metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊1D4, C4⋊2D28, D14⋊2D4, C4⋊C4⋊3D7, D14⋊C4⋊8C2, (C2×D28)⋊4C2, C7⋊2(C4⋊D4), C2.13(D4×D7), C2.9(C2×D28), 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 C28⋊1D4
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 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×10], C7, C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×6], C23 [×3], D7 [×4], C14 [×3], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic7, C28 [×2], C28 [×2], D14 [×2], D14 [×8], C2×C14, C4⋊D4, C4×D7 [×2], D28 [×6], C2×Dic7, C2×C28, C2×C28 [×2], C22×D7, C22×D7 [×2], D14⋊C4 [×2], C7×C4⋊C4, C2×C4×D7, C2×D28, C2×D28 [×2], C28⋊1D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D7, C2×D4 [×2], C4○D4, D14 [×3], C4⋊D4, D28 [×2], C22×D7, C2×D28, D4×D7, Q8⋊2D7, C28⋊1D4
►On 112 points
Generators in S112
(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 47)(2 77 95 34)(3 64 96 49)(4 79 97 36)(5 66 98 51)(6 81 99 38)(7 68 100 53)(8 83 101 40)(9 70 102 55)(10 57 103 42)(11 72 104 29)(12 59 105 44)(13 74 106 31)(14 61 107 46)(15 76 108 33)(16 63 109 48)(17 78 110 35)(18 65 111 50)(19 80 112 37)(20 67 85 52)(21 82 86 39)(22 69 87 54)(23 84 88 41)(24 71 89 56)(25 58 90 43)(26 73 91 30)(27 60 92 45)(28 75 93 32)
(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 37)(30 36)(31 35)(32 34)(38 56)(39 55)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(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), (1,62,94,47)(2,77,95,34)(3,64,96,49)(4,79,97,36)(5,66,98,51)(6,81,99,38)(7,68,100,53)(8,83,101,40)(9,70,102,55)(10,57,103,42)(11,72,104,29)(12,59,105,44)(13,74,106,31)(14,61,107,46)(15,76,108,33)(16,63,109,48)(17,78,110,35)(18,65,111,50)(19,80,112,37)(20,67,85,52)(21,82,86,39)(22,69,87,54)(23,84,88,41)(24,71,89,56)(25,58,90,43)(26,73,91,30)(27,60,92,45)(28,75,93,32), (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)(29,37)(30,36)(31,35)(32,34)(38,56)(39,55)(40,54)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(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), (1,62,94,47)(2,77,95,34)(3,64,96,49)(4,79,97,36)(5,66,98,51)(6,81,99,38)(7,68,100,53)(8,83,101,40)(9,70,102,55)(10,57,103,42)(11,72,104,29)(12,59,105,44)(13,74,106,31)(14,61,107,46)(15,76,108,33)(16,63,109,48)(17,78,110,35)(18,65,111,50)(19,80,112,37)(20,67,85,52)(21,82,86,39)(22,69,87,54)(23,84,88,41)(24,71,89,56)(25,58,90,43)(26,73,91,30)(27,60,92,45)(28,75,93,32), (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)(29,37)(30,36)(31,35)(32,34)(38,56)(39,55)(40,54)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(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)], [(1,62,94,47),(2,77,95,34),(3,64,96,49),(4,79,97,36),(5,66,98,51),(6,81,99,38),(7,68,100,53),(8,83,101,40),(9,70,102,55),(10,57,103,42),(11,72,104,29),(12,59,105,44),(13,74,106,31),(14,61,107,46),(15,76,108,33),(16,63,109,48),(17,78,110,35),(18,65,111,50),(19,80,112,37),(20,67,85,52),(21,82,86,39),(22,69,87,54),(23,84,88,41),(24,71,89,56),(25,58,90,43),(26,73,91,30),(27,60,92,45),(28,75,93,32)], [(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),(29,37),(30,36),(31,35),(32,34),(38,56),(39,55),(40,54),(41,53),(42,52),(43,51),(44,50),(45,49),(46,48),(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)])
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 | C28⋊1D4 | 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 C28⋊1D4 ►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] >;
C28⋊1D4 in GAP, Magma, Sage, TeX
C_{28}\rtimes_1D_4 % in TeX
G:=Group("C28:1D4"); // 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