metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊15D14, Q16⋊13D14, D28.45D4, SD16⋊11D14, D56⋊19C22, C56.38C23, C28.16C24, Dic14.45D4, D28.11C23, C4○D8⋊4D7, (D7×D8)⋊7C2, C7⋊3(D4○D8), C4○D4⋊1D14, (C2×C8)⋊13D14, C7⋊D4.1D4, C7⋊C8.7C23, (C2×D56)⋊23C2, D56⋊C2⋊6C2, D4⋊D7⋊3C22, C4.143(D4×D7), (C8×D7)⋊8C22, (D4×D7)⋊2C22, Q8⋊D7⋊2C22, Q8.D14⋊7C2, D4⋊D14⋊7C2, D4⋊8D14⋊5C2, C22.8(D4×D7), (C2×C56)⋊12C22, D14.29(C2×D4), C28.349(C2×D4), (C7×D8)⋊13C22, (C4×D7).9C23, C8.16(C22×D7), C4.16(C23×D7), D28.2C4⋊7C2, (C2×D28)⋊34C22, C8⋊D7⋊11C22, Dic7.34(C2×D4), (C7×Q16)⋊11C22, Q8⋊2D7⋊2C22, D4.10(C22×D7), (C7×D4).10C23, (C7×Q8).10C23, Q8.10(C22×D7), (C2×C28).533C23, (C7×SD16)⋊11C22, C4○D28.54C22, C14.117(C22×D4), C4.Dic7⋊30C22, C2.90(C2×D4×D7), (C7×C4○D8)⋊5C2, (C2×C14).13(C2×D4), (C7×C4○D4)⋊3C22, (C2×C4).232(C22×D7), SmallGroup(448,1222)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1620 in 268 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2 [×9], C4 [×2], C4 [×4], C22, C22 [×14], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×8], D4 [×2], D4 [×19], Q8 [×2], Q8, C23 [×6], D7 [×6], C14, C14 [×3], C2×C8, C2×C8 [×2], M4(2) [×3], D8, D8 [×8], SD16 [×2], SD16 [×4], Q16, C2×D4 [×12], C4○D4 [×2], C4○D4 [×7], Dic7 [×2], C28 [×2], C28 [×2], D14 [×2], D14 [×10], C2×C14, C2×C14 [×2], C8○D4, C2×D8 [×3], C4○D8, C4○D8 [×2], C8⋊C22 [×6], 2+ (1+4) [×2], C7⋊C8 [×2], C56 [×2], Dic14, C4×D7 [×2], C4×D7 [×4], D28, D28 [×4], D28 [×6], C7⋊D4 [×2], C7⋊D4 [×4], C2×C28, C2×C28 [×2], C7×D4 [×2], C7×D4 [×2], C7×Q8 [×2], C22×D7 [×6], D4○D8, C8×D7 [×2], C8⋊D7 [×2], D56 [×4], C4.Dic7, D4⋊D7 [×4], Q8⋊D7 [×4], C2×C56, C7×D8, C7×SD16 [×2], C7×Q16, C2×D28 [×2], C2×D28 [×2], C4○D28, C4○D28 [×2], D4×D7 [×4], D4×D7 [×4], Q8⋊2D7 [×4], C7×C4○D4 [×2], D28.2C4, C2×D56, D7×D8 [×2], D56⋊C2 [×4], Q8.D14 [×2], D4⋊D14 [×2], C7×C4○D8, D4⋊8D14 [×2], D8⋊15D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, C22×D7 [×7], D4○D8, D4×D7 [×2], C23×D7, C2×D4×D7, D8⋊15D14
Generators and relations
G = < a,b,c,d | a8=b2=c14=d2=1, bab=dad=a-1, ac=ca, cbc-1=a4b, dbd=a2b, dcd=c-1 >
►On 112 points
(1 66 47 73 107 41 91 19)(2 67 48 74 108 42 92 20)(3 68 49 75 109 29 93 21)(4 69 50 76 110 30 94 22)(5 70 51 77 111 31 95 23)(6 57 52 78 112 32 96 24)(7 58 53 79 99 33 97 25)(8 59 54 80 100 34 98 26)(9 60 55 81 101 35 85 27)(10 61 56 82 102 36 86 28)(11 62 43 83 103 37 87 15)(12 63 44 84 104 38 88 16)(13 64 45 71 105 39 89 17)(14 65 46 72 106 40 90 18)
(2 108)(4 110)(6 112)(8 100)(10 102)(12 104)(14 106)(15 62)(16 38)(17 64)(18 40)(19 66)(20 42)(21 68)(22 30)(23 70)(24 32)(25 58)(26 34)(27 60)(28 36)(29 75)(31 77)(33 79)(35 81)(37 83)(39 71)(41 73)(43 87)(45 89)(47 91)(49 93)(51 95)(53 97)(55 85)(57 78)(59 80)(61 82)(63 84)(65 72)(67 74)(69 76)
(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 84)(11 83)(12 82)(13 81)(14 80)(15 103)(16 102)(17 101)(18 100)(19 99)(20 112)(21 111)(22 110)(23 109)(24 108)(25 107)(26 106)(27 105)(28 104)(29 95)(30 94)(31 93)(32 92)(33 91)(34 90)(35 89)(36 88)(37 87)(38 86)(39 85)(40 98)(41 97)(42 96)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)
G:=sub<Sym(112)| (1,66,47,73,107,41,91,19)(2,67,48,74,108,42,92,20)(3,68,49,75,109,29,93,21)(4,69,50,76,110,30,94,22)(5,70,51,77,111,31,95,23)(6,57,52,78,112,32,96,24)(7,58,53,79,99,33,97,25)(8,59,54,80,100,34,98,26)(9,60,55,81,101,35,85,27)(10,61,56,82,102,36,86,28)(11,62,43,83,103,37,87,15)(12,63,44,84,104,38,88,16)(13,64,45,71,105,39,89,17)(14,65,46,72,106,40,90,18), (2,108)(4,110)(6,112)(8,100)(10,102)(12,104)(14,106)(15,62)(16,38)(17,64)(18,40)(19,66)(20,42)(21,68)(22,30)(23,70)(24,32)(25,58)(26,34)(27,60)(28,36)(29,75)(31,77)(33,79)(35,81)(37,83)(39,71)(41,73)(43,87)(45,89)(47,91)(49,93)(51,95)(53,97)(55,85)(57,78)(59,80)(61,82)(63,84)(65,72)(67,74)(69,76), (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)(10,84)(11,83)(12,82)(13,81)(14,80)(15,103)(16,102)(17,101)(18,100)(19,99)(20,112)(21,111)(22,110)(23,109)(24,108)(25,107)(26,106)(27,105)(28,104)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,98)(41,97)(42,96)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)>;
G:=Group( (1,66,47,73,107,41,91,19)(2,67,48,74,108,42,92,20)(3,68,49,75,109,29,93,21)(4,69,50,76,110,30,94,22)(5,70,51,77,111,31,95,23)(6,57,52,78,112,32,96,24)(7,58,53,79,99,33,97,25)(8,59,54,80,100,34,98,26)(9,60,55,81,101,35,85,27)(10,61,56,82,102,36,86,28)(11,62,43,83,103,37,87,15)(12,63,44,84,104,38,88,16)(13,64,45,71,105,39,89,17)(14,65,46,72,106,40,90,18), (2,108)(4,110)(6,112)(8,100)(10,102)(12,104)(14,106)(15,62)(16,38)(17,64)(18,40)(19,66)(20,42)(21,68)(22,30)(23,70)(24,32)(25,58)(26,34)(27,60)(28,36)(29,75)(31,77)(33,79)(35,81)(37,83)(39,71)(41,73)(43,87)(45,89)(47,91)(49,93)(51,95)(53,97)(55,85)(57,78)(59,80)(61,82)(63,84)(65,72)(67,74)(69,76), (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)(10,84)(11,83)(12,82)(13,81)(14,80)(15,103)(16,102)(17,101)(18,100)(19,99)(20,112)(21,111)(22,110)(23,109)(24,108)(25,107)(26,106)(27,105)(28,104)(29,95)(30,94)(31,93)(32,92)(33,91)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,98)(41,97)(42,96)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63) );
G=PermutationGroup([(1,66,47,73,107,41,91,19),(2,67,48,74,108,42,92,20),(3,68,49,75,109,29,93,21),(4,69,50,76,110,30,94,22),(5,70,51,77,111,31,95,23),(6,57,52,78,112,32,96,24),(7,58,53,79,99,33,97,25),(8,59,54,80,100,34,98,26),(9,60,55,81,101,35,85,27),(10,61,56,82,102,36,86,28),(11,62,43,83,103,37,87,15),(12,63,44,84,104,38,88,16),(13,64,45,71,105,39,89,17),(14,65,46,72,106,40,90,18)], [(2,108),(4,110),(6,112),(8,100),(10,102),(12,104),(14,106),(15,62),(16,38),(17,64),(18,40),(19,66),(20,42),(21,68),(22,30),(23,70),(24,32),(25,58),(26,34),(27,60),(28,36),(29,75),(31,77),(33,79),(35,81),(37,83),(39,71),(41,73),(43,87),(45,89),(47,91),(49,93),(51,95),(53,97),(55,85),(57,78),(59,80),(61,82),(63,84),(65,72),(67,74),(69,76)], [(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),(10,84),(11,83),(12,82),(13,81),(14,80),(15,103),(16,102),(17,101),(18,100),(19,99),(20,112),(21,111),(22,110),(23,109),(24,108),(25,107),(26,106),(27,105),(28,104),(29,95),(30,94),(31,93),(32,92),(33,91),(34,90),(35,89),(36,88),(37,87),(38,86),(39,85),(40,98),(41,97),(42,96),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57),(49,70),(50,69),(51,68),(52,67),(53,66),(54,65),(55,64),(56,63)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 82 | 0 | 31 | 0 |
| 0 | 82 | 0 | 31 |
| 82 | 0 | 82 | 0 |
| 0 | 82 | 0 | 82 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 0 | 94 | 19 |
| 0 | 0 | 94 | 13 |
| 19 | 94 | 0 | 0 |
| 19 | 100 | 0 | 0 |
| 16 | 16 | 97 | 97 |
| 37 | 97 | 76 | 16 |
| 97 | 97 | 97 | 97 |
| 76 | 16 | 76 | 16 |
G:=sub<GL(4,GF(113))| [82,0,82,0,0,82,0,82,31,0,82,0,0,31,0,82],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[0,0,19,19,0,0,94,100,94,94,0,0,19,13,0,0],[16,37,97,76,16,97,97,16,97,76,97,76,97,16,97,16] >;
64 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 | 14E | 14F | 14G | ··· | 14L | 28A | ··· | 28F | 28G | 28H | 28I | 28J | ··· | 28O | 56A | ··· | 56L |
| 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 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | D4○D8 | D4×D7 | D4×D7 | D8⋊15D14 |
| kernel | D8⋊15D14 | D28.2C4 | C2×D56 | D7×D8 | D56⋊C2 | Q8.D14 | D4⋊D14 | C7×C4○D8 | D4⋊8D14 | Dic14 | D28 | C7⋊D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 3 | 3 | 3 | 6 | 3 | 6 | 2 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
D_8\rtimes_{15}D_{14} % in TeX
G:=Group("D8:15D14"); // GroupNames label
G:=SmallGroup(448,1222);
// by ID
G=gap.SmallGroup(448,1222);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,387,570,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations