metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊5C4, Dic7⋊5D4, C7⋊3(C4×D4), C4⋊C4⋊8D7, C4⋊1(C4×D7), C28⋊2(C2×C4), C2.4(D4×D7), D14⋊3(C2×C4), D14⋊C4⋊12C2, (C4×Dic7)⋊3C2, (C2×D28).7C2, C14.24(C2×D4), (C2×C4).31D14, C14.33(C4○D4), (C2×C14).34C23, (C2×C28).24C22, C14.11(C22×C4), C2.2(Q8⋊2D7), C22.18(C22×D7), (C2×Dic7).49C22, (C22×D7).20C22, (C7×C4⋊C4)⋊4C2, (C2×C4×D7)⋊12C2, C2.13(C2×C4×D7), SmallGroup(224,88)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28⋊C4
G = < a,b,c | a28=b2=c4=1, bab=a-1, cac-1=a15, cbc-1=a14b >
Subgroups: 406 in 94 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C4×D4, C4×D7, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C4×Dic7, D14⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, D28⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, C22×D7, C2×C4×D7, D4×D7, Q8⋊2D7, D28⋊C4
►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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 35)(30 34)(31 33)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(70 84)(71 83)(72 82)(73 81)(74 80)(75 79)(76 78)(85 87)(88 112)(89 111)(90 110)(91 109)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)
(1 50 74 90)(2 37 75 105)(3 52 76 92)(4 39 77 107)(5 54 78 94)(6 41 79 109)(7 56 80 96)(8 43 81 111)(9 30 82 98)(10 45 83 85)(11 32 84 100)(12 47 57 87)(13 34 58 102)(14 49 59 89)(15 36 60 104)(16 51 61 91)(17 38 62 106)(18 53 63 93)(19 40 64 108)(20 55 65 95)(21 42 66 110)(22 29 67 97)(23 44 68 112)(24 31 69 99)(25 46 70 86)(26 33 71 101)(27 48 72 88)(28 35 73 103)
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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,35)(30,34)(31,33)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(70,84)(71,83)(72,82)(73,81)(74,80)(75,79)(76,78)(85,87)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101), (1,50,74,90)(2,37,75,105)(3,52,76,92)(4,39,77,107)(5,54,78,94)(6,41,79,109)(7,56,80,96)(8,43,81,111)(9,30,82,98)(10,45,83,85)(11,32,84,100)(12,47,57,87)(13,34,58,102)(14,49,59,89)(15,36,60,104)(16,51,61,91)(17,38,62,106)(18,53,63,93)(19,40,64,108)(20,55,65,95)(21,42,66,110)(22,29,67,97)(23,44,68,112)(24,31,69,99)(25,46,70,86)(26,33,71,101)(27,48,72,88)(28,35,73,103)>;
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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,35)(30,34)(31,33)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(70,84)(71,83)(72,82)(73,81)(74,80)(75,79)(76,78)(85,87)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101), (1,50,74,90)(2,37,75,105)(3,52,76,92)(4,39,77,107)(5,54,78,94)(6,41,79,109)(7,56,80,96)(8,43,81,111)(9,30,82,98)(10,45,83,85)(11,32,84,100)(12,47,57,87)(13,34,58,102)(14,49,59,89)(15,36,60,104)(16,51,61,91)(17,38,62,106)(18,53,63,93)(19,40,64,108)(20,55,65,95)(21,42,66,110)(22,29,67,97)(23,44,68,112)(24,31,69,99)(25,46,70,86)(26,33,71,101)(27,48,72,88)(28,35,73,103) );
