metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.29D4, C28.9C24, SD16⋊13D14, D56⋊21C22, C56.36C23, D28.5C23, Dic14.29D4, Dic28⋊18C22, Dic14.5C23, (C2×C8)⋊11D14, C4.76(D4×D7), C7⋊D4.9D4, C7⋊C8.3C23, D56⋊C2⋊5C2, (C2×C56)⋊6C22, D4⋊D7⋊2C22, (C2×Q8)⋊11D14, (D7×SD16)⋊5C2, (C2×SD16)⋊6D7, C28.84(C2×D4), C7⋊2(D4○SD16), (C8×D7)⋊9C22, Q8⋊D7⋊1C22, D4⋊6D14⋊6C2, D56⋊7C2⋊8C2, (Q8×D7)⋊1C22, C4.9(C23×D7), (C14×SD16)⋊2C2, D14.27(C2×D4), SD16⋊D7⋊5C2, C4○D28⋊4C22, D4.D7⋊2C22, (D4×D7).1C22, D4.7(C22×D7), (C7×D4).7C23, (C4×D7).5C23, C7⋊Q16⋊1C22, C8.12(C22×D7), C22.21(D4×D7), SD16⋊3D7⋊5C2, (C2×D4).116D14, D4.D14⋊8C2, D28.2C4⋊5C2, (C7×Q8).3C23, Q8.3(C22×D7), C28.C23⋊7C2, C56⋊C2⋊19C22, C8⋊D7⋊10C22, Dic7.32(C2×D4), (Q8×C14)⋊19C22, Q8⋊2D7⋊1C22, (C2×C28).526C23, Q8.10D14⋊3C2, (C7×SD16)⋊14C22, D4⋊2D7.1C22, C14.110(C22×D4), C4.Dic7⋊29C22, (D4×C14).167C22, C2.83(C2×D4×D7), (C2×C14).399(C2×D4), (C2×C4).230(C22×D7), SmallGroup(448,1215)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1316 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×9], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4 [×2], D4 [×14], Q8 [×2], Q8 [×6], C23 [×3], D7 [×4], C14, C14 [×3], C2×C8, C2×C8 [×2], M4(2) [×3], D8 [×3], SD16 [×4], SD16 [×6], Q16 [×3], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8 [×3], C4○D4 [×11], Dic7 [×2], Dic7 [×2], C28 [×2], C28 [×2], D14 [×2], D14 [×4], C2×C14, C2×C14 [×3], C8○D4, C2×SD16, C2×SD16 [×2], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ (1+4), 2- (1+4), C7⋊C8 [×2], C56 [×2], Dic14 [×3], Dic14 [×2], C4×D7 [×2], C4×D7 [×6], D28 [×3], D28 [×2], C2×Dic7 [×2], C7⋊D4 [×2], C7⋊D4 [×6], C2×C28, C2×C28, C7×D4 [×2], C7×D4, C7×Q8 [×2], C7×Q8, C22×D7 [×2], C22×C14, D4○SD16, C8×D7 [×2], C8⋊D7 [×2], C56⋊C2 [×2], D56, Dic28, C4.Dic7, D4⋊D7 [×2], D4.D7 [×2], Q8⋊D7 [×2], C7⋊Q16 [×2], C2×C56, C7×SD16 [×4], C4○D28 [×3], C4○D28 [×2], D4×D7 [×2], D4×D7, D4⋊2D7 [×2], D4⋊2D7, Q8×D7 [×2], Q8×D7, Q8⋊2D7 [×2], Q8⋊2D7, C2×C7⋊D4 [×2], D4×C14, Q8×C14, D28.2C4, D56⋊7C2, D7×SD16 [×2], D56⋊C2 [×2], SD16⋊D7 [×2], SD16⋊3D7 [×2], D4.D14, C28.C23, C14×SD16, D4⋊6D14, Q8.10D14, D28.29D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, C22×D7 [×7], D4○SD16, D4×D7 [×2], C23×D7, C2×D4×D7, D28.29D4
Generators and relations
G = < a,b,c,d | a28=b2=d2=1, c4=a14, bab=a-1, ac=ca, dad=a15, bc=cb, dbd=a14b, dcd=c3 >
►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 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 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(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 94 38 75 15 108 52 61)(2 95 39 76 16 109 53 62)(3 96 40 77 17 110 54 63)(4 97 41 78 18 111 55 64)(5 98 42 79 19 112 56 65)(6 99 43 80 20 85 29 66)(7 100 44 81 21 86 30 67)(8 101 45 82 22 87 31 68)(9 102 46 83 23 88 32 69)(10 103 47 84 24 89 33 70)(11 104 48 57 25 90 34 71)(12 105 49 58 26 91 35 72)(13 106 50 59 27 92 36 73)(14 107 51 60 28 93 37 74)
(1 8)(2 23)(3 10)(4 25)(5 12)(6 27)(7 14)(9 16)(11 18)(13 20)(15 22)(17 24)(19 26)(21 28)(29 36)(30 51)(31 38)(32 53)(33 40)(34 55)(35 42)(37 44)(39 46)(41 48)(43 50)(45 52)(47 54)(49 56)(57 111)(58 98)(59 85)(60 100)(61 87)(62 102)(63 89)(64 104)(65 91)(66 106)(67 93)(68 108)(69 95)(70 110)(71 97)(72 112)(73 99)(74 86)(75 101)(76 88)(77 103)(78 90)(79 105)(80 92)(81 107)(82 94)(83 109)(84 96)
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)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(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), (1,94,38,75,15,108,52,61)(2,95,39,76,16,109,53,62)(3,96,40,77,17,110,54,63)(4,97,41,78,18,111,55,64)(5,98,42,79,19,112,56,65)(6,99,43,80,20,85,29,66)(7,100,44,81,21,86,30,67)(8,101,45,82,22,87,31,68)(9,102,46,83,23,88,32,69)(10,103,47,84,24,89,33,70)(11,104,48,57,25,90,34,71)(12,105,49,58,26,91,35,72)(13,106,50,59,27,92,36,73)(14,107,51,60,28,93,37,74), (1,8)(2,23)(3,10)(4,25)(5,12)(6,27)(7,14)(9,16)(11,18)(13,20)(15,22)(17,24)(19,26)(21,28)(29,36)(30,51)(31,38)(32,53)(33,40)(34,55)(35,42)(37,44)(39,46)(41,48)(43,50)(45,52)(47,54)(49,56)(57,111)(58,98)(59,85)(60,100)(61,87)(62,102)(63,89)(64,104)(65,91)(66,106)(67,93)(68,108)(69,95)(70,110)(71,97)(72,112)(73,99)(74,86)(75,101)(76,88)(77,103)(78,90)(79,105)(80,92)(81,107)(82,94)(83,109)(84,96)>;
