metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.37C24, D28.32C23, 2+ (1+4)⋊3D7, Dic14.32C23, C7⋊5(D4○D8), C4○D4⋊5D14, (C2×D4)⋊16D14, (C7×D4).36D4, C7⋊C8.16C23, (C7×Q8).36D4, D4⋊8D14⋊9C2, D4⋊D7⋊20C22, C28.269(C2×D4), Q8⋊D7⋊19C22, D4⋊D14⋊11C2, C4.37(C23×D7), D4.8D14⋊8C2, D4.18(C7⋊D4), C4○D28⋊10C22, (D4×C14)⋊24C22, (C2×D28)⋊39C22, D4.Dic7⋊10C2, D4.D7⋊19C22, Q8.18(C7⋊D4), D4.25(C22×D7), C7⋊Q16⋊21C22, (C7×D4).25C23, D4.D14⋊11C2, Q8.25(C22×D7), (C7×Q8).25C23, (C2×C28).118C23, C14.171(C22×D4), C4.Dic7⋊16C22, (C7×2+ (1+4))⋊2C2, (C2×D4⋊D7)⋊32C2, (C2×C7⋊C8)⋊24C22, C4.75(C2×C7⋊D4), (C2×C14).85(C2×D4), (C7×C4○D4)⋊8C22, C22.6(C2×C7⋊D4), C2.44(C22×C7⋊D4), (C2×C4).102(C22×D7), SmallGroup(448,1288)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1236 in 268 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×9], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C7, C8 [×4], C2×C4 [×3], C2×C4 [×6], D4 [×6], D4 [×15], Q8 [×2], Q8, C23 [×6], D7 [×3], C14, C14 [×6], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×3], C2×D4 [×9], C4○D4, C4○D4 [×3], C4○D4 [×5], Dic7, C28, C28 [×3], C28, D14 [×6], C2×C14 [×3], C2×C14 [×6], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ (1+4), 2+ (1+4), C7⋊C8, C7⋊C8 [×3], Dic14, C4×D7 [×3], D28 [×3], D28 [×3], C7⋊D4 [×3], C2×C28 [×3], C2×C28 [×3], C7×D4 [×6], C7×D4 [×6], C7×Q8 [×2], C22×D7 [×3], C22×C14 [×3], D4○D8, C2×C7⋊C8 [×3], C4.Dic7 [×3], D4⋊D7 [×9], D4.D7 [×3], Q8⋊D7 [×3], C7⋊Q16, C2×D28 [×3], C4○D28 [×3], D4×D7 [×3], Q8⋊2D7, D4×C14 [×3], D4×C14 [×3], C7×C4○D4, C7×C4○D4 [×3], C7×C4○D4, C2×D4⋊D7 [×3], D4.D14 [×3], D4.Dic7, D4⋊D14 [×3], D4.8D14 [×3], D4⋊8D14, C7×2+ (1+4), D28.32C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, C7⋊D4 [×4], C22×D7 [×7], D4○D8, C2×C7⋊D4 [×6], C23×D7, C22×C7⋊D4, D28.32C23
Generators and relations
G = < a,b,c,d,e | a28=b2=c2=d2=e2=1, bab=dad=a-1, ac=ca, eae=a15, cbc=a14b, dbd=a26b, ebe=a7b, cd=dc, ce=ec, ede=a21d >
►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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(54 56)(57 74)(58 73)(59 72)(60 71)(61 70)(62 69)(63 68)(64 67)(65 66)(75 84)(76 83)(77 82)(78 81)(79 80)(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 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 105)(41 106)(42 107)(43 108)(44 109)(45 110)(46 111)(47 112)(48 85)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)
(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(55 56)(57 61)(58 60)(62 84)(63 83)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)
(1 31)(2 46)(3 33)(4 48)(5 35)(6 50)(7 37)(8 52)(9 39)(10 54)(11 41)(12 56)(13 43)(14 30)(15 45)(16 32)(17 47)(18 34)(19 49)(20 36)(21 51)(22 38)(23 53)(24 40)(25 55)(26 42)(27 29)(28 44)(57 108)(58 95)(59 110)(60 97)(61 112)(62 99)(63 86)(64 101)(65 88)(66 103)(67 90)(68 105)(69 92)(70 107)(71 94)(72 109)(73 96)(74 111)(75 98)(76 85)(77 100)(78 87)(79 102)(80 89)(81 104)(82 91)(83 106)(84 93)
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)(75,84)(76,83)(77,82)(78,81)(79,80)(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), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93), (2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(55,56)(57,61)(58,60)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107), (1,31)(2,46)(3,33)(4,48)(5,35)(6,50)(7,37)(8,52)(9,39)(10,54)(11,41)(12,56)(13,43)(14,30)(15,45)(16,32)(17,47)(18,34)(19,49)(20,36)(21,51)(22,38)(23,53)(24,40)(25,55)(26,42)(27,29)(28,44)(57,108)(58,95)(59,110)(60,97)(61,112)(62,99)(63,86)(64,101)(65,88)(66,103)(67,90)(68,105)(69,92)(70,107)(71,94)(72,109)(73,96)(74,111)(75,98)(76,85)(77,100)(78,87)(79,102)(80,89)(81,104)(82,91)(83,106)(84,93)>;
