direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8○D8, D8⋊5C28, Q16⋊5C28, C56.77D4, SD16⋊3C28, C4≀C2⋊7C14, (C4×C56)⋊26C2, (C4×C8)⋊10C14, C8○D4⋊6C14, (C7×D8)⋊11C4, C8.30(C7×D4), C8.11(C2×C28), C56.67(C2×C4), (C7×Q16)⋊11C4, C4○D8.5C14, (C7×SD16)⋊7C4, D4.3(C2×C28), C2.18(D4×C28), C4.82(D4×C14), Q8.3(C2×C28), C8.C4⋊8C14, C14.120(C4×D4), C28.487(C2×D4), C42.73(C2×C14), C4.15(C22×C28), (C4×C28).358C22, C28.160(C22×C4), (C2×C56).433C22, (C2×C28).910C23, M4(2).11(C2×C14), (C7×M4(2)).45C22, (C7×C4≀C2)⋊15C2, (C7×C8○D4)⋊15C2, (C7×C4○D8).10C2, C4○D4.8(C2×C14), (C7×D4).20(C2×C4), (C7×Q8).21(C2×C4), (C7×C8.C4)⋊17C2, C22.1(C7×C4○D4), (C2×C8).101(C2×C14), (C2×C14).49(C4○D4), (C2×C4).85(C22×C14), (C7×C4○D4).53C22, SmallGroup(448,851)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8○D8
G = < a,b,c,d | a7=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c3 >
Subgroups: 154 in 106 conjugacy classes, 66 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, 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, C4×C8, C4≀C2, C8.C4, C8○D4, C4○D8, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C8○D8, C4×C28, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C7×D8, C7×SD16, C7×Q16, C7×C4○D4, C4×C56, C7×C4≀C2, C7×C8.C4, C7×C8○D4, C7×C4○D8, C7×C8○D8
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○D8, C22×C28, D4×C14, C7×C4○D4, D4×C28, C7×C8○D8
►On 112 points
Generators in S112
(1 59 23 55 10 47 35)(2 60 24 56 11 48 36)(3 61 17 49 12 41 37)(4 62 18 50 13 42 38)(5 63 19 51 14 43 39)(6 64 20 52 15 44 40)(7 57 21 53 16 45 33)(8 58 22 54 9 46 34)(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)
(1 6 3 8 5 2 7 4)(9 14 11 16 13 10 15 12)(17 22 19 24 21 18 23 20)(25 28 31 26 29 32 27 30)(33 38 35 40 37 34 39 36)(41 46 43 48 45 42 47 44)(49 54 51 56 53 50 55 52)(57 62 59 64 61 58 63 60)(65 68 71 66 69 72 67 70)(73 76 79 74 77 80 75 78)(81 84 87 82 85 88 83 86)(89 92 95 90 93 96 91 94)(97 100 103 98 101 104 99 102)(105 108 111 106 109 112 107 110)
(1 103)(2 104)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 93)(10 94)(11 95)(12 96)(13 89)(14 90)(15 91)(16 92)(17 76)(18 77)(19 78)(20 79)(21 80)(22 73)(23 74)(24 75)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 72)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 85)(42 86)(43 87)(44 88)(45 81)(46 82)(47 83)(48 84)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)
G:=sub<Sym(112)| (1,59,23,55,10,47,35)(2,60,24,56,11,48,36)(3,61,17,49,12,41,37)(4,62,18,50,13,42,38)(5,63,19,51,14,43,39)(6,64,20,52,15,44,40)(7,57,21,53,16,45,33)(8,58,22,54,9,46,34)(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), (1,6,3,8,5,2,7,4)(9,14,11,16,13,10,15,12)(17,22,19,24,21,18,23,20)(25,28,31,26,29,32,27,30)(33,38,35,40,37,34,39,36)(41,46,43,48,45,42,47,44)(49,54,51,56,53,50,55,52)(57,62,59,64,61,58,63,60)(65,68,71,66,69,72,67,70)(73,76,79,74,77,80,75,78)(81,84,87,82,85,88,83,86)(89,92,95,90,93,96,91,94)(97,100,103,98,101,104,99,102)(105,108,111,106,109,112,107,110), (1,103)(2,104)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,93)(10,94)(11,95)(12,96)(13,89)(14,90)(15,91)(16,92)(17,76)(18,77)(19,78)(20,79)(21,80)(22,73)(23,74)(24,75)(25,61)(26,62)(27,63)(28,64)(29,57)(30,58)(31,59)(32,60)(33,72)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,85)(42,86)(43,87)(44,88)(45,81)(46,82)(47,83)(48,84)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)>;