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)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(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), (1,94,38,75,15,108,52,61)(2,95,39,76,16,109,53,62)(3,96,40,77,17,110,54,63)(4,97,41,78,18,111,55,64)(5,98,42,79,19,112,56,65)(6,99,43,80,20,85,29,66)(7,100,44,81,21,86,30,67)(8,101,45,82,22,87,31,68)(9,102,46,83,23,88,32,69)(10,103,47,84,24,89,33,70)(11,104,48,57,25,90,34,71)(12,105,49,58,26,91,35,72)(13,106,50,59,27,92,36,73)(14,107,51,60,28,93,37,74), (1,8)(2,23)(3,10)(4,25)(5,12)(6,27)(7,14)(9,16)(11,18)(13,20)(15,22)(17,24)(19,26)(21,28)(29,36)(30,51)(31,38)(32,53)(33,40)(34,55)(35,42)(37,44)(39,46)(41,48)(43,50)(45,52)(47,54)(49,56)(57,111)(58,98)(59,85)(60,100)(61,87)(62,102)(63,89)(64,104)(65,91)(66,106)(67,93)(68,108)(69,95)(70,110)(71,97)(72,112)(73,99)(74,86)(75,101)(76,88)(77,103)(78,90)(79,105)(80,92)(81,107)(82,94)(83,109)(84,96) );
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),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(54,56),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(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)], [(1,94,38,75,15,108,52,61),(2,95,39,76,16,109,53,62),(3,96,40,77,17,110,54,63),(4,97,41,78,18,111,55,64),(5,98,42,79,19,112,56,65),(6,99,43,80,20,85,29,66),(7,100,44,81,21,86,30,67),(8,101,45,82,22,87,31,68),(9,102,46,83,23,88,32,69),(10,103,47,84,24,89,33,70),(11,104,48,57,25,90,34,71),(12,105,49,58,26,91,35,72),(13,106,50,59,27,92,36,73),(14,107,51,60,28,93,37,74)], [(1,8),(2,23),(3,10),(4,25),(5,12),(6,27),(7,14),(9,16),(11,18),(13,20),(15,22),(17,24),(19,26),(21,28),(29,36),(30,51),(31,38),(32,53),(33,40),(34,55),(35,42),(37,44),(39,46),(41,48),(43,50),(45,52),(47,54),(49,56),(57,111),(58,98),(59,85),(60,100),(61,87),(62,102),(63,89),(64,104),(65,91),(66,106),(67,93),(68,108),(69,95),(70,110),(71,97),(72,112),(73,99),(74,86),(75,101),(76,88),(77,103),(78,90),(79,105),(80,92),(81,107),(82,94),(83,109),(84,96)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 0 | 0 | 54 | 82 |
| 0 | 0 | 31 | 91 |
| 59 | 31 | 0 | 0 |
| 82 | 22 | 0 | 0 |
| 88 | 79 | 0 | 0 |
| 25 | 25 | 0 | 0 |
| 0 | 0 | 88 | 79 |
| 0 | 0 | 25 | 25 |
| 100 | 0 | 13 | 0 |
| 0 | 100 | 0 | 13 |
| 100 | 0 | 100 | 0 |
| 0 | 100 | 0 | 100 |
| 0 | 0 | 22 | 101 |
| 0 | 0 | 12 | 91 |
| 22 | 101 | 0 | 0 |
| 12 | 91 | 0 | 0 |
G:=sub<GL(4,GF(113))| [0,0,59,82,0,0,31,22,54,31,0,0,82,91,0,0],[88,25,0,0,79,25,0,0,0,0,88,25,0,0,79,25],[100,0,100,0,0,100,0,100,13,0,100,0,0,13,0,100],[0,0,22,12,0,0,101,91,22,12,0,0,101,91,0,0] >;
64 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 | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| 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 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D4○SD16 | D4×D7 | D4×D7 | D28.29D4 |
| kernel | D28.29D4 | D28.2C4 | D56⋊7C2 | D7×SD16 | D56⋊C2 | SD16⋊D7 | SD16⋊3D7 | D4.D14 | C28.C23 | C14×SD16 | D4⋊6D14 | Q8.10D14 | Dic14 | D28 | C7⋊D4 | C2×SD16 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 12 | 3 | 3 | 2 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
D_{28}._{29}D_4 % in TeX
G:=Group("D28.29D4"); // GroupNames label
G:=SmallGroup(448,1215);
// by ID
G=gap.SmallGroup(448,1215);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=d^2=1,c^4=a^14,b*a*b=a^-1,a*c=c*a,d*a*d=a^15,b*c=c*b,d*b*d=a^14*b,d*c*d=c^3>;
// generators/relations