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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,35),(30,34),(31,33),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(57,69),(58,68),(59,67),(60,66),(61,65),(62,64),(70,84),(71,83),(72,82),(73,81),(74,80),(75,79),(76,78),(85,87),(88,112),(89,111),(90,110),(91,109),(92,108),(93,107),(94,106),(95,105),(96,104),(97,103),(98,102),(99,101)], [(1,50,74,90),(2,37,75,105),(3,52,76,92),(4,39,77,107),(5,54,78,94),(6,41,79,109),(7,56,80,96),(8,43,81,111),(9,30,82,98),(10,45,83,85),(11,32,84,100),(12,47,57,87),(13,34,58,102),(14,49,59,89),(15,36,60,104),(16,51,61,91),(17,38,62,106),(18,53,63,93),(19,40,64,108),(20,55,65,95),(21,42,66,110),(22,29,67,97),(23,44,68,112),(24,31,69,99),(25,46,70,86),(26,33,71,101),(27,48,72,88),(28,35,73,103)]])
D28⋊C4 is a maximal subgroup of
Dic7⋊4D8 D4⋊D7⋊C4 D28⋊3D4 D28.D4 Dic7⋊7SD16 Q8⋊D7⋊C4 Dic7⋊SD16 D28.12D4 Dic7⋊8SD16 D56⋊9C4 D28⋊Q8 D28.Q8 Dic7⋊5D8 C56⋊C2⋊C4 D28⋊2Q8 D28.2Q8 C14.82+ 1+4 C14.2- 1+4 C14.112+ 1+4 C42⋊7D14 C42.188D14 C42.95D14 C42.97D14 C4×D4×D7 C42⋊11D14 Dic14⋊24D4 C42.114D14 C42.122D14 C4×Q8⋊2D7 C42.126D14 C42.136D14 C28⋊(C4○D4) D28⋊19D4 D28⋊20D4 C14.472+ 1+4 C22⋊Q8⋊25D7 C4⋊C4⋊26D14 D28⋊21D4 D28⋊22D4 Dic14⋊22D4 C14.532+ 1+4 C14.202- 1+4 C14.242- 1+4 C14.1212+ 1+4 C4⋊C4⋊28D14 C14.612+ 1+4 C14.642+ 1+4 D28⋊7Q8 C42.237D14 C42.150D14 C42.151D14 C42.153D14 C42.156D14 C42⋊23D14 C42⋊24D14 C42.189D14 C42.163D14 C42.240D14 D28⋊8Q8 D28⋊9Q8 C42.178D14
D28⋊C4 is a maximal quotient of
C4⋊Dic7⋊8C4 (C2×C4)⋊9D28 D14⋊C42 D14⋊C4⋊5C4 D28⋊C8 D14⋊3M4(2) C28⋊2M4(2) Dic7⋊8SD16 Dic28⋊9C4 D56⋊9C4 Dic7⋊5D8 Dic28⋊6C4 C56⋊C2⋊C4 D56⋊10C4 D56⋊7C4 C28⋊(C4⋊C4) C4⋊C4×Dic7 (C2×D28)⋊10C4 D14⋊C4⋊6C4 D14⋊C4⋊7C4
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 7 | 7 | 7 | 7 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | C4×D7 | D4×D7 | Q8⋊2D7 |
| kernel | D28⋊C4 | C4×Dic7 | D14⋊C4 | C7×C4⋊C4 | C2×C4×D7 | C2×D28 | D28 | Dic7 | C4⋊C4 | C14 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 2 | 3 | 2 | 9 | 12 | 3 | 3 |
Matrix representation of D28⋊C4 ►in GL4(𝔽29) generated by
| 10 | 7 | 0 | 0 |
| 22 | 1 | 0 | 0 |
| 0 | 0 | 24 | 11 |
| 0 | 0 | 24 | 5 |
| 19 | 22 | 0 | 0 |
| 10 | 10 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 7 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 27 | 16 |
| 0 | 0 | 16 | 2 |
G:=sub<GL(4,GF(29))| [10,22,0,0,7,1,0,0,0,0,24,24,0,0,11,5],[19,10,0,0,22,10,0,0,0,0,28,7,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,27,16,0,0,16,2] >;
D28⋊C4 in GAP, Magma, Sage, TeX
D_{28}\rtimes C_4 % in TeX
G:=Group("D28:C4"); // GroupNames label
G:=SmallGroup(224,88);
// by ID
G=gap.SmallGroup(224,88);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,103,188,50,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^14*b>;
// generators/relations