direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D4○SD16, C56.53C23, C28.86C24, 2+ 1+4⋊4C14, 2- 1+4⋊2C14, C4○D8⋊5C14, C8○D4⋊4C14, D8⋊5(C2×C14), C8⋊C22⋊5C14, Q16⋊5(C2×C14), (C7×D4).46D4, D4.12(C7×D4), C4.46(D4×C14), Q8.12(C7×D4), (C7×Q8).46D4, (C2×C56)⋊32C22, (C2×SD16)⋊6C14, SD16⋊6(C2×C14), C28.407(C2×D4), (C7×D8)⋊22C22, C8.C22⋊4C14, C4.9(C23×C14), C22.8(D4×C14), (C14×SD16)⋊17C2, M4(2)⋊7(C2×C14), C8.13(C22×C14), (C7×Q16)⋊19C22, (Q8×C14)⋊32C22, D4.6(C22×C14), (C7×D4).39C23, (C7×Q8).40C23, Q8.6(C22×C14), (C2×C28).688C23, (C7×SD16)⋊21C22, C14.207(C22×D4), (C7×2- 1+4)⋊7C2, (D4×C14).226C22, (C7×M4(2))⋊33C22, (C7×2+ 1+4)⋊10C2, (C2×C8)⋊5(C2×C14), C2.31(D4×C2×C14), (C7×C4○D8)⋊12C2, (C7×C8○D4)⋊13C2, C4○D4⋊2(C2×C14), (C2×Q8)⋊7(C2×C14), (C7×C8⋊C22)⋊12C2, (C2×D4).39(C2×C14), (C2×C14).185(C2×D4), (C7×C4○D4)⋊15C22, (C7×C8.C22)⋊11C2, (C2×C4).49(C22×C14), SmallGroup(448,1360)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4○SD16
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=d3 >
Subgroups: 410 in 258 conjugacy classes, 158 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, Q8, C23, C14, C14, C2×C8, M4(2), D8, SD16, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, C28, C28, C28, C2×C14, C2×C14, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×D4, C7×Q8, C7×Q8, C7×Q8, C22×C14, D4○SD16, C2×C56, C7×M4(2), C7×D8, C7×SD16, C7×SD16, C7×Q16, D4×C14, D4×C14, Q8×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C7×C4○D4, C7×C8○D4, C14×SD16, C7×C4○D8, C7×C8⋊C22, C7×C8.C22, C7×2+ 1+4, C7×2- 1+4, C7×D4○SD16
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C22×D4, C7×D4, C22×C14, D4○SD16, D4×C14, C23×C14, D4×C2×C14, C7×D4○SD16
►On 112 points
(1 77 42 20 91 69 34)(2 78 43 21 92 70 35)(3 79 44 22 93 71 36)(4 80 45 23 94 72 37)(5 73 46 24 95 65 38)(6 74 47 17 96 66 39)(7 75 48 18 89 67 40)(8 76 41 19 90 68 33)(9 112 82 55 25 104 59)(10 105 83 56 26 97 60)(11 106 84 49 27 98 61)(12 107 85 50 28 99 62)(13 108 86 51 29 100 63)(14 109 87 52 30 101 64)(15 110 88 53 31 102 57)(16 111 81 54 32 103 58)
(1 99 5 103)(2 100 6 104)(3 101 7 97)(4 102 8 98)(9 43 13 47)(10 44 14 48)(11 45 15 41)(12 46 16 42)(17 112 21 108)(18 105 22 109)(19 106 23 110)(20 107 24 111)(25 35 29 39)(26 36 30 40)(27 37 31 33)(28 38 32 34)(49 72 53 68)(50 65 54 69)(51 66 55 70)(52 67 56 71)(57 76 61 80)(58 77 62 73)(59 78 63 74)(60 79 64 75)(81 91 85 95)(82 92 86 96)(83 93 87 89)(84 94 88 90)
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(81 85)(82 86)(83 87)(84 88)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 19)(18 22)(21 23)(25 27)(26 30)(29 31)(33 39)(35 37)(36 40)(41 47)(43 45)(44 48)(49 55)(51 53)(52 56)(57 63)(59 61)(60 64)(66 68)(67 71)(70 72)(74 76)(75 79)(78 80)(82 84)(83 87)(86 88)(89 93)(90 96)(92 94)(97 101)(98 104)(100 102)(105 109)(106 112)(108 110)
