direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×C8.C4, M4(2).24D14, (C8×D7).1C4, C8.31(C4×D7), C56.18(C2×C4), (C4×D7).34D4, C4.210(D4×D7), C56.C4⋊6C2, C28.369(C2×D4), (C2×C8).251D14, C22.3(Q8×D7), D14.12(C4⋊C4), (C22×D7).8Q8, C28.53D4⋊7C2, Dic7.8(C4⋊C4), (C2×C56).39C22, C28.52(C22×C4), (C2×Dic7).13Q8, (D7×M4(2)).2C2, (C2×C28).308C23, C4.Dic7.12C22, (C7×M4(2)).18C22, (D7×C2×C8).1C2, C4.82(C2×C4×D7), C7⋊1(C2×C8.C4), C7⋊C8.18(C2×C4), C2.17(D7×C4⋊C4), C14.16(C2×C4⋊C4), (C2×C14).1(C2×Q8), (C7×C8.C4)⋊2C2, (C4×D7).29(C2×C4), (C2×C7⋊C8).237C22, (C2×C4×D7).238C22, (C2×C4).411(C22×D7), SmallGroup(448,426)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×C8.C4
G = < a,b,c,d | a7=b2=c8=1, d4=c4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 412 in 106 conjugacy classes, 53 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, D7, C14, C14, C2×C8, C2×C8, M4(2), M4(2), C22×C4, Dic7, C28, D14, D14, C2×C14, C8.C4, C8.C4, C22×C8, C2×M4(2), C7⋊C8, C7⋊C8, C56, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C2×C8.C4, C8×D7, C8×D7, C8⋊D7, C2×C7⋊C8, C4.Dic7, C2×C56, C7×M4(2), C2×C4×D7, C56.C4, C28.53D4, C7×C8.C4, D7×C2×C8, D7×M4(2), D7×C8.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, D14, C8.C4, C2×C4⋊C4, C4×D7, C22×D7, C2×C8.C4, C2×C4×D7, D4×D7, Q8×D7, D7×C4⋊C4, D7×C8.C4
►On 112 points
Generators in S112
(1 90 31 83 99 63 107)(2 91 32 84 100 64 108)(3 92 25 85 101 57 109)(4 93 26 86 102 58 110)(5 94 27 87 103 59 111)(6 95 28 88 104 60 112)(7 96 29 81 97 61 105)(8 89 30 82 98 62 106)(9 37 52 46 74 71 20)(10 38 53 47 75 72 21)(11 39 54 48 76 65 22)(12 40 55 41 77 66 23)(13 33 56 42 78 67 24)(14 34 49 43 79 68 17)(15 35 50 44 80 69 18)(16 36 51 45 73 70 19)
(1 107)(2 108)(3 109)(4 110)(5 111)(6 112)(7 105)(8 106)(9 71)(10 72)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(25 101)(26 102)(27 103)(28 104)(29 97)(30 98)(31 99)(32 100)(33 78)(34 79)(35 80)(36 73)(37 74)(38 75)(39 76)(40 77)(41 55)(42 56)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(57 92)(58 93)(59 94)(60 95)(61 96)(62 89)(63 90)(64 91)
(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 43 3 41 5 47 7 45)(2 42 4 48 6 46 8 44)(9 98 15 100 13 102 11 104)(10 97 16 99 14 101 12 103)(17 85 23 87 21 81 19 83)(18 84 24 86 22 88 20 82)(25 66 27 72 29 70 31 68)(26 65 28 71 30 69 32 67)(33 58 39 60 37 62 35 64)(34 57 40 59 38 61 36 63)(49 109 55 111 53 105 51 107)(50 108 56 110 54 112 52 106)(73 90 79 92 77 94 75 96)(74 89 80 91 78 93 76 95)
G:=sub<Sym(112)| (1,90,31,83,99,63,107)(2,91,32,84,100,64,108)(3,92,25,85,101,57,109)(4,93,26,86,102,58,110)(5,94,27,87,103,59,111)(6,95,28,88,104,60,112)(7,96,29,81,97,61,105)(8,89,30,82,98,62,106)(9,37,52,46,74,71,20)(10,38,53,47,75,72,21)(11,39,54,48,76,65,22)(12,40,55,41,77,66,23)(13,33,56,42,78,67,24)(14,34,49,43,79,68,17)(15,35,50,44,80,69,18)(16,36,51,45,73,70,19), (1,107)(2,108)(3,109)(4,110)(5,111)(6,112)(7,105)(8,106)(9,71)(10,72)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(25,101)(26,102)(27,103)(28,104)(29,97)(30,98)(31,99)(32,100)(33,78)(34,79)(35,80)(36,73)(37,74)(38,75)(39,76)(40,77)(41,55)(42,56)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(57,92)(58,93)(59,94)(60,95)(61,96)(62,89)(63,90)(64,91), (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), (1,43,3,41,5,47,7,45)(2,42,4,48,6,46,8,44)(9,98,15,100,13,102,11,104)(10,97,16,99,14,101,12,103)(17,85,23,87,21,81,19,83)(18,84,24,86,22,88,20,82)(25,66,27,72,29,70,31,68)(26,65,28,71,30,69,32,67)(33,58,39,60,37,62,35,64)(34,57,40,59,38,61,36,63)(49,109,55,111,53,105,51,107)(50,108,56,110,54,112,52,106)(73,90,79,92,77,94,75,96)(74,89,80,91,78,93,76,95)>;
