metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.39C24, D28.34C23, 2- (1+4)⋊2D7, Dic14.34C23, C4○D4⋊6D14, (C2×Q8)⋊13D14, (C7×D4).38D4, C7⋊7(D4○SD16), C7⋊C8.18C23, (C7×Q8).38D4, D4⋊D7⋊22C22, C28.271(C2×D4), Q8⋊D7⋊20C22, D4⋊8D14⋊10C2, D4⋊D14⋊12C2, C4.39(C23×D7), D4.20(C7⋊D4), D4.Dic7⋊12C2, D4.D7⋊22C22, Q8.20(C7⋊D4), (Q8×C14)⋊23C22, C7⋊Q16⋊19C22, D4.27(C22×D7), (C7×D4).27C23, D4.8D14⋊10C2, (C7×Q8).27C23, Q8.27(C22×D7), C28.C23⋊11C2, (C2×C28).120C23, C4○D28.33C22, C14.173(C22×D4), C4.Dic7⋊18C22, (C7×2- (1+4))⋊2C2, (C2×D28).186C22, (C2×C7⋊C8)⋊26C22, (C2×Q8⋊D7)⋊32C2, C4.77(C2×C7⋊D4), (C2×C14).87(C2×D4), (C7×C4○D4)⋊9C22, C22.8(C2×C7⋊D4), C2.46(C22×C7⋊D4), (C2×C4).104(C22×D7), SmallGroup(448,1290)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1172 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×7], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], C7, C8 [×4], C2×C4 [×3], C2×C4 [×9], D4, D4 [×3], D4 [×12], Q8, Q8 [×3], Q8 [×4], C23 [×3], D7 [×3], C14, C14 [×4], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×6], C2×Q8 [×3], C2×Q8, C4○D4, C4○D4 [×3], C4○D4 [×7], Dic7, C28, C28 [×3], C28 [×3], D14 [×6], C2×C14 [×3], C2×C14, C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 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 [×6], C7×D4, C7×D4 [×3], C7×D4 [×3], C7×Q8, C7×Q8 [×3], C7×Q8 [×3], C22×D7 [×3], D4○SD16, C2×C7⋊C8 [×3], C4.Dic7 [×3], D4⋊D7 [×3], D4.D7, Q8⋊D7 [×9], C7⋊Q16 [×3], C2×D28 [×3], C4○D28 [×3], D4×D7 [×3], Q8⋊2D7, Q8×C14 [×3], Q8×C14, C7×C4○D4, C7×C4○D4 [×3], C7×C4○D4 [×3], C2×Q8⋊D7 [×3], C28.C23 [×3], D4.Dic7, D4⋊D14 [×3], D4.8D14 [×3], D4⋊8D14, C7×2- (1+4), D28.34C23
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○SD16, C2×C7⋊D4 [×6], C23×D7, C22×C7⋊D4, D28.34C23
Generators and relations
G = < a,b,c,d,e | a28=b2=c2=d2=1, e2=a14, bab=dad=a-1, ac=ca, eae-1=a15, cbc=a14b, dbd=a26b, ebe-1=a7b, cd=dc, ce=ec, ede-1=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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(52 56)(53 55)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(85 109)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 100)(95 99)(96 98)(110 112)
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)
(1 84)(2 83)(3 82)(4 81)(5 80)(6 79)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 64)(22 63)(23 62)(24 61)(25 60)(26 59)(27 58)(28 57)(29 109)(30 108)(31 107)(32 106)(33 105)(34 104)(35 103)(36 102)(37 101)(38 100)(39 99)(40 98)(41 97)(42 96)(43 95)(44 94)(45 93)(46 92)(47 91)(48 90)(49 89)(50 88)(51 87)(52 86)(53 85)(54 112)(55 111)(56 110)
(1 51 15 37)(2 38 16 52)(3 53 17 39)(4 40 18 54)(5 55 19 41)(6 42 20 56)(7 29 21 43)(8 44 22 30)(9 31 23 45)(10 46 24 32)(11 33 25 47)(12 48 26 34)(13 35 27 49)(14 50 28 36)(57 109 71 95)(58 96 72 110)(59 111 73 97)(60 98 74 112)(61 85 75 99)(62 100 76 86)(63 87 77 101)(64 102 78 88)(65 89 79 103)(66 104 80 90)(67 91 81 105)(68 106 82 92)(69 93 83 107)(70 108 84 94)
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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(52,56)(53,55)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(110,112), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106), (1,84)(2,83)(3,82)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,109)(30,108)(31,107)(32,106)(33,105)(34,104)(35,103)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,112)(55,111)(56,110), (1,51,15,37)(2,38,16,52)(3,53,17,39)(4,40,18,54)(5,55,19,41)(6,42,20,56)(7,29,21,43)(8,44,22,30)(9,31,23,45)(10,46,24,32)(11,33,25,47)(12,48,26,34)(13,35,27,49)(14,50,28,36)(57,109,71,95)(58,96,72,110)(59,111,73,97)(60,98,74,112)(61,85,75,99)(62,100,76,86)(63,87,77,101)(64,102,78,88)(65,89,79,103)(66,104,80,90)(67,91,81,105)(68,106,82,92)(69,93,83,107)(70,108,84,94)>;
