direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D4○D8, C28.85C24, C56.52C23, 2+ (1+4)⋊3C14, C4○D8⋊4C14, C8○D4⋊3C14, D8⋊7(C2×C14), (C14×D8)⋊26C2, (C2×D8)⋊12C14, C8⋊C22⋊4C14, Q16⋊7(C2×C14), (C7×D4).45D4, D4.11(C7×D4), C4.45(D4×C14), (C7×Q8).45D4, Q8.11(C7×D4), (C2×C56)⋊31C22, SD16⋊4(C2×C14), C28.406(C2×D4), (C7×D8)⋊21C22, C4.8(C23×C14), C22.7(D4×C14), (D4×C14)⋊40C22, M4(2)⋊6(C2×C14), C8.10(C22×C14), (C7×Q16)⋊21C22, D4.5(C22×C14), (C7×D4).38C23, Q8.5(C22×C14), (C7×Q8).39C23, (C2×C28).687C23, (C7×SD16)⋊20C22, C14.206(C22×D4), (C7×2+ (1+4))⋊9C2, (C7×M4(2))⋊32C22, (C2×C8)⋊4(C2×C14), C2.30(D4×C2×C14), (C7×C8○D4)⋊12C2, C4○D4⋊1(C2×C14), (C7×C4○D8)⋊11C2, (C2×D4)⋊7(C2×C14), (C7×C8⋊C22)⋊11C2, (C2×C14).184(C2×D4), (C7×C4○D4)⋊14C22, (C2×C4).48(C22×C14), SmallGroup(448,1359)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 474 in 268 conjugacy classes, 158 normal (18 characteristic)
C1, C2, C2 [×9], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C7, C8, C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×9], D4 [×12], Q8, Q8 [×2], C23 [×6], C14, C14 [×9], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×6], C2×D4 [×6], C4○D4, C4○D4 [×6], C4○D4 [×2], C28, C28 [×3], C28 [×2], C2×C14 [×3], C2×C14 [×12], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ (1+4) [×2], C56, C56 [×3], C2×C28 [×3], C2×C28 [×6], C7×D4 [×9], C7×D4 [×12], C7×Q8, C7×Q8 [×2], C22×C14 [×6], D4○D8, C2×C56 [×3], C7×M4(2) [×3], C7×D8 [×9], C7×SD16 [×6], C7×Q16, D4×C14 [×6], D4×C14 [×6], C7×C4○D4, C7×C4○D4 [×6], C7×C4○D4 [×2], C7×C8○D4, C14×D8 [×3], C7×C4○D8 [×3], C7×C8⋊C22 [×6], C7×2+ (1+4) [×2], C7×D4○D8
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C24, C2×C14 [×35], C22×D4, C7×D4 [×4], C22×C14 [×15], D4○D8, D4×C14 [×6], C23×C14, D4×C2×C14, C7×D4○D8
Generators and relations
G = < a,b,c,d,e | a7=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >
►On 112 points
Generators in S112
(1 82 58 109 99 31 35)(2 83 59 110 100 32 36)(3 84 60 111 101 25 37)(4 85 61 112 102 26 38)(5 86 62 105 103 27 39)(6 87 63 106 104 28 40)(7 88 64 107 97 29 33)(8 81 57 108 98 30 34)(9 46 76 17 54 95 68)(10 47 77 18 55 96 69)(11 48 78 19 56 89 70)(12 41 79 20 49 90 71)(13 42 80 21 50 91 72)(14 43 73 22 51 92 65)(15 44 74 23 52 93 66)(16 45 75 24 53 94 67)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)(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 83 85 87)(82 84 86 88)(89 95 93 91)(90 96 94 92)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)
(1 46)(2 47)(3 48)(4 41)(5 42)(6 43)(7 44)(8 45)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 33)(16 34)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 57)(25 70)(26 71)(27 72)(28 65)(29 66)(30 67)(31 68)(32 69)(49 112)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(73 87)(74 88)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(89 101)(90 102)(91 103)(92 104)(93 97)(94 98)(95 99)(96 100)
(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 44)(2 43)(3 42)(4 41)(5 48)(6 47)(7 46)(8 45)(9 33)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 64)(18 63)(19 62)(20 61)(21 60)(22 59)(23 58)(24 57)(25 72)(26 71)(27 70)(28 69)(29 68)(30 67)(31 66)(32 65)(49 112)(50 111)(51 110)(52 109)(53 108)(54 107)(55 106)(56 105)(73 83)(74 82)(75 81)(76 88)(77 87)(78 86)(79 85)(80 84)(89 103)(90 102)(91 101)(92 100)(93 99)(94 98)(95 97)(96 104)
G:=sub<Sym(112)| (1,82,58,109,99,31,35)(2,83,59,110,100,32,36)(3,84,60,111,101,25,37)(4,85,61,112,102,26,38)(5,86,62,105,103,27,39)(6,87,63,106,104,28,40)(7,88,64,107,97,29,33)(8,81,57,108,98,30,34)(9,46,76,17,54,95,68)(10,47,77,18,55,96,69)(11,48,78,19,56,89,70)(12,41,79,20,49,90,71)(13,42,80,21,50,91,72)(14,43,73,22,51,92,65)(15,44,74,23,52,93,66)(16,45,75,24,53,94,67), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,27,29,31)(26,28,30,32)(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)(81,83,85,87)(82,84,86,88)(89,95,93,91)(90,96,94,92)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112), (1,46)(2,47)(3,48)(4,41)(5,42)(6,43)(7,44)(8,45)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,33)(16,34)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,57)(25,70)(26,71)(27,72)(28,65)(29,66)(30,67)(31,68)(32,69)(49,112)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(73,87)(74,88)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(89,101)(90,102)(91,103)(92,104)(93,97)(94,98)(95,99)(96,100), (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), (1,44)(2,43)(3,42)(4,41)(5,48)(6,47)(7,46)(8,45)(9,33)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(49,112)(50,111)(51,110)(52,109)(53,108)(54,107)(55,106)(56,105)(73,83)(74,82)(75,81)(76,88)(77,87)(78,86)(79,85)(80,84)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(96,104)>;
G:=Group( (1,82,58,109,99,31,35)(2,83,59,110,100,32,36)(3,84,60,111,101,25,37)(4,85,61,112,102,26,38)(5,86,62,105,103,27,39)(6,87,63,106,104,28,40)(7,88,64,107,97,29,33)(8,81,57,108,98,30,34)(9,46,76,17,54,95,68)(10,47,77,18,55,96,69)(11,48,78,19,56,89,70)(12,41,79,20,49,90,71)(13,42,80,21,50,91,72)(14,43,73,22,51,92,65)(15,44,74,23,52,93,66)(16,45,75,24,53,94,67), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,27,29,31)(26,28,30,32)(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)(81,83,85,87)(82,84,86,88)(89,95,93,91)(90,96,94,92)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112), (1,46)(2,47)(3,48)(4,41)(5,42)(6,43)(7,44)(8,45)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,33)(16,34)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,57)(25,70)(26,71)(27,72)(28,65)(29,66)(30,67)(31,68)(32,69)(49,112)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(73,87)(74,88)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(89,101)(90,102)(91,103)(92,104)(93,97)(94,98)(95,99)(96,100), (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), (1,44)(2,43)(3,42)(4,41)(5,48)(6,47)(7,46)(8,45)(9,33)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(49,112)(50,111)(51,110)(52,109)(53,108)(54,107)(55,106)(56,105)(73,83)(74,82)(75,81)(76,88)(77,87)(78,86)(79,85)(80,84)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(96,104) );
G=PermutationGroup([(1,82,58,109,99,31,35),(2,83,59,110,100,32,36),(3,84,60,111,101,25,37),(4,85,61,112,102,26,38),(5,86,62,105,103,27,39),(6,87,63,106,104,28,40),(7,88,64,107,97,29,33),(8,81,57,108,98,30,34),(9,46,76,17,54,95,68),(10,47,77,18,55,96,69),(11,48,78,19,56,89,70),(12,41,79,20,49,90,71),(13,42,80,21,50,91,72),(14,43,73,22,51,92,65),(15,44,74,23,52,93,66),(16,45,75,24,53,94,67)], [(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,27,29,31),(26,28,30,32),(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),(81,83,85,87),(82,84,86,88),(89,95,93,91),(90,96,94,92),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112)], [(1,46),(2,47),(3,48),(4,41),(5,42),(6,43),(7,44),(8,45),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,33),(16,34),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,57),(25,70),(26,71),(27,72),(28,65),(29,66),(30,67),(31,68),(32,69),(49,112),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(73,87),(74,88),(75,81),(76,82),(77,83),(78,84),(79,85),(80,86),(89,101),(90,102),(91,103),(92,104),(93,97),(94,98),(95,99),(96,100)], [(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)], [(1,44),(2,43),(3,42),(4,41),(5,48),(6,47),(7,46),(8,45),(9,33),(10,40),(11,39),(12,38),(13,37),(14,36),(15,35),(16,34),(17,64),(18,63),(19,62),(20,61),(21,60),(22,59),(23,58),(24,57),(25,72),(26,71),(27,70),(28,69),(29,68),(30,67),(31,66),(32,65),(49,112),(50,111),(51,110),(52,109),(53,108),(54,107),(55,106),(56,105),(73,83),(74,82),(75,81),(76,88),(77,87),(78,86),(79,85),(80,84),(89,103),(90,102),(91,101),(92,100),(93,99),(94,98),(95,97),(96,104)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 |
| 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 82 | 31 | 0 | 0 |
| 82 | 82 | 0 | 0 |
| 0 | 0 | 82 | 31 |
| 0 | 0 | 82 | 82 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
G:=sub<GL(4,GF(113))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,1,0,0,112,0,0,0,0,0,0,112,0,0,1,0],[0,0,0,1,0,0,112,0,0,112,0,0,1,0,0,0],[82,82,0,0,31,82,0,0,0,0,82,82,0,0,31,82],[0,0,1,0,0,0,0,112,1,0,0,0,0,112,0,0] >;
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 14A | ··· | 14F | 14G | ··· | 14X | 14Y | ··· | 14BH | 28A | ··· | 28X | 28Y | ··· | 28AJ | 56A | ··· | 56L | 56M | ··· | 56AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | C7×D4 | C7×D4 | D4○D8 | C7×D4○D8 |
| kernel | C7×D4○D8 | C7×C8○D4 | C14×D8 | C7×C4○D8 | C7×C8⋊C22 | C7×2+ (1+4) | D4○D8 | C8○D4 | C2×D8 | C4○D8 | C8⋊C22 | 2+ (1+4) | C7×D4 | C7×Q8 | D4 | Q8 | C7 | C1 |
| # reps | 1 | 1 | 3 | 3 | 6 | 2 | 6 | 6 | 18 | 18 | 36 | 12 | 3 | 1 | 18 | 6 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_7\times D_4\circ D_8
% in TeX
G:=Group("C7xD4oD8"); // GroupNames label
G:=SmallGroup(448,1359);
// by ID
G=gap.SmallGroup(448,1359);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,1641,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=e^2=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