direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D4.3D4, C56.103D4, C8○D4⋊1C14, C8⋊C22.C14, D4.3(C7×D4), C8.23(C7×D4), Q8.3(C7×D4), (C7×D4).28D4, C4.37(D4×C14), (C7×Q8).28D4, C8.C4⋊5C14, (C2×SD16)⋊2C14, C4.D4⋊3C14, C28.398(C2×D4), C8.C22⋊3C14, (C14×SD16)⋊13C2, C4.10D4⋊3C14, (C2×C28).613C23, (C2×C56).272C22, M4(2).3(C2×C14), C14.154(C4⋊D4), (D4×C14).193C22, (Q8×C14).167C22, (C7×M4(2)).47C22, (C7×C8○D4)⋊10C2, (C2×C8).24(C2×C14), (C7×C4.D4)⋊9C2, C2.23(C7×C4⋊D4), (C7×C8⋊C22).2C2, (C7×C8.C4)⋊14C2, C22.6(C7×C4○D4), C4○D4.10(C2×C14), (C2×D4).16(C2×C14), (C7×C4.10D4)⋊9C2, (C7×C8.C22)⋊10C2, (C2×C4).8(C22×C14), (C2×Q8).11(C2×C14), (C7×C4○D4).55C22, (C2×C14).115(C4○D4), SmallGroup(448,879)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4.3D4
G = < a,b,c,d,e | a7=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d3 >
Subgroups: 194 in 104 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C14, C14, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, D4.3D4, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C7×D8, C7×SD16, C7×Q16, D4×C14, Q8×C14, C7×C4○D4, C7×C4.D4, C7×C4.10D4, C7×C8.C4, C7×C8○D4, C14×SD16, C7×C8⋊C22, C7×C8.C22, C7×D4.3D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C4⋊D4, C7×D4, C22×C14, D4.3D4, D4×C14, C7×C4○D4, C7×C4⋊D4, C7×D4.3D4
►On 112 points
Generators in S112
(1 25 43 73 91 17 35)(2 26 44 74 92 18 36)(3 27 45 75 93 19 37)(4 28 46 76 94 20 38)(5 29 47 77 95 21 39)(6 30 48 78 96 22 40)(7 31 41 79 89 23 33)(8 32 42 80 90 24 34)(9 108 68 52 82 100 63)(10 109 69 53 83 101 64)(11 110 70 54 84 102 57)(12 111 71 55 85 103 58)(13 112 72 56 86 104 59)(14 105 65 49 87 97 60)(15 106 66 50 88 98 61)(16 107 67 51 81 99 62)
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(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 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 60)(26 61)(27 62)(28 63)(29 64)(30 57)(31 58)(32 59)(33 85)(34 86)(35 87)(36 88)(37 81)(38 82)(39 83)(40 84)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 89)(72 90)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)
(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)
(1 4 5 8)(2 7 6 3)(9 16 13 12)(10 11 14 15)(17 20 21 24)(18 23 22 19)(25 28 29 32)(26 31 30 27)(33 40 37 36)(34 35 38 39)(41 48 45 44)(42 43 46 47)(49 50 53 54)(51 56 55 52)(57 60 61 64)(58 63 62 59)(65 66 69 70)(67 72 71 68)(73 76 77 80)(74 79 78 75)(81 86 85 82)(83 84 87 88)(89 96 93 92)(90 91 94 95)(97 98 101 102)(99 104 103 100)(105 106 109 110)(107 112 111 108)
G:=sub<Sym(112)| (1,25,43,73,91,17,35)(2,26,44,74,92,18,36)(3,27,45,75,93,19,37)(4,28,46,76,94,20,38)(5,29,47,77,95,21,39)(6,30,48,78,96,22,40)(7,31,41,79,89,23,33)(8,32,42,80,90,24,34)(9,108,68,52,82,100,63)(10,109,69,53,83,101,64)(11,110,70,54,84,102,57)(12,111,71,55,85,103,58)(13,112,72,56,86,104,59)(14,105,65,49,87,97,60)(15,106,66,50,88,98,61)(16,107,67,51,81,99,62), (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(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), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,85)(34,86)(35,87)(36,88)(37,81)(38,82)(39,83)(40,84)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,89)(72,90)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112), (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), (1,4,5,8)(2,7,6,3)(9,16,13,12)(10,11,14,15)(17,20,21,24)(18,23,22,19)(25,28,29,32)(26,31,30,27)(33,40,37,36)(34,35,38,39)(41,48,45,44)(42,43,46,47)(49,50,53,54)(51,56,55,52)(57,60,61,64)(58,63,62,59)(65,66,69,70)(67,72,71,68)(73,76,77,80)(74,79,78,75)(81,86,85,82)(83,84,87,88)(89,96,93,92)(90,91,94,95)(97,98,101,102)(99,104,103,100)(105,106,109,110)(107,112,111,108)>;
G:=Group( (1,25,43,73,91,17,35)(2,26,44,74,92,18,36)(3,27,45,75,93,19,37)(4,28,46,76,94,20,38)(5,29,47,77,95,21,39)(6,30,48,78,96,22,40)(7,31,41,79,89,23,33)(8,32,42,80,90,24,34)(9,108,68,52,82,100,63)(10,109,69,53,83,101,64)(11,110,70,54,84,102,57)(12,111,71,55,85,103,58)(13,112,72,56,86,104,59)(14,105,65,49,87,97,60)(15,106,66,50,88,98,61)(16,107,67,51,81,99,62), (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(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), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,85)(34,86)(35,87)(36,88)(37,81)(38,82)(39,83)(40,84)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,89)(72,90)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112), (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), (1,4,5,8)(2,7,6,3)(9,16,13,12)(10,11,14,15)(17,20,21,24)(18,23,22,19)(25,28,29,32)(26,31,30,27)(33,40,37,36)(34,35,38,39)(41,48,45,44)(42,43,46,47)(49,50,53,54)(51,56,55,52)(57,60,61,64)(58,63,62,59)(65,66,69,70)(67,72,71,68)(73,76,77,80)(74,79,78,75)(81,86,85,82)(83,84,87,88)(89,96,93,92)(90,91,94,95)(97,98,101,102)(99,104,103,100)(105,106,109,110)(107,112,111,108) );
G=PermutationGroup([[(1,25,43,73,91,17,35),(2,26,44,74,92,18,36),(3,27,45,75,93,19,37),(4,28,46,76,94,20,38),(5,29,47,77,95,21,39),(6,30,48,78,96,22,40),(7,31,41,79,89,23,33),(8,32,42,80,90,24,34),(9,108,68,52,82,100,63),(10,109,69,53,83,101,64),(11,110,70,54,84,102,57),(12,111,71,55,85,103,58),(13,112,72,56,86,104,59),(14,105,65,49,87,97,60),(15,106,66,50,88,98,61),(16,107,67,51,81,99,62)], [(1,7,5,3),(2,8,6,4),(9,11,13,15),(10,12,14,16),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,39,37,35),(34,40,38,36),(41,47,45,43),(42,48,46,44),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(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)], [(1,97),(2,98),(3,99),(4,100),(5,101),(6,102),(7,103),(8,104),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,60),(26,61),(27,62),(28,63),(29,64),(30,57),(31,58),(32,59),(33,85),(34,86),(35,87),(36,88),(37,81),(38,82),(39,83),(40,84),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,89),(72,90),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112)], [(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)], [(1,4,5,8),(2,7,6,3),(9,16,13,12),(10,11,14,15),(17,20,21,24),(18,23,22,19),(25,28,29,32),(26,31,30,27),(33,40,37,36),(34,35,38,39),(41,48,45,44),(42,43,46,47),(49,50,53,54),(51,56,55,52),(57,60,61,64),(58,63,62,59),(65,66,69,70),(67,72,71,68),(73,76,77,80),(74,79,78,75),(81,86,85,82),(83,84,87,88),(89,96,93,92),(90,91,94,95),(97,98,101,102),(99,104,103,100),(105,106,109,110),(107,112,111,108)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14R | 14S | ··· | 14X | 28A | ··· | 28L | 28M | ··· | 28R | 28S | ··· | 28X | 56A | ··· | 56L | 56M | ··· | 56AD | 56AE | ··· | 56AP |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 8 | 2 | 2 | 4 | 8 | 1 | ··· | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 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 | D4 | C4○D4 | C7×D4 | C7×D4 | C7×D4 | C7×C4○D4 | D4.3D4 | C7×D4.3D4 |
| kernel | C7×D4.3D4 | C7×C4.D4 | C7×C4.10D4 | C7×C8.C4 | C7×C8○D4 | C14×SD16 | C7×C8⋊C22 | C7×C8.C22 | D4.3D4 | C4.D4 | C4.10D4 | C8.C4 | C8○D4 | C2×SD16 | C8⋊C22 | C8.C22 | C56 | C7×D4 | C7×Q8 | C2×C14 | C8 | D4 | Q8 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 2 | 12 | 6 | 6 | 12 | 2 | 12 |
Matrix representation of C7×D4.3D4 ►in GL4(𝔽113) generated by
| 106 | 0 | 0 | 0 |
| 0 | 106 | 0 | 0 |
| 0 | 0 | 106 | 0 |
| 0 | 0 | 0 | 106 |
| 1 | 111 | 0 | 0 |
| 1 | 112 | 0 | 0 |
| 112 | 1 | 0 | 112 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 2 |
| 1 | 0 | 1 | 1 |
| 112 | 1 | 0 | 112 |
| 0 | 0 | 0 | 112 |
| 87 | 26 | 0 | 0 |
| 100 | 0 | 0 | 0 |
| 0 | 100 | 100 | 100 |
| 13 | 100 | 13 | 100 |
| 87 | 26 | 0 | 0 |
| 100 | 26 | 0 | 0 |
| 13 | 100 | 13 | 100 |
| 0 | 100 | 100 | 100 |
G:=sub<GL(4,GF(113))| [106,0,0,0,0,106,0,0,0,0,106,0,0,0,0,106],[1,1,112,0,111,112,1,1,0,0,0,1,0,0,112,0],[1,1,112,0,0,0,1,0,0,1,0,0,2,1,112,112],[87,100,0,13,26,0,100,100,0,0,100,13,0,0,100,100],[87,100,13,0,26,26,100,100,0,0,13,100,0,0,100,100] >;
C7×D4.3D4 in GAP, Magma, Sage, TeX
C_7\times D_4._3D_4
% in TeX
G:=Group("C7xD4.3D4"); // GroupNames label
G:=SmallGroup(448,879);
// by ID
G=gap.SmallGroup(448,879);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,400,2438,9804,172,14117,3547,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=1,d^4=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^3>;
// generators/relations