metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.38C24, D28.33C23, 2+ (1+4)⋊4D7, Dic14.33C23, (C7×D4).37D4, C7⋊6(D4○SD16), C7⋊C8.17C23, (C7×Q8).37D4, D4⋊D7⋊21C22, C4○D4.16D14, C28.270(C2×D4), Q8⋊D7⋊22C22, C4.38(C23×D7), D4.8D14⋊9C2, (C2×D4).118D14, D4.19(C7⋊D4), C4○D28⋊11C22, D4.Dic7⋊11C2, D4.D7⋊21C22, Q8.19(C7⋊D4), (C7×D4).26C23, C7⋊Q16⋊18C22, D4.26(C22×D7), D4.10D14⋊9C2, D4.D14⋊12C2, D4.9D14⋊11C2, (C7×Q8).26C23, Q8.26(C22×D7), (C2×C28).119C23, C14.172(C22×D4), C4.Dic7⋊17C22, (C7×2+ (1+4))⋊3C2, (C2×Dic14)⋊43C22, (D4×C14).169C22, (C2×C7⋊C8)⋊25C22, C4.76(C2×C7⋊D4), (C2×D4.D7)⋊32C2, (C2×C14).86(C2×D4), C22.7(C2×C7⋊D4), C2.45(C22×C7⋊D4), (C7×C4○D4).29C22, (C2×C4).103(C22×D7), SmallGroup(448,1289)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 980 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 [×6], D4 [×10], Q8 [×2], Q8 [×6], C23 [×3], D7, C14, C14 [×6], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×3], C2×D4 [×3], C2×Q8 [×4], C4○D4, C4○D4 [×3], C4○D4 [×7], Dic7 [×3], C28, C28 [×3], C28, D14, C2×C14 [×3], C2×C14 [×6], 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 [×3], Dic14 [×3], C4×D7 [×3], D28, C2×Dic7 [×3], C7⋊D4 [×3], C2×C28 [×3], C2×C28 [×3], C7×D4 [×6], C7×D4 [×6], C7×Q8 [×2], C22×C14 [×3], D4○SD16, C2×C7⋊C8 [×3], C4.Dic7 [×3], D4⋊D7 [×3], D4.D7 [×9], Q8⋊D7, C7⋊Q16 [×3], C2×Dic14 [×3], C4○D28 [×3], D4⋊2D7 [×3], Q8×D7, D4×C14 [×3], D4×C14 [×3], C7×C4○D4, C7×C4○D4 [×3], C7×C4○D4, D4.D14 [×3], C2×D4.D7 [×3], D4.Dic7, D4.8D14 [×3], D4.9D14 [×3], D4.10D14, C7×2+ (1+4), D28.33C23
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, D28.33C23
Generators and relations
G = < a,b,c,d,e | a28=b2=1, c2=d2=e2=a14, bab=a-1, ac=ca, ad=da, eae-1=a15, bc=cb, bd=db, ebe-1=a7b, dcd-1=a14c, ce=ec, de=ed >
►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 79)(2 78)(3 77)(4 76)(5 75)(6 74)(7 73)(8 72)(9 71)(10 70)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 60)(21 59)(22 58)(23 57)(24 84)(25 83)(26 82)(27 81)(28 80)(29 95)(30 94)(31 93)(32 92)(33 91)(34 90)(35 89)(36 88)(37 87)(38 86)(39 85)(40 112)(41 111)(42 110)(43 109)(44 108)(45 107)(46 106)(47 105)(48 104)(49 103)(50 102)(51 101)(52 100)(53 99)(54 98)(55 97)(56 96)
(1 43 15 29)(2 44 16 30)(3 45 17 31)(4 46 18 32)(5 47 19 33)(6 48 20 34)(7 49 21 35)(8 50 22 36)(9 51 23 37)(10 52 24 38)(11 53 25 39)(12 54 26 40)(13 55 27 41)(14 56 28 42)(57 87 71 101)(58 88 72 102)(59 89 73 103)(60 90 74 104)(61 91 75 105)(62 92 76 106)(63 93 77 107)(64 94 78 108)(65 95 79 109)(66 96 80 110)(67 97 81 111)(68 98 82 112)(69 99 83 85)(70 100 84 86)
(1 50 15 36)(2 51 16 37)(3 52 17 38)(4 53 18 39)(5 54 19 40)(6 55 20 41)(7 56 21 42)(8 29 22 43)(9 30 23 44)(10 31 24 45)(11 32 25 46)(12 33 26 47)(13 34 27 48)(14 35 28 49)(57 108 71 94)(58 109 72 95)(59 110 73 96)(60 111 74 97)(61 112 75 98)(62 85 76 99)(63 86 77 100)(64 87 78 101)(65 88 79 102)(66 89 80 103)(67 90 81 104)(68 91 82 105)(69 92 83 106)(70 93 84 107)
(1 29 15 43)(2 44 16 30)(3 31 17 45)(4 46 18 32)(5 33 19 47)(6 48 20 34)(7 35 21 49)(8 50 22 36)(9 37 23 51)(10 52 24 38)(11 39 25 53)(12 54 26 40)(13 41 27 55)(14 56 28 42)(57 94 71 108)(58 109 72 95)(59 96 73 110)(60 111 74 97)(61 98 75 112)(62 85 76 99)(63 100 77 86)(64 87 78 101)(65 102 79 88)(66 89 80 103)(67 104 81 90)(68 91 82 105)(69 106 83 92)(70 93 84 107)
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), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,72)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,84)(25,83)(26,82)(27,81)(28,80)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,112)(41,111)(42,110)(43,109)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,100)(53,99)(54,98)(55,97)(56,96), (1,43,15,29)(2,44,16,30)(3,45,17,31)(4,46,18,32)(5,47,19,33)(6,48,20,34)(7,49,21,35)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,87,71,101)(58,88,72,102)(59,89,73,103)(60,90,74,104)(61,91,75,105)(62,92,76,106)(63,93,77,107)(64,94,78,108)(65,95,79,109)(66,96,80,110)(67,97,81,111)(68,98,82,112)(69,99,83,85)(70,100,84,86), (1,50,15,36)(2,51,16,37)(3,52,17,38)(4,53,18,39)(5,54,19,40)(6,55,20,41)(7,56,21,42)(8,29,22,43)(9,30,23,44)(10,31,24,45)(11,32,25,46)(12,33,26,47)(13,34,27,48)(14,35,28,49)(57,108,71,94)(58,109,72,95)(59,110,73,96)(60,111,74,97)(61,112,75,98)(62,85,76,99)(63,86,77,100)(64,87,78,101)(65,88,79,102)(66,89,80,103)(67,90,81,104)(68,91,82,105)(69,92,83,106)(70,93,84,107), (1,29,15,43)(2,44,16,30)(3,31,17,45)(4,46,18,32)(5,33,19,47)(6,48,20,34)(7,35,21,49)(8,50,22,36)(9,37,23,51)(10,52,24,38)(11,39,25,53)(12,54,26,40)(13,41,27,55)(14,56,28,42)(57,94,71,108)(58,109,72,95)(59,96,73,110)(60,111,74,97)(61,98,75,112)(62,85,76,99)(63,100,77,86)(64,87,78,101)(65,102,79,88)(66,89,80,103)(67,104,81,90)(68,91,82,105)(69,106,83,92)(70,93,84,107)>;
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), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,72)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,84)(25,83)(26,82)(27,81)(28,80)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,112)(41,111)(42,110)(43,109)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,100)(53,99)(54,98)(55,97)(56,96), (1,43,15,29)(2,44,16,30)(3,45,17,31)(4,46,18,32)(5,47,19,33)(6,48,20,34)(7,49,21,35)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,87,71,101)(58,88,72,102)(59,89,73,103)(60,90,74,104)(61,91,75,105)(62,92,76,106)(63,93,77,107)(64,94,78,108)(65,95,79,109)(66,96,80,110)(67,97,81,111)(68,98,82,112)(69,99,83,85)(70,100,84,86), (1,50,15,36)(2,51,16,37)(3,52,17,38)(4,53,18,39)(5,54,19,40)(6,55,20,41)(7,56,21,42)(8,29,22,43)(9,30,23,44)(10,31,24,45)(11,32,25,46)(12,33,26,47)(13,34,27,48)(14,35,28,49)(57,108,71,94)(58,109,72,95)(59,110,73,96)(60,111,74,97)(61,112,75,98)(62,85,76,99)(63,86,77,100)(64,87,78,101)(65,88,79,102)(66,89,80,103)(67,90,81,104)(68,91,82,105)(69,92,83,106)(70,93,84,107), (1,29,15,43)(2,44,16,30)(3,31,17,45)(4,46,18,32)(5,33,19,47)(6,48,20,34)(7,35,21,49)(8,50,22,36)(9,37,23,51)(10,52,24,38)(11,39,25,53)(12,54,26,40)(13,41,27,55)(14,56,28,42)(57,94,71,108)(58,109,72,95)(59,96,73,110)(60,111,74,97)(61,98,75,112)(62,85,76,99)(63,100,77,86)(64,87,78,101)(65,102,79,88)(66,89,80,103)(67,104,81,90)(68,91,82,105)(69,106,83,92)(70,93,84,107) );
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)], [(1,79),(2,78),(3,77),(4,76),(5,75),(6,74),(7,73),(8,72),(9,71),(10,70),(11,69),(12,68),(13,67),(14,66),(15,65),(16,64),(17,63),(18,62),(19,61),(20,60),(21,59),(22,58),(23,57),(24,84),(25,83),(26,82),(27,81),(28,80),(29,95),(30,94),(31,93),(32,92),(33,91),(34,90),(35,89),(36,88),(37,87),(38,86),(39,85),(40,112),(41,111),(42,110),(43,109),(44,108),(45,107),(46,106),(47,105),(48,104),(49,103),(50,102),(51,101),(52,100),(53,99),(54,98),(55,97),(56,96)], [(1,43,15,29),(2,44,16,30),(3,45,17,31),(4,46,18,32),(5,47,19,33),(6,48,20,34),(7,49,21,35