | 0 | 24 | 8 |
| 21 | 6 | 0 | 0 |
| 23 | 8 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 1 | 1 |
| 19 | 7 | 0 | 0 |
| 19 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 28 | 28 |
G:=sub<GL(4,GF(29))| [19,22,0,0,7,28,0,0,0,0,21,24,0,0,13,8],[21,23,0,0,6,8,0,0,0,0,28,1,0,0,0,1],[19,19,0,0,7,10,0,0,0,0,1,28,0,0,0,28] >;
C4⋊D28 in GAP, Magma, Sage, TeX
C_4\rtimes D_{28} % in TeX
G:=Group("C4:D28"); // GroupNames label
G:=SmallGroup(224,90);
// by ID
G=gap.SmallGroup(224,90);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,50,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=c^2=1,b*a*b^-1=a^15,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations