metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊21D4, C14.1172+ (1+4), C4⋊C4⋊10D14, C22⋊Q8⋊7D7, C7⋊7(D4⋊5D4), (C2×Q8)⋊16D14, C4.111(D4×D7), C28⋊1D4⋊25C2, D14.20(C2×D4), C28.234(C2×D4), D28⋊C4⋊25C2, C22⋊D28⋊16C2, (Q8×C14)⋊7C22, D14⋊C4⋊20C22, (C2×D28)⋊25C22, (C22×D28)⋊16C2, (C2×C28).54C23, C22⋊C4.57D14, C14.76(C22×D4), D14.5D4⋊17C2, C28.23D4⋊12C2, (C2×C14).174C24, Dic7⋊C4⋊53C22, C22⋊2(Q8⋊2D7), (C4×Dic7)⋊28C22, (C22×C4).236D14, C2.34(D4⋊8D14), (C23×D7).52C22, C23.189(C22×D7), C22.195(C23×D7), (C22×C14).202C23, (C22×C28).254C22, (C2×Dic7).233C23, (C22×D7).196C23, C23.D7.115C22, C2.49(C2×D4×D7), (C4×C7⋊D4)⋊22C2, (D7×C22⋊C4)⋊8C2, (C2×C4×D7)⋊18C22, (C2×C14)⋊7(C4○D4), (C7×C4⋊C4)⋊19C22, (C2×Q8⋊2D7)⋊7C2, (C7×C22⋊Q8)⋊10C2, C14.114(C2×C4○D4), C2.17(C2×Q8⋊2D7), (C2×C4).47(C22×D7), (C2×C7⋊D4).122C22, (C7×C22⋊C4).29C22, SmallGroup(448,1083)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1980 in 334 conjugacy classes, 107 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×27], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], D4 [×18], Q8 [×2], C23, C23 [×15], D7 [×7], C14 [×3], C14 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4, C22×C4, C22×C4 [×5], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24 [×2], Dic7 [×3], C28 [×2], C28 [×5], D14 [×4], D14 [×21], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×D7 [×8], D28 [×4], D28 [×12], C2×Dic7 [×3], C7⋊D4 [×2], C2×C28 [×2], C2×C28 [×4], C2×C28 [×2], C7×Q8 [×2], C22×D7, C22×D7 [×4], C22×D7 [×10], C22×C14, D4⋊5D4, C4×Dic7, Dic7⋊C4, D14⋊C4, D14⋊C4 [×8], C23.D7, C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×C4×D7, C2×C4×D7 [×4], C2×D28 [×2], C2×D28 [×6], C2×D28 [×4], Q8⋊2D7 [×4], C2×C7⋊D4, C22×C28, Q8×C14, C23×D7 [×2], D7×C22⋊C4 [×2], C22⋊D28 [×2], D28⋊C4, D14.5D4 [×2], C28⋊1D4, C28⋊1D4 [×2], C4×C7⋊D4, C28.23D4, C7×C22⋊Q8, C22×D28, C2×Q8⋊2D7, D28⋊21D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×2], C24, D14 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C22×D7 [×7], D4⋊5D4, D4×D7 [×2], Q8⋊2D7 [×2], C23×D7, C2×D4×D7, C2×Q8⋊2D7, D4⋊8D14, D28⋊21D4
Generators and relations
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, cac-1=dad=a13, cbc-1=dbd=a26b, dcd=c-1 >
►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 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(57 59)(60 84)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(86 112)(87 111)(88 110)(89 109)(90 108)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)
(1 76 39 89)(2 61 40 102)(3 74 41 87)(4 59 42 100)(5 72 43 85)(6 57 44 98)(7 70 45 111)(8 83 46 96)(9 68 47 109)(10 81 48 94)(11 66 49 107)(12 79 50 92)(13 64 51 105)(14 77 52 90)(15 62 53 103)(16 75 54 88)(17 60 55 101)(18 73 56 86)(19 58 29 99)(20 71 30 112)(21 84 31 97)(22 69 32 110)(23 82 33 95)(24 67 34 108)(25 80 35 93)(26 65 36 106)(27 78 37 91)(28 63 38 104)
(1 76)(2 61)(3 74)(4 59)(5 72)(6 57)(7 70)(8 83)(9 68)(10 81)(11 66)(12 79)(13 64)(14 77)(15 62)(16 75)(17 60)(18 73)(19 58)(20 71)(21 84)(22 69)(23 82)(24 67)(25 80)(26 65)(27 78)(28 63)(29 99)(30 112)(31 97)(32 110)(33 95)(34 108)(35 93)(36 106)(37 91)(38 104)(39 89)(40 102)(41 87)(42 100)(43 85)(44 98)(45 111)(46 96)(47 109)(48 94)(49 107)(50 92)(51 105)(52 90)(53 103)(54 88)(55 101)(56 86)
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)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(57,59)(60,84)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100), (1,76,39,89)(2,61,40,102)(3,74,41,87)(4,59,42,100)(5,72,43,85)(6,57,44,98)(7,70,45,111)(8,83,46,96)(9,68,47,109)(10,81,48,94)(11,66,49,107)(12,79,50,92)(13,64,51,105)(14,77,52,90)(15,62,53,103)(16,75,54,88)(17,60,55,101)(18,73,56,86)(19,58,29,99)(20,71,30,112)(21,84,31,97)(22,69,32,110)(23,82,33,95)(24,67,34,108)(25,80,35,93)(26,65,36,106)(27,78,37,91)(28,63,38,104), (1,76)(2,61)(3,74)(4,59)(5,72)(6,57)(7,70)(8,83)(9,68)(10,81)(11,66)(12,79)(13,64)(14,77)(15,62)(16,75)(17,60)(18,73)(19,58)(20,71)(21,84)(22,69)(23,82)(24,67)(25,80)(26,65)(27,78)(28,63)(29,99)(30,112)(31,97)(32,110)(33,95)(34,108)(35,93)(36,106)(37,91)(38,104)(39,89)(40,102)(41,87)(42,100)(43,85)(44,98)(45,111)(46,96)(47,109)(48,94)(49,107)(50,92)(51,105)(52,90)(53,103)(54,88)(55,101)(56,86)>;
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)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(57,59)(60,84)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100), (1,76,39,89)(2,61,40,102)(3,74,41,87)(4,59,42,100)(5,72,43,85)(6,57,44,98)(7,70,45,111)(8,83,46,96)(9,68,47,109)(10,81,48,94)(11,66,49,107)(12,79,50,92)(13,64,51,105)(14,77,52,90)(15,62,53,103)(16,75,54,88)(17,60,55,101)(18,73,56,86)(19,58,29,99)(20,71,30,112)(21,84,31,97)(22,69,32,110)(23,82,33,95)(24,67,34,108)(25,80,35,93)(26,65,36,106)(27,78,37,91)(28,63,38,104), (1,76)(2,61)(3,74)(4,59)(5,72)(6,57)(7,70)(8,83)(9,68)(10,81)(11,66)(12,79)(13,64)(14,77)(15,62)(16,75)(17,60)(18,73)(19,58)(20,71)(21,84)(22,69)(23,82)(24,67)(25,80)(26,65)(27,78)(28,63)(29,99)(30,112)(31,97)(32,110)(33,95)(34,108)(35,93)(36,106)(37,91)(38,104)(39,89)(40,102)(41,87)(42,100)(43,85)(44,98)(45,111)(46,96)(47,109)(48,94)(49,107)(50,92)(51,105)(52,90)(53,103)(54,88)(55,101)(56,86) );
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),(29,55),(30,54),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(57,59),(60,84),(61,83),(62,82),(63,81),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(86,112),(87,111),(88,110),(89,109),(90,108),(91,107),(92,106),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100)], [(1,76,39,89),(2,61,40,102),(3,74,41,87),(4,59,42,100),(5,72,43,85),(6,57,44,98),(7,70,45,111),(8,83,46,96),(9,68,47,109),(10,81,48,94),(11,66,49,107),(12,79,50,92),(13,64,51,105),(14,77,52,90),(15,62,53,103),(16,75,54,88),(17,60,55,101),(18,73,56,86),(19,58,29,99),(20,71,30,112),(21,84,31,97),(22,69,32,110),(23,82,33,95),(24,67,34,108),(25,80,35,93),(26,65,36,106),(27,78,37,91),(28,63,38,104)], [(1,76),(2,61),(3,74),(4,59),(5,72),(6,57),(7,70),(8,83),(9,68),(10,81),(11,66),(12,79),(13,64),(14,77),(15,62),(16,75),(17,60),(18,73),(19,58),(20,71),(21,84),(22,69),(23,82),(24,67),(25,80),(26,65),(27,78),(28,63),(29,99),(30,112),(31,97),(32,110),(33,95),(34,108),(35,93),(36,106),(37,91),(38,104),(39,89),(40,102),(41,87),(42,100),(43,85),(44,98),(45,111),(46,96),(47,109),(48,94),(49,107),(50,92),(51,105),(52,90),(53,103),(54,88),(55,101),(56,86)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 22 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 16 |
| 0 | 0 | 0 | 0 | 18 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 22 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 |
| 12 | 21 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 22 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 16 | 17 |
| 17 | 8 | 0 | 0 | 0 | 0 |
| 22 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 22 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 16 | 17 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,22,0,0,0,0,7,1,0,0,0,0,0,0,28,18,0,0,0,0,16,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,10,0,0,0,0,22,10,0,0,0,0,0,0,28,18,0,0,0,0,0,1],[12,0,0,0,0,0,21,17,0,0,0,0,0,0,28,22,0,0,0,0,0,1,0,0,0,0,0,0,12,16,0,0,0,0,11,17],[17,22,0,0,0,0,8,12,0,0,0,0,0,0,28,22,0,0,0,0,0,1,0,0,0,0,0,0,12,16,0,0,0,0,11,17] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4×D7 | Q8⋊2D7 | D4⋊8D14 |
| kernel | D28⋊21D4 | D7×C22⋊C4 | C22⋊D28 | D28⋊C4 | D14.5D4 | C28⋊1D4 | C4×C7⋊D4 | C28.23D4 | C7×C22⋊Q8 | C22×D28 | C2×Q8⋊2D7 | D28 | C22⋊Q8 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 6 | 9 | 3 | 3 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
D_{28}\rtimes_{21}D_4 % in TeX
G:=Group("D28:21D4"); // GroupNames label
G:=SmallGroup(448,1083);
// by ID
G=gap.SmallGroup(448,1083);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,1571,297,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^13,c*b*c^-1=d*b*d=a^26*b,d*c*d=c^-1>;
// generators/relations