G:=sub<Sym(112)| (1,77,42,20,91,69,34)(2,78,43,21,92,70,35)(3,79,44,22,93,71,36)(4,80,45,23,94,72,37)(5,73,46,24,95,65,38)(6,74,47,17,96,66,39)(7,75,48,18,89,67,40)(8,76,41,19,90,68,33)(9,112,82,55,25,104,59)(10,105,83,56,26,97,60)(11,106,84,49,27,98,61)(12,107,85,50,28,99,62)(13,108,86,51,29,100,63)(14,109,87,52,30,101,64)(15,110,88,53,31,102,57)(16,111,81,54,32,103,58), (1,99,5,103)(2,100,6,104)(3,101,7,97)(4,102,8,98)(9,43,13,47)(10,44,14,48)(11,45,15,41)(12,46,16,42)(17,112,21,108)(18,105,22,109)(19,106,23,110)(20,107,24,111)(25,35,29,39)(26,36,30,40)(27,37,31,33)(28,38,32,34)(49,72,53,68)(50,65,54,69)(51,66,55,70)(52,67,56,71)(57,76,61,80)(58,77,62,73)(59,78,63,74)(60,79,64,75)(81,91,85,95)(82,92,86,96)(83,93,87,89)(84,94,88,90), (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(81,85)(82,86)(83,87)(84,88)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,19)(18,22)(21,23)(25,27)(26,30)(29,31)(33,39)(35,37)(36,40)(41,47)(43,45)(44,48)(49,55)(51,53)(52,56)(57,63)(59,61)(60,64)(66,68)(67,71)(70,72)(74,76)(75,79)(78,80)(82,84)(83,87)(86,88)(89,93)(90,96)(92,94)(97,101)(98,104)(100,102)(105,109)(106,112)(108,110)>;
G:=Group( (1,77,42,20,91,69,34)(2,78,43,21,92,70,35)(3,79,44,22,93,71,36)(4,80,45,23,94,72,37)(5,73,46,24,95,65,38)(6,74,47,17,96,66,39)(7,75,48,18,89,67,40)(8,76,41,19,90,68,33)(9,112,82,55,25,104,59)(10,105,83,56,26,97,60)(11,106,84,49,27,98,61)(12,107,85,50,28,99,62)(13,108,86,51,29,100,63)(14,109,87,52,30,101,64)(15,110,88,53,31,102,57)(16,111,81,54,32,103,58), (1,99,5,103)(2,100,6,104)(3,101,7,97)(4,102,8,98)(9,43,13,47)(10,44,14,48)(11,45,15,41)(12,46,16,42)(17,112,21,108)(18,105,22,109)(19,106,23,110)(20,107,24,111)(25,35,29,39)(26,36,30,40)(27,37,31,33)(28,38,32,34)(49,72,53,68)(50,65,54,69)(51,66,55,70)(52,67,56,71)(57,76,61,80)(58,77,62,73)(59,78,63,74)(60,79,64,75)(81,91,85,95)(82,92,86,96)(83,93,87,89)(84,94,88,90), (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(81,85)(82,86)(83,87)(84,88)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,19)(18,22)(21,23)(25,27)(26,30)(29,31)(33,39)(35,37)(36,40)(41,47)(43,45)(44,48)(49,55)(51,53)(52,56)(57,63)(59,61)(60,64)(66,68)(67,71)(70,72)(74,76)(75,79)(78,80)(82,84)(83,87)(86,88)(89,93)(90,96)(92,94)(97,101)(98,104)(100,102)(105,109)(106,112)(108,110) );