G:=Group( (1,90,31,83,99,63,107)(2,91,32,84,100,64,108)(3,92,25,85,101,57,109)(4,93,26,86,102,58,110)(5,94,27,87,103,59,111)(6,95,28,88,104,60,112)(7,96,29,81,97,61,105)(8,89,30,82,98,62,106)(9,37,52,46,74,71,20)(10,38,53,47,75,72,21)(11,39,54,48,76,65,22)(12,40,55,41,77,66,23)(13,33,56,42,78,67,24)(14,34,49,43,79,68,17)(15,35,50,44,80,69,18)(16,36,51,45,73,70,19), (1,107)(2,108)(3,109)(4,110)(5,111)(6,112)(7,105)(8,106)(9,71)(10,72)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(25,101)(26,102)(27,103)(28,104)(29,97)(30,98)(31,99)(32,100)(33,78)(34,79)(35,80)(36,73)(37,74)(38,75)(39,76)(40,77)(41,55)(42,56)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(57,92)(58,93)(59,94)(60,95)(61,96)(62,89)(63,90)(64,91), (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), (1,43,3,41,5,47,7,45)(2,42,4,48,6,46,8,44)(9,98,15,100,13,102,11,104)(10,97,16,99,14,101,12,103)(17,85,23,87,21,81,19,83)(18,84,24,86,22,88,20,82)(25,66,27,72,29,70,31,68)(26,65,28,71,30,69,32,67)(33,58,39,60,37,62,35,64)(34,57,40,59,38,61,36,63)(49,109,55,111,53,105,51,107)(50,108,56,110,54,112,52,106)(73,90,79,92,77,94,75,96)(74,89,80,91,78,93,76,95) );
G=PermutationGroup([[(1,90,31,83,99,63,107),(2,91,32,84,100,64,108),(3,92,25,85,101,57,109),(4,93,26,86,102,58,110),(5,94,27,87,103,59,111),(6,95,28,88,104,60,112),(7,96,29,81,97,61,105),(8,89,30,82,98,62,106),(9,37,52,46,74,71,20),(10,38,53,47,75,72,21),(11,39,54,48,76,65,22),(12,40,55,41,77,66,23),(13,33,56,42,78,67,24),(14,34,49,43,79,68,17),(15,35,50,44,80,69,18),(16,36,51,45,73,70,19)], [(1,107),(2,108),(3,109),(4,110),(5,111),(6,112),(7,105),(8,106),(9,71),(10,72),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(25,101),(26,102),(27,103),(28,104),(29,97),(30,98),(31,99),(32,100),(33,78),(34,79),(35,80),(36,73),(37,74),(38,75),(39,76),(40,77),(41,55),(42,56),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54),(57,92),(58,93),(59,94),(60,95),(61,96),(62,89),(63,90),(64,91)], [(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)], [(1,43,3,41,5,47,7,45),(2,42,4,48,6,46,8,44),(9,98,15,100,13,102,11,104),(10,97,16,99,14,101,12,103),(17,85,23,87,21,81,19,83),(18,84,24,86,22,88,20,82),(25,66,27,72,29,70,31,68),(26,65,28,71,30,69,32,67),(33,58,39,60,37,62,35,64),(34,57,40,59,38,61,36,63),(49,109,55,111,53,105,51,107),(50,108,56,110,54,112,52,106),(73,90,79,92,77,94,75,96),(74,89,80,91,78,93,76,95)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 8O | 8P | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 7 | 7 | 14 | 1 | 1 | 2 | 7 | 7 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | Q8 | D7 | D14 | D14 | C8.C4 | C4×D7 | D4×D7 | Q8×D7 | D7×C8.C4 |
| kernel | D7×C8.C4 | C56.C4 | C28.53D4 | C7×C8.C4 | D7×C2×C8 | D7×M4(2) | C8×D7 | C4×D7 | C2×Dic7 | C22×D7 | C8.C4 | C2×C8 | M4(2) | D7 | C8 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 8 | 2 | 1 | 1 | 3 | 3 | 6 | 8 | 12 | 3 | 3 | 12 |
Matrix representation of D7×C8.C4 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 104 | 1 |
| 0 | 0 | 41 | 33 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 33 | 112 |
| 0 | 0 | 71 | 80 |
| 44 | 0 | 0 | 0 |
| 13 | 18 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 111 | 0 | 0 |
| 106 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,104,41,0,0,1,33],[1,0,0,0,0,1,0,0,0,0,33,71,0,0,112,80],[44,13,0,0,0,18,0,0,0,0,112,0,0,0,0,112],[1,106,0,0,111,112,0,0,0,0,1,0,0,0,0,1] >;
D7×C8.C4 in GAP, Magma, Sage, TeX
D_7\times C_8.C_4
% in TeX
G:=Group("D7xC8.C4"); // GroupNames label
G:=SmallGroup(448,426);
// by ID
G=gap.SmallGroup(448,426);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,58,136,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^8=1,d^4=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations