direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×C4○D8, D8⋊14D14, Q16⋊12D14, SD16⋊14D14, D56⋊18C22, C56.45C23, C28.14C24, D28.9C23, Dic28⋊16C22, Dic14.9C23, (D7×D8)⋊8C2, C4○D4⋊7D14, (C2×C8)⋊27D14, (D7×Q16)⋊8C2, D8⋊3D7⋊8C2, (C2×C56)⋊4C22, (D7×SD16)⋊7C2, (C4×D7).54D4, C4.221(D4×D7), C7⋊C8.24C23, Q8.D14⋊8C2, D56⋊7C2⋊6C2, C22.4(D4×D7), D4⋊D7⋊12C22, D14.66(C2×D4), C28.380(C2×D4), C4○D28⋊5C22, (C8×D7)⋊16C22, (C7×D8)⋊12C22, Q8⋊D7⋊11C22, (C7×D4).8C23, D4.8(C22×D7), (D4×D7).6C22, C4.14(C23×D7), C8.42(C22×D7), SD16⋊3D7⋊7C2, D4.8D14⋊1C2, (C7×Q8).8C23, Q8.8(C22×D7), (Q8×D7).5C22, D4⋊2D7⋊8C22, C56⋊C2⋊20C22, D4.D7⋊11C22, Dic7.71(C2×D4), Q8⋊2D7⋊8C22, (C7×Q16)⋊10C22, C7⋊Q16⋊10C22, (C4×D7).29C23, (C22×D7).64D4, (C2×C28).531C23, (C2×Dic7).124D4, (C7×SD16)⋊15C22, C14.115(C22×D4), (D7×C2×C8)⋊1C2, C7⋊5(C2×C4○D8), C2.88(C2×D4×D7), (C7×C4○D8)⋊2C2, (D7×C4○D4)⋊1C2, (C2×C7⋊C8)⋊37C22, (C2×C14).11(C2×D4), (C7×C4○D4)⋊1C22, (C2×C4×D7).261C22, (C2×C4).618(C22×D7), SmallGroup(448,1220)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×C4○D8
G = < a,b,c,d,e | a7=b2=c4=e2=1, d4=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d3 >
Subgroups: 1332 in 266 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, D7, C14, C14, C2×C8, C2×C8, D8, D8, SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, C2×C4○D8, C8×D7, C56⋊C2, D56, Dic28, C2×C7⋊C8, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C2×C56, C7×D8, C7×SD16, C7×Q16, C2×C4×D7, C2×C4×D7, C4○D28, C4○D28, D4×D7, D4×D7, D4⋊2D7, D4⋊2D7, Q8×D7, Q8⋊2D7, C7×C4○D4, D7×C2×C8, D56⋊7C2, D7×D8, D8⋊3D7, D7×SD16, SD16⋊3D7, D7×Q16, Q8.D14, D4.8D14, C7×C4○D8, D7×C4○D4, D7×C4○D8
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C4○D8, C22×D4, C22×D7, C2×C4○D8, D4×D7, C23×D7, C2×D4×D7, D7×C4○D8
►On 112 points
(1 102 14 111 93 40 57)(2 103 15 112 94 33 58)(3 104 16 105 95 34 59)(4 97 9 106 96 35 60)(5 98 10 107 89 36 61)(6 99 11 108 90 37 62)(7 100 12 109 91 38 63)(8 101 13 110 92 39 64)(17 84 29 74 50 66 41)(18 85 30 75 51 67 42)(19 86 31 76 52 68 43)(20 87 32 77 53 69 44)(21 88 25 78 54 70 45)(22 81 26 79 55 71 46)(23 82 27 80 56 72 47)(24 83 28 73 49 65 48)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 96)(10 89)(11 90)(12 91)(13 92)(14 93)(15 94)(16 95)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 73)(25 88)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 103)(34 104)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 49)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 35 37 39)(34 36 38 40)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 59 61 63)(58 60 62 64)(65 71 69 67)(66 72 70 68)(73 79 77 75)(74 80 78 76)(81 87 85 83)(82 88 86 84)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)
(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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 50)(10 49)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 35)(18 34)(19 33)(20 40)(21 39)(22 38)(23 37)(24 36)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(57 87)(58 86)(59 85)(60 84)(61 83)(62 82)(63 81)(64 88)(65 107)(66 106)(67 105)(68 112)(69 111)(70 110)(71 109)(72 108)(73 98)(74 97)(75 104)(76 103)(77 102)(78 101)(79 100)(80 99)
G:=sub<Sym(112)| (1,102,14,111,93,40,57)(2,103,15,112,94,33,58)(3,104,16,105,95,34,59)(4,97,9,106,96,35,60)(5,98,10,107,89,36,61)(6,99,11,108,90,37,62)(7,100,12,109,91,38,63)(8,101,13,110,92,39,64)(17,84,29,74,50,66,41)(18,85,30,75,51,67,42)(19,86,31,76,52,68,43)(20,87,32,77,53,69,44)(21,88,25,78,54,70,45)(22,81,26,79,55,71,46)(23,82,27,80,56,72,47)(24,83,28,73,49,65,48), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,96)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,73)(25,88)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,103)(34,104)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,49), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,59,61,63)(58,60,62,64)(65,71,69,67)(66,72,70,68)(73,79,77,75)(74,80,78,76)(81,87,85,83)(82,88,86,84)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,50)(10,49)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,35)(18,34)(19,33)(20,40)(21,39)(22,38)(23,37)(24,36)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(57,87)(58,86)(59,85)(60,84)(61,83)(62,82)(63,81)(64,88)(65,107)(66,106)(67,105)(68,112)(69,111)(70,110)(71,109)(72,108)(73,98)(74,97)(75,104)(76,103)(77,102)(78,101)(79,100)(80,99)>;
G:=Group( (1,102,14,111,93,40,57)(2,103,15,112,94,33,58)(3,104,16,105,95,34,59)(4,97,9,106,96,35,60)(5,98,10,107,89,36,61)(6,99,11,108,90,37,62)(7,100,12,109,91,38,63)(8,101,13,110,92,39,64)(17,84,29,74,50,66,41)(18,85,30,75,51,67,42)(19,86,31,76,52,68,43)(20,87,32,77,53,69,44)(21,88,25,78,54,70,45)(22,81,26,79,55,71,46)(23,82,27,80,56,72,47)(24,83,28,73,49,65,48), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,96)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,73)(25,88)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,103)(34,104)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,49), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,59,61,63)(58,60,62,64)(65,71,69,67)(66,72,70,68)(73,79,77,75)(74,80,78,76)(81,87,85,83)(82,88,86,84)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,50)(10,49)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,35)(18,34)(19,33)(20,40)(21,39)(22,38)(23,37)(24,36)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(57,87)(58,86)(59,85)(60,84)(61,83)(62,82)(63,81)(64,88)(65,107)(66,106)(67,105)(68,112)(69,111)(70,110)(71,109)(72,108)(73,98)(74,97)(75,104)(76,103)(77,102)(78,101)(79,100)(80,99) );
G=PermutationGroup([[(1,102,14,111,93,40,57),(2,103,15,112,94,33,58),(3,104,16,105,95,34,59),(4,97,9,106,96,35,60),(5,98,10,107,89,36,61),(6,99,11,108,90,37,62),(7,100,12,109,91,38,63),(8,101,13,110,92,39,64),(17,84,29,74,50,66,41),(18,85,30,75,51,67,42),(19,86,31,76,52,68,43),(20,87,32,77,53,69,44),(21,88,25,78,54,70,45),(22,81,26,79,55,71,46),(23,82,27,80,56,72,47),(24,83,28,73,49,65,48)], [(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,96),(10,89),(11,90),(12,91),(13,92),(14,93),(15,94),(16,95),(17,74),(18,75),(19,76),(20,77),(21,78),(22,79),(23,80),(24,73),(25,88),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,103),(34,104),(35,97),(36,98),(37,99),(38,100),(39,101),(40,102),(41,50),(42,51),(43,52),(44,53),(45,54),(46,55),(47,56),(48,49)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,35,37,39),(34,36,38,40),(41,47,45,43),(42,48,46,44),(49,55,53,51),(50,56,54,52),(57,59,61,63),(58,60,62,64),(65,71,69,67),(66,72,70,68),(73,79,77,75),(74,80,78,76),(81,87,85,83),(82,88,86,84),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112)], [(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)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,50),(10,49),(11,56),(12,55),(13,54),(14,53),(15,52),(16,51),(17,35),(18,34),(19,33),(20,40),(21,39),(22,38),(23,37),(24,36),(41,96),(42,95),(43,94),(44,93),(45,92),(46,91),(47,90),(48,89),(57,87),(58,86),(59,85),(60,84),(61,83),(62,82),(63,81),(64,88),(65,107),(66,106),(67,105),(68,112),(69,111),(70,110),(71,109),(72,108),(73,98),(74,97),(75,104),(76,103),(77,102),(78,101),(79,100),(80,99)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 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 | 7 | 7 | 14 | 28 | 28 | 1 | 1 | 2 | 4 | 4 | 7 | 7 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | D14 | C4○D8 | D4×D7 | D4×D7 | D7×C4○D8 |
| kernel | D7×C4○D8 | D7×C2×C8 | D56⋊7C2 | D7×D8 | D8⋊3D7 | D7×SD16 | SD16⋊3D7 | D7×Q16 | Q8.D14 | D4.8D14 | C7×C4○D8 | D7×C4○D4 | C4×D7 | C2×Dic7 | C22×D7 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | D7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 3 | 6 | 3 | 6 | 8 | 3 | 3 | 12 |
Matrix representation of D7×C4○D8 ►in GL4(𝔽113) generated by
| 0 | 1 | 0 | 0 |
| 112 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 95 | 0 |
| 0 | 0 | 100 | 69 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 44 | 88 |
| 0 | 0 | 100 | 69 |
G:=sub<GL(4,GF(113))| [0,112,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,98,0,0,0,0,98],[1,0,0,0,0,1,0,0,0,0,95,100,0,0,0,69],[1,0,0,0,0,1,0,0,0,0,44,100,0,0,88,69] >;
D7×C4○D8 in GAP, Magma, Sage, TeX
D_7\times C_4\circ D_8
% in TeX
G:=Group("D7xC4oD8"); // GroupNames label
G:=SmallGroup(448,1220);
// by ID
G=gap.SmallGroup(448,1220);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,570,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^4=e^2=1,d^4=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d^3>;
// generators/relations