G=PermutationGroup([[(1,77,42,20,91,69,34),(2,78,43,21,92,70,35),(3,79,44,22,93,71,36),(4,80,45,23,94,72,37),(5,73,46,24,95,65,38),(6,74,47,17,96,66,39),(7,75,48,18,89,67,40),(8,76,41,19,90,68,33),(9,112,82,55,25,104,59),(10,105,83,56,26,97,60),(11,106,84,49,27,98,61),(12,107,85,50,28,99,62),(13,108,86,51,29,100,63),(14,109,87,52,30,101,64),(15,110,88,53,31,102,57),(16,111,81,54,32,103,58)], [(1,99,5,103),(2,100,6,104),(3,101,7,97),(4,102,8,98),(9,43,13,47),(10,44,14,48),(11,45,15,41),(12,46,16,42),(17,112,21,108),(18,105,22,109),(19,106,23,110),(20,107,24,111),(25,35,29,39),(26,36,30,40),(27,37,31,33),(28,38,32,34),(49,72,53,68),(50,65,54,69),(51,66,55,70),(52,67,56,71),(57,76,61,80),(58,77,62,73),(59,78,63,74),(60,79,64,75),(81,91,85,95),(82,92,86,96),(83,93,87,89),(84,94,88,90)], [(9,13),(10,14),(11,15),(12,16),(25,29),(26,30),(27,31),(28,32),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(81,85),(82,86),(83,87),(84,88),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112)], [(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)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,19),(18,22),(21,23),(25,27),(26,30),(29,31),(33,39),(35,37),(36,40),(41,47),(43,45),(44,48),(49,55),(51,53),(52,56),(57,63),(59,61),(60,64),(66,68),(67,71),(70,72),(74,76),(75,79),(78,80),(82,84),(83,87),(86,88),(89,93),(90,96),(92,94),(97,101),(98,104),(100,102),(105,109),(106,112),(108,110)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 14A | ··· | 14F | 14G | ··· | 14X | 14Y | ··· | 14AV | 28A | ··· | 28X | 28Y | ··· | 28AV | 56A | ··· | 56L | 56M | ··· | 56AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | C7×D4 | C7×D4 | D4○SD16 | C7×D4○SD16 |
| kernel | C7×D4○SD16 | C7×C8○D4 | C14×SD16 | C7×C4○D8 | C7×C8⋊C22 | C7×C8.C22 | C7×2+ 1+4 | C7×2- 1+4 | D4○SD16 | C8○D4 | C2×SD16 | C4○D8 | C8⋊C22 | C8.C22 | 2+ 1+4 | 2- 1+4 | C7×D4 | C7×Q8 | D4 | Q8 | C7 | C1 |
| # reps | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 6 | 6 | 18 | 18 | 18 | 18 | 6 | 6 | 3 | 1 | 18 | 6 | 2 | 12 |
Matrix representation of C7×D4○SD16 ►in GL4(𝔽113) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 112 | 0 |
| 112 | 1 | 112 | 2 |
| 1 | 0 | 0 | 0 |
| 1 | 112 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 1 | 112 | 0 | 112 |
| 100 | 13 | 0 | 0 |
| 100 | 100 | 0 | 0 |
| 13 | 100 | 0 | 87 |
| 13 | 0 | 13 | 87 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 1 | 112 |
G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[0,112,1,1,0,1,0,112,112,112,0,0,0,2,0,112],[1,0,0,1,0,1,0,112,0,0,112,0,0,0,0,112],[100,100,13,13,13,100,100,0,0,0,0,13,0,0,87,87],[1,0,0,1,0,112,0,0,0,0,1,1,0,0,0,112] >;
C7×D4○SD16 in GAP, Magma, Sage, TeX
C_7\times D_4\circ {\rm SD}_{16} % in TeX
G:=Group("C7xD4oSD16"); // GroupNames label
G:=SmallGroup(448,1360);
// by ID
G=gap.SmallGroup(448,1360);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,1641,14117,7068,124]);
// 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=d^3>;
// generators/relations