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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(52,56)(53,55)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(85,109)(86,108)(87,107)(88,106)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(110,112), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106), (1,84)(2,83)(3,82)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,109)(30,108)(31,107)(32,106)(33,105)(34,104)(35,103)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,112)(55,111)(56,110), (1,51,15,37)(2,38,16,52)(3,53,17,39)(4,40,18,54)(5,55,19,41)(6,42,20,56)(7,29,21,43)(8,44,22,30)(9,31,23,45)(10,46,24,32)(11,33,25,47)(12,48,26,34)(13,35,27,49)(14,50,28,36)(57,109,71,95)(58,96,72,110)(59,111,73,97)(60,98,74,112)(61,85,75,99)(62,100,76,86)(63,87,77,101)(64,102,78,88)(65,89,79,103)(66,104,80,90)(67,91,81,105)(68,106,82,92)(69,93,83,107)(70,108,84,94) );
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)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(52,56),(53,55),(57,68),(58,67),(59,66),(60,65),(61,64),(62,63),(69,84),(70,83),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77),(85,109),(86,108),(87,107),(88,106),(89,105),(90,104),(91,103),(92,102),(93,101),(94,100),(95,99),(96,98),(110,112)], [(1,77),(2,78),(3,79),(4,80),(5,81),(6,82),(7,83),(8,84),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,107),(30,108),(31,109),(32,110),(33,111),(34,112),(35,85),(36,86),(37,87),(38,88),(39,89),(40,90),(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,97),(48,98),(49,99),(50,100),(51,101),(52,102),(53,103),(54,104),(55,105),(56,106)], [(1,84),(2,83),(3,82),(4,81),(5,80),(6,79),(7,78),(8,77),(9,76),(10,75),(11,74),(12,73),(13,72),(14,71),(15,70),(16,69),(17,68),(18,67),(19,66),(20,65),(21,64),(22,63),(23,62),(24,61),(25,60),(26,59),(27,58),(28,57),(29,109),(30,108),(31,107),(32,106),(33,105),(34,104),(35,103),(36,102),(37,101),(38,100),(39,99),(40,98),(41,97),(42,96),(43,95),(44,94),(45,93),(46,92),(47,91),(48,90),(49,89),(50,88),(51,87),(52,86),(53,85),(54,112),(55,111),(56,110)], [(1,51,15,37),(2,38,16,52),(3,53,17,39),(4,40,18,54),(5,55,19,41),(6,42,20,56),(7,29,21,43),(8,44,22,30),(9,31,23,45),(10,46,24,32),(11,33,25,47),(12,48,26,34),(13,35,27,49),(14,50,28,36),(57,109,71,95),(58,96,72,110),(59,111,73,97),(60,98,74,112),(61,85,75,99),(62,100,76,86),(63,87,77,101),(64,102,78,88),(65,89,79,103),(66,104,80,90),(67,91,81,105),(68,106,82,92),(69,93,83,107),(70,108,84,94)])
Matrix representation ►G ⊆ GL6(𝔽113)
| 25 | 34 | 0 | 0 | 0 | 0 |
| 79 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 112 | 1 | 2 |
| 0 | 0 | 0 | 1 | 112 | 112 |
| 25 | 34 | 0 | 0 | 0 | 0 |
| 88 | 88 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 112 | 112 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 100 | 100 | 26 | 26 |
| 0 | 0 | 100 | 100 | 0 | 26 |
| 0 | 0 | 100 | 13 | 0 | 0 |
| 0 | 0 | 100 | 87 | 13 | 26 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 34 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 100 | 100 | 0 | 26 |
| 0 | 0 | 100 | 100 | 26 | 26 |
| 0 | 0 | 13 | 100 | 0 | 0 |
| 0 | 0 | 87 | 100 | 13 | 26 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 112 | 111 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 112 |
G:=sub<GL(6,GF(113))| [25,79,0,0,0,0,34,112,0,0,0,0,0,0,0,112,112,0,0,0,1,0,112,1,0,0,0,0,1,112,0,0,0,0,2,112],[25,88,0,0,0,0,34,88,0,0,0,0,0,0,0,112,0,112,0,0,112,0,0,112,0,0,0,0,112,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,100,100,100,100,0,0,100,100,13,87,0,0,26,0,0,13,0,0,26,26,0,26],[112,34,0,0,0,0,0,1,0,0,0,0,0,0,100,100,13,87,0,0,100,100,100,100,0,0,0,26,0,13,0,0,26,26,0,26],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,112,1,0,0,0,1,0,1,0,0,1,112,0,0,0,0,0,111,0,112] >;
73 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | ··· | 14R | 28A | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 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○SD16 | D28.34C23 |
| kernel | D28.34C23 | C2×Q8⋊D7 | C28.C23 | D4.Dic7 | D4⋊D14 | D4.8D14 | D4⋊8D14 | C7×2- (1+4) | C7×D4 | C7×Q8 | 2- (1+4) | C2×Q8 | 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}._{34}C_2^3 % in TeX
G:=Group("D28.34C2^3"); // GroupNames label
G:=SmallGroup(448,1290);
// by ID
G=gap.SmallGroup(448,1290);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,675,136,1684,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^28=b^2=c^2=d^2=1,e^2=a^14,b*a*b=d*a*d=a^-1,a*c=c*a,e*a*e^-1=a^15,c*b*c=a^14*b,d*b*d=a^26*b,e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^21*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