),(8,50,22,36),(9,51,23,37),(10,52,24,38),(11,53,25,39),(12,54,26,40),(13,55,27,41),(14,56,28,42),(57,87,71,101),(58,88,72,102),(59,89,73,103),(60,90,74,104),(61,91,75,105),(62,92,76,106),(63,93,77,107),(64,94,78,108),(65,95,79,109),(66,96,80,110),(67,97,81,111),(68,98,82,112),(69,99,83,85),(70,100,84,86)], [(1,50,15,36),(2,51,16,37),(3,52,17,38),(4,53,18,39),(5,54,19,40),(6,55,20,41),(7,56,21,42),(8,29,22,43),(9,30,23,44),(10,31,24,45),(11,32,25,46),(12,33,26,47),(13,34,27,48),(14,35,28,49),(57,108,71,94),(58,109,72,95),(59,110,73,96),(60,111,74,97),(61,112,75,98),(62,85,76,99),(63,86,77,100),(64,87,78,101),(65,88,79,102),(66,89,80,103),(67,90,81,104),(68,91,82,105),(69,92,83,106),(70,93,84,107)], [(1,29,15,43),(2,44,16,30),(3,31,17,45),(4,46,18,32),(5,33,19,47),(6,48,20,34),(7,35,21,49),(8,50,22,36),(9,37,23,51),(10,52,24,38),(11,39,25,53),(12,54,26,40),(13,41,27,55),(14,56,28,42),(57,94,71,108),(58,109,72,95),(59,96,73,110),(60,111,74,97),(61,98,75,112),(62,85,76,99),(63,100,77,86),(64,87,78,101),(65,102,79,88),(66,89,80,103),(67,104,81,90),(68,91,82,105),(69,106,83,92),(70,93,84,107)])
Matrix representation ►G ⊆ GL6(𝔽113)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 8 | 109 | 0 | 0 | 0 | 0 |
| 0 | 0 | 57 | 90 | 0 | 0 |
| 0 | 0 | 102 | 56 | 0 | 0 |
| 0 | 0 | 28 | 89 | 112 | 77 |
| 0 | 0 | 76 | 71 | 44 | 1 |
| 84 | 3 | 0 | 0 | 0 | 0 |
| 59 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 84 | 17 | 0 | 22 |
| 0 | 0 | 24 | 92 | 47 | 26 |
| 0 | 0 | 65 | 62 | 0 | 16 |
| 0 | 0 | 46 | 90 | 51 | 50 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 72 | 0 |
| 0 | 0 | 45 | 25 | 44 | 39 |
| 0 | 0 | 91 | 0 | 1 | 0 |
| 0 | 0 | 29 | 39 | 102 | 88 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 103 | 41 | 7 |
| 0 | 0 | 45 | 44 | 90 | 37 |
| 0 | 0 | 28 | 89 | 112 | 77 |
| 0 | 0 | 49 | 64 | 41 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 41 | 0 |
| 0 | 0 | 104 | 25 | 90 | 39 |
| 0 | 0 | 22 | 0 | 112 | 0 |
| 0 | 0 | 103 | 39 | 41 | 88 |
G:=sub<GL(6,GF(113))| [28,8,0,0,0,0,0,109,0,0,0,0,0,0,57,102,28,76,0,0,90,56,89,71,0,0,0,0,112,44,0,0,0,0,77,1],[84,59,0,0,0,0,3,29,0,0,0,0,0,0,84,24,65,46,0,0,17,92,62,90,0,0,0,47,0,51,0,0,22,26,16,50],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,45,91,29,0,0,0,25,0,39,0,0,72,44,1,102,0,0,0,39,0,88],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,38,45,28,49,0,0,103,44,89,64,0,0,41,90,112,41,0,0,7,37,77,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,104,22,103,0,0,0,25,0,39,0,0,41,90,112,41,0,0,0,39,0,88] >;
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 | ··· | 14AD | 28A | ··· | 28R |
| 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 | 4 | 4 | 28 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 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 | D28.33C23 |
| kernel | D28.33C23 | D4.D14 | C2×D4.D7 | D4.Dic7 | D4.8D14 | D4.9D14 | D4.10D14 | 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}._{33}C_2^3 % in TeX
G:=Group("D28.33C2^3"); // GroupNames label
G:=SmallGroup(448,1289);
// by ID
G=gap.SmallGroup(448,1289);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,184,675,136,1684,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^28=b^2=1,c^2=d^2=e^2=a^14,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^15,b*c=c*b,b*d=d*b,e*b*e^-1=a^7*b,d*c*d^-1=a^14*c,c*e=e*c,d*e=e*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