direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8.26D4, D8⋊4C28, Q16⋊4C28, SD16⋊2C28, C56.105D4, C4≀C2⋊6C14, C8○D4⋊7C14, (C7×D8)⋊10C4, C8.6(C2×C28), C8⋊C4⋊3C14, C8.25(C7×D4), C56.47(C2×C4), (C7×Q16)⋊10C4, (C7×SD16)⋊6C4, C4○D8.3C14, D4.4(C2×C28), C4.83(D4×C14), C2.19(D4×C28), Q8.4(C2×C28), C8.C4⋊4C14, C14.121(C4×D4), C28.488(C2×D4), C42.11(C2×C14), C4.16(C22×C28), (C2×C28).911C23, (C2×C56).271C22, (C4×C28).252C22, C28.161(C22×C4), M4(2).12(C2×C14), (C7×M4(2)).46C22, (C7×C4≀C2)⋊14C2, (C7×C8○D4)⋊16C2, (C7×C4○D8).8C2, (C7×C8⋊C4)⋊12C2, (C2×C8).54(C2×C14), C4○D4.9(C2×C14), (C7×D4).21(C2×C4), (C7×Q8).22(C2×C4), (C7×C8.C4)⋊13C2, C22.2(C7×C4○D4), (C2×C14).50(C4○D4), (C2×C4).86(C22×C14), (C7×C4○D4).54C22, SmallGroup(448,852)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8.26D4
G = < a,b,c,d | a7=b8=c4=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b2c-1 >
Subgroups: 154 in 104 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C14, C14, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C28, C28, C2×C14, C2×C14, C8⋊C4, C4≀C2, C8.C4, C8○D4, C4○D8, C56, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C8.26D4, C4×C28, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C7×D8, C7×SD16, C7×Q16, C7×C4○D4, C7×C8⋊C4, C7×C4≀C2, C7×C8.C4, C7×C8○D4, C7×C4○D8, C7×C8.26D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22×C4, C2×D4, C4○D4, C28, C2×C14, C4×D4, C2×C28, C7×D4, C22×C14, C8.26D4, C22×C28, D4×C14, C7×C4○D4, D4×C28, C7×C8.26D4
►On 112 points
(1 59 38 55 23 47 11)(2 60 39 56 24 48 12)(3 61 40 49 17 41 13)(4 62 33 50 18 42 14)(5 63 34 51 19 43 15)(6 64 35 52 20 44 16)(7 57 36 53 21 45 9)(8 58 37 54 22 46 10)(25 76 105 96 85 68 97)(26 77 106 89 86 69 98)(27 78 107 90 87 70 99)(28 79 108 91 88 71 100)(29 80 109 92 81 72 101)(30 73 110 93 82 65 102)(31 74 111 94 83 66 103)(32 75 112 95 84 67 104)
(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 6)(4 8)(10 14)(12 16)(18 22)(20 24)(25 27 29 31)(26 32 30 28)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(58 62)(60 64)(65 71 69 67)(66 68 70 72)(73 79 77 75)(74 76 78 80)(81 83 85 87)(82 88 86 84)(89 95 93 91)(90 92 94 96)(97 99 101 103)(98 104 102 100)(105 107 109 111)(106 112 110 108)
(1 103 3 97 5 99 7 101)(2 100 4 102 6 104 8 98)(9 72 11 66 13 68 15 70)(10 69 12 71 14 65 16 67)(17 96 19 90 21 92 23 94)(18 93 20 95 22 89 24 91)(25 63 27 57 29 59 31 61)(26 60 28 62 30 64 32 58)(33 73 35 75 37 77 39 79)(34 78 36 80 38 74 40 76)(41 85 43 87 45 81 47 83)(42 82 44 84 46 86 48 88)(49 105 51 107 53 109 55 111)(50 110 52 112 54 106 56 108)
G:=sub<Sym(112)| (1,59,38,55,23,47,11)(2,60,39,56,24,48,12)(3,61,40,49,17,41,13)(4,62,33,50,18,42,14)(5,63,34,51,19,43,15)(6,64,35,52,20,44,16)(7,57,36,53,21,45,9)(8,58,37,54,22,46,10)(25,76,105,96,85,68,97)(26,77,106,89,86,69,98)(27,78,107,90,87,70,99)(28,79,108,91,88,71,100)(29,80,109,92,81,72,101)(30,73,110,93,82,65,102)(31,74,111,94,83,66,103)(32,75,112,95,84,67,104), (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(25,27,29,31)(26,32,30,28)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64)(65,71,69,67)(66,68,70,72)(73,79,77,75)(74,76,78,80)(81,83,85,87)(82,88,86,84)(89,95,93,91)(90,92,94,96)(97,99,101,103)(98,104,102,100)(105,107,109,111)(106,112,110,108), (1,103,3,97,5,99,7,101)(2,100,4,102,6,104,8,98)(9,72,11,66,13,68,15,70)(10,69,12,71,14,65,16,67)(17,96,19,90,21,92,23,94)(18,93,20,95,22,89,24,91)(25,63,27,57,29,59,31,61)(26,60,28,62,30,64,32,58)(33,73,35,75,37,77,39,79)(34,78,36,80,38,74,40,76)(41,85,43,87,45,81,47,83)(42,82,44,84,46,86,48,88)(49,105,51,107,53,109,55,111)(50,110,52,112,54,106,56,108)>;