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)(75,84)(76,83)(77,82)(78,81)(79,80)(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), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93), (2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(55,56)(57,61)(58,60)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107), (1,31)(2,46)(3,33)(4,48)(5,35)(6,50)(7,37)(8,52)(9,39)(10,54)(11,41)(12,56)(13,43)(14,30)(15,45)(16,32)(17,47)(18,34)(19,49)(20,36)(21,51)(22,38)(23,53)(24,40)(25,55)(26,42)(27,29)(28,44)(57,108)(58,95)(59,110)(60,97)(61,112)(62,99)(63,86)(64,101)(65,88)(66,103)(67,90)(68,105)(69,92)(70,107)(71,94)(72,109)(73,96)(74,111)(75,98)(76,85)(77,100)(78,87)(79,102)(80,89)(81,104)(82,91)(83,106)(84,93) );
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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(54,56),(57,74),(58,73),(59,72),(60,71),(61,70),(62,69),(63,68),(64,67),(65,66),(75,84),(76,83),(77,82),(78,81),(79,80),(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)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,57),(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,94),(30,95),(31,96),(32,97),(33,98),(34,99),(35,100),(36,101),(37,102),(38,103),(39,104),(40,105),(41,106),(42,107),(43,108),(44,109),(45,110),(46,111),(47,112),(48,85),(49,86),(50,87),(51,88),(52,89),(53,90),(54,91),(55,92),(56,93)], [(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(29,54),(30,53),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(41,42),(55,56),(57,61),(58,60),(62,84),(63,83),(64,82),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107)], [(1,31),(2,46),(3,33),(4,48),(5,35),(6,50),(7,37),(8,52),(9,39),(10,54),(11,41),(12,56),(13,43),(14,30),(15,45),(16,32),(17,47),(18,34),(19,49),(20,36),(21,51),(22,38),(23,53),(24,40),(25,55),(26,42),(27,29),(28,44),(57,108),(58,95),(59,110),(60,97),(61,112),(62,99),(63,86),(64,101),(65,88),(66,103),(67,90),(68,105),(69,92),(70,107),(71,94),(72,109),(73,96),(74,111),(75,98),(76,85),(77,100),(78,87),(79,102),(80,89),(81,104),(82,91),(83,106),(84,93)])
Matrix representation ►G ⊆ GL6(𝔽113)
| 80 | 104 | 0 | 0 | 0 | 0 |
| 9 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 80 | 104 | 0 | 0 | 0 | 0 |
| 33 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 81 | 0 | 112 |
| 0 | 0 | 81 | 0 | 112 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 106 | 0 |
| 0 | 0 | 0 | 1 | 0 | 106 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 104 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 82 | 82 | 0 | 0 |
| 0 | 0 | 82 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 82 | 82 |
| 0 | 0 | 0 | 0 | 82 | 31 |
G:=sub<GL(6,GF(113))| [80,9,0,0,0,0,104,112,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[80,33,0,0,0,0,104,33,0,0,0,0,0,0,0,1,0,81,0,0,1,0,81,0,0,0,0,0,0,112,0,0,0,0,112,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,106,0,112,0,0,0,0,106,0,112],[112,104,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,82,82,0,0,0,0,82,31,0,0,0,0,0,0,82,82,0,0,0,0,82,31] >;
73 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | ··· | 14AD | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 28 | 2 | 2 | 2 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
73 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | C7⋊D4 | C7⋊D4 | D4○D8 | D28.32C23 |
| kernel | D28.32C23 | C2×D4⋊D7 | D4.D14 | D4.Dic7 | D4⋊D14 | D4.8D14 | D4⋊8D14 | C7×2+ (1+4) | C7×D4 | C7×Q8 | 2+ (1+4) | C2×D4 | C4○D4 | D4 | Q8 | C7 | C1 |
| # reps | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 1 | 3 | 1 | 3 | 9 | 12 | 18 | 6 | 2 | 3 |
In GAP, Magma, Sage, TeX
D_{28}._{32}C_2^3 % in TeX
G:=Group("D28.32C2^3"); // GroupNames label
G:=SmallGroup(448,1288);
// by ID
G=gap.SmallGroup(448,1288);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,675,1684,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^28=b^2=c^2=d^2=e^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,e*a*e=a^15,c*b*c=a^14*b,d*b*d=a^26*b,e*b*e=a^7*b,c*d=d*c,c*e=e*c,e*d*e=a^21*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