G:=Group( (1,59,23,55,10,47,35)(2,60,24,56,11,48,36)(3,61,17,49,12,41,37)(4,62,18,50,13,42,38)(5,63,19,51,14,43,39)(6,64,20,52,15,44,40)(7,57,21,53,16,45,33)(8,58,22,54,9,46,34)(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), (1,6,3,8,5,2,7,4)(9,14,11,16,13,10,15,12)(17,22,19,24,21,18,23,20)(25,28,31,26,29,32,27,30)(33,38,35,40,37,34,39,36)(41,46,43,48,45,42,47,44)(49,54,51,56,53,50,55,52)(57,62,59,64,61,58,63,60)(65,68,71,66,69,72,67,70)(73,76,79,74,77,80,75,78)(81,84,87,82,85,88,83,86)(89,92,95,90,93,96,91,94)(97,100,103,98,101,104,99,102)(105,108,111,106,109,112,107,110), (1,103)(2,104)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,93)(10,94)(11,95)(12,96)(13,89)(14,90)(15,91)(16,92)(17,76)(18,77)(19,78)(20,79)(21,80)(22,73)(23,74)(24,75)(25,61)(26,62)(27,63)(28,64)(29,57)(30,58)(31,59)(32,60)(33,72)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,85)(42,86)(43,87)(44,88)(45,81)(46,82)(47,83)(48,84)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112) );
G=PermutationGroup([[(1,59,23,55,10,47,35),(2,60,24,56,11,48,36),(3,61,17,49,12,41,37),(4,62,18,50,13,42,38),(5,63,19,51,14,43,39),(6,64,20,52,15,44,40),(7,57,21,53,16,45,33),(8,58,22,54,9,46,34),(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)], [(1,6,3,8,5,2,7,4),(9,14,11,16,13,10,15,12),(17,22,19,24,21,18,23,20),(25,28,31,26,29,32,27,30),(33,38,35,40,37,34,39,36),(41,46,43,48,45,42,47,44),(49,54,51,56,53,50,55,52),(57,62,59,64,61,58,63,60),(65,68,71,66,69,72,67,70),(73,76,79,74,77,80,75,78),(81,84,87,82,85,88,83,86),(89,92,95,90,93,96,91,94),(97,100,103,98,101,104,99,102),(105,108,111,106,109,112,107,110)], [(1,103),(2,104),(3,97),(4,98),(5,99),(6,100),(7,101),(8,102),(9,93),(10,94),(11,95),(12,96),(13,89),(14,90),(15,91),(16,92),(17,76),(18,77),(19,78),(20,79),(21,80),(22,73),(23,74),(24,75),(25,61),(26,62),(27,63),(28,64),(29,57),(30,58),(31,59),(32,60),(33,72),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71),(41,85),(42,86),(43,87),(44,88),(45,81),(46,82),(47,83),(48,84),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14X | 28A | ··· | 28L | 28M | ··· | 28AP | 28AQ | ··· | 28BB | 56A | ··· | 56X | 56Y | ··· | 56BH | 56BI | ··· | 56CF |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
196 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 | 2 | 2 |
| 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 | C8○D8 | C7×C4○D4 | C7×C8○D8 |
| kernel | C7×C8○D8 | C4×C56 | C7×C4≀C2 | C7×C8.C4 | C7×C8○D4 | C7×C4○D8 | C7×D8 | C7×SD16 | C7×Q16 | C8○D8 | C4×C8 | C4≀C2 | C8.C4 | C8○D4 | C4○D8 | D8 | SD16 | Q16 | C56 | C2×C14 | C8 | C7 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 6 | 6 | 12 | 6 | 12 | 6 | 12 | 24 | 12 | 2 | 2 | 12 | 8 | 12 | 48 |
Matrix representation of C7×C8○D8 ►in GL3(𝔽113) generated by
| 30 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 112 | 0 | 0 |
| 0 | 95 | 0 |
| 0 | 13 | 69 |
| 112 | 0 | 0 |
| 0 | 1 | 111 |
| 0 | 0 | 112 |
G:=sub<GL(3,GF(113))| [30,0,0,0,1,0,0,0,1],[1,0,0,0,18,0,0,0,18],[112,0,0,0,95,13,0,0,69],[112,0,0,0,1,0,0,111,112] >;
C7×C8○D8 in GAP, Magma, Sage, TeX
C_7\times C_8\circ D_8
% in TeX
G:=Group("C7xC8oD8"); // GroupNames label
G:=SmallGroup(448,851);
// by ID
G=gap.SmallGroup(448,851);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,604,9804,4911,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c^3>;
// generators/relations