G:=Group( (1,59,38,55,23,47,11)(2,60,39,56,24,48,12)(3,61,40,49,17,41,13)(4,62,33,50,18,42,14)(5,63,34,51,19,43,15)(6,64,35,52,20,44,16)(7,57,36,53,21,45,9)(8,58,37,54,22,46,10)(25,76,105,96,85,68,97)(26,77,106,89,86,69,98)(27,78,107,90,87,70,99)(28,79,108,91,88,71,100)(29,80,109,92,81,72,101)(30,73,110,93,82,65,102)(31,74,111,94,83,66,103)(32,75,112,95,84,67,104), (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(25,27,29,31)(26,32,30,28)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64)(65,71,69,67)(66,68,70,72)(73,79,77,75)(74,76,78,80)(81,83,85,87)(82,88,86,84)(89,95,93,91)(90,92,94,96)(97,99,101,103)(98,104,102,100)(105,107,109,111)(106,112,110,108), (1,103,3,97,5,99,7,101)(2,100,4,102,6,104,8,98)(9,72,11,66,13,68,15,70)(10,69,12,71,14,65,16,67)(17,96,19,90,21,92,23,94)(18,93,20,95,22,89,24,91)(25,63,27,57,29,59,31,61)(26,60,28,62,30,64,32,58)(33,73,35,75,37,77,39,79)(34,78,36,80,38,74,40,76)(41,85,43,87,45,81,47,83)(42,82,44,84,46,86,48,88)(49,105,51,107,53,109,55,111)(50,110,52,112,54,106,56,108) );
G=PermutationGroup([[(1,59,38,55,23,47,11),(2,60,39,56,24,48,12),(3,61,40,49,17,41,13),(4,62,33,50,18,42,14),(5,63,34,51,19,43,15),(6,64,35,52,20,44,16),(7,57,36,53,21,45,9),(8,58,37,54,22,46,10),(25,76,105,96,85,68,97),(26,77,106,89,86,69,98),(27,78,107,90,87,70,99),(28,79,108,91,88,71,100),(29,80,109,92,81,72,101),(30,73,110,93,82,65,102),(31,74,111,94,83,66,103),(32,75,112,95,84,67,104)], [(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)], [(2,6),(4,8),(10,14),(12,16),(18,22),(20,24),(25,27,29,31),(26,32,30,28),(33,37),(35,39),(42,46),(44,48),(50,54),(52,56),(58,62),(60,64),(65,71,69,67),(66,68,70,72),(73,79,77,75),(74,76,78,80),(81,83,85,87),(82,88,86,84),(89,95,93,91),(90,92,94,96),(97,99,101,103),(98,104,102,100),(105,107,109,111),(106,112,110,108)], [(1,103,3,97,5,99,7,101),(2,100,4,102,6,104,8,98),(9,72,11,66,13,68,15,70),(10,69,12,71,14,65,16,67),(17,96,19,90,21,92,23,94),(18,93,20,95,22,89,24,91),(25,63,27,57,29,59,31,61),(26,60,28,62,30,64,32,58),(33,73,35,75,37,77,39,79),(34,78,36,80,38,74,40,76),(41,85,43,87,45,81,47,83),(42,82,44,84,46,86,48,88),(49,105,51,107,53,109,55,111),(50,110,52,112,54,106,56,108)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14X | 28A | ··· | 28L | 28M | ··· | 28R | 28S | ··· | 28AP | 56A | ··· | 56X | 56Y | ··· | 56BH |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C7 | C14 | C14 | C14 | C14 | C14 | C28 | C28 | C28 | D4 | C4○D4 | C7×D4 | C7×C4○D4 | C8.26D4 | C7×C8.26D4 |
| kernel | C7×C8.26D4 | C7×C8⋊C4 | C7×C4≀C2 | C7×C8.C4 | C7×C8○D4 | C7×C4○D8 | C7×D8 | C7×SD16 | C7×Q16 | C8.26D4 | C8⋊C4 | C4≀C2 | C8.C4 | C8○D4 | C4○D8 | D8 | SD16 | Q16 | C56 | C2×C14 | C8 | C22 | C7 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 6 | 6 | 12 | 6 | 12 | 6 | 12 | 24 | 12 | 2 | 2 | 12 | 12 | 2 | 12 |
Matrix representation of C7×C8.26D4 ►in GL4(𝔽113) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 98 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 15 |
| 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],[0,98,0,0,1,0,0,0,0,0,0,98,0,0,1,0],[1,0,0,0,0,112,0,0,0,0,98,0,0,0,0,15],[0,0,1,0,0,0,0,112,98,0,0,0,0,15,0,0] >;
C7×C8.26D4 in GAP, Magma, Sage, TeX
C_7\times C_8._{26}D_4 % in TeX
G:=Group("C7xC8.26D4"); // GroupNames label
G:=SmallGroup(448,852);
// by ID
G=gap.SmallGroup(448,852);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,4790,604,9804,4911,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^2*c^-1>;
// generators/relations