direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D4.4D4, C56.99D4, C8○D4⋊2C14, (C2×D8)⋊8C14, D4.4(C7×D4), C8.19(C7×D4), Q8.4(C7×D4), (C14×D8)⋊22C2, C8⋊C22⋊3C14, (C7×D4).29D4, C4.38(D4×C14), (C7×Q8).29D4, C8.C4⋊6C14, C28.399(C2×D4), C4.D4⋊4C14, (C2×C56).273C22, (C2×C28).614C23, M4(2).4(C2×C14), C14.155(C4⋊D4), (D4×C14).194C22, (C7×M4(2)).48C22, (C7×C8○D4)⋊11C2, (C7×C8⋊C22)⋊10C2, (C2×C8).25(C2×C14), C2.24(C7×C4⋊D4), (C7×C8.C4)⋊15C2, C22.7(C7×C4○D4), C4○D4.11(C2×C14), (C2×D4).17(C2×C14), (C7×C4.D4)⋊10C2, (C2×C4).9(C22×C14), (C7×C4○D4).56C22, (C2×C14).116(C4○D4), SmallGroup(448,880)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4.4D4
G = < a,b,c,d,e | a7=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=b2d3 >
Subgroups: 226 in 108 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C14, C14, C2×C8, C2×C8, M4(2), M4(2), M4(2), D8, SD16, C2×D4, C4○D4, C28, C28, C2×C14, C2×C14, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×C14, D4.4D4, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C7×M4(2), C7×D8, C7×SD16, D4×C14, C7×C4○D4, C7×C4.D4, C7×C8.C4, C7×C8○D4, C14×D8, C7×C8⋊C22, C7×D4.4D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C4⋊D4, C7×D4, C22×C14, D4.4D4, D4×C14, C7×C4○D4, C7×C4⋊D4, C7×D4.4D4
►On 112 points
Generators in S112
(1 31 47 79 95 23 39)(2 32 48 80 96 24 40)(3 25 41 73 89 17 33)(4 26 42 74 90 18 34)(5 27 43 75 91 19 35)(6 28 44 76 92 20 36)(7 29 45 77 93 21 37)(8 30 46 78 94 22 38)(9 106 66 50 82 98 62)(10 107 67 51 83 99 63)(11 108 68 52 84 100 64)(12 109 69 53 85 101 57)(13 110 70 54 86 102 58)(14 111 71 55 87 103 59)(15 112 72 56 88 104 60)(16 105 65 49 81 97 61)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)(65 71 69 67)(66 72 70 68)(73 75 77 79)(74 76 78 80)(81 87 85 83)(82 88 86 84)(89 91 93 95)(90 92 94 96)(97 103 101 99)(98 104 102 100)(105 111 109 107)(106 112 110 108)
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 81)(40 82)(65 95)(66 96)(67 89)(68 90)(69 91)(70 92)(71 93)(72 94)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 105)(80 106)
(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 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)(37 40)(38 39)(41 44)(42 43)(45 48)(46 47)(49 50)(51 56)(52 55)(53 54)(57 58)(59 64)(60 63)(61 62)(65 66)(67 72)(68 71)(69 70)(73 76)(74 75)(77 80)(78 79)(81 82)(83 88)(84 87)(85 86)(89 92)(90 91)(93 96)(94 95)(97 98)(99 104)(100 103)(101 102)(105 106)(107 112)(108 111)(109 110)
G:=sub<Sym(112)| (1,31,47,79,95,23,39)(2,32,48,80,96,24,40)(3,25,41,73,89,17,33)(4,26,42,74,90,18,34)(5,27,43,75,91,19,35)(6,28,44,76,92,20,36)(7,29,45,77,93,21,37)(8,30,46,78,94,22,38)(9,106,66,50,82,98,62)(10,107,67,51,83,99,63)(11,108,68,52,84,100,64)(12,109,69,53,85,101,57)(13,110,70,54,86,102,58)(14,111,71,55,87,103,59)(15,112,72,56,88,104,60)(16,105,65,49,81,97,61), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,75,77,79)(74,76,78,80)(81,87,85,83)(82,88,86,84)(89,91,93,95)(90,92,94,96)(97,103,101,99)(98,104,102,100)(105,111,109,107)(106,112,110,108), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,81)(40,82)(65,95)(66,96)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,105)(80,106), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43)(45,48)(46,47)(49,50)(51,56)(52,55)(53,54)(57,58)(59,64)(60,63)(61,62)(65,66)(67,72)(68,71)(69,70)(73,76)(74,75)(77,80)(78,79)(81,82)(83,88)(84,87)(85,86)(89,92)(90,91)(93,96)(94,95)(97,98)(99,104)(100,103)(101,102)(105,106)(107,112)(108,111)(109,110)>;
G:=Group( (1,31,47,79,95,23,39)(2,32,48,80,96,24,40)(3,25,41,73,89,17,33)(4,26,42,74,90,18,34)(5,27,43,75,91,19,35)(6,28,44,76,92,20,36)(7,29,45,77,93,21,37)(8,30,46,78,94,22,38)(9,106,66,50,82,98,62)(10,107,67,51,83,99,63)(11,108,68,52,84,100,64)(12,109,69,53,85,101,57)(13,110,70,54,86,102,58)(14,111,71,55,87,103,59)(15,112,72,56,88,104,60)(16,105,65,49,81,97,61), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,75,77,79)(74,76,78,80)(81,87,85,83)(82,88,86,84)(89,91,93,95)(90,92,94,96)(97,103,101,99)(98,104,102,100)(105,111,109,107)(106,112,110,108), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,81)(40,82)(65,95)(66,96)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,105)(80,106), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43)(45,48)(46,47)(49,50)(51,56)(52,55)(53,54)(57,58)(59,64)(60,63)(61,62)(65,66)(67,72)(68,71)(69,70)(73,76)(74,75)(77,80)(78,79)(81,82)(83,88)(84,87)(85,86)(89,92)(90,91)(93,96)(94,95)(97,98)(99,104)(100,103)(101,102)(105,106)(107,112)(108,111)(109,110) );
G=PermutationGroup([[(1,31,47,79,95,23,39),(2,32,48,80,96,24,40),(3,25,41,73,89,17,33),(4,26,42,74,90,18,34),(5,27,43,75,91,19,35),(6,28,44,76,92,20,36),(7,29,45,77,93,21,37),(8,30,46,78,94,22,38),(9,106,66,50,82,98,62),(10,107,67,51,83,99,63),(11,108,68,52,84,100,64),(12,109,69,53,85,101,57),(13,110,70,54,86,102,58),(14,111,71,55,87,103,59),(15,112,72,56,88,104,60),(16,105,65,49,81,97,61)], [(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60),(65,71,69,67),(66,72,70,68),(73,75,77,79),(74,76,78,80),(81,87,85,83),(82,88,86,84),(89,91,93,95),(90,92,94,96),(97,103,101,99),(98,104,102,100),(105,111,109,107),(106,112,110,108)], [(1,97),(2,98),(3,99),(4,100),(5,101),(6,102),(7,103),(8,104),(9,48),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,49),(24,50),(25,63),(26,64),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,83),(34,84),(35,85),(36,86),(37,87),(38,88),(39,81),(40,82),(65,95),(66,96),(67,89),(68,90),(69,91),(70,92),(71,93),(72,94),(73,107),(74,108),(75,109),(76,110),(77,111),(78,112),(79,105),(80,106)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(17,20),(18,19),(21,24),(22,23),(25,28),(26,27),(29,32),(30,31),(33,36),(34,35),(37,40),(38,39),(41,44),(42,43),(45,48),(46,47),(49,50),(51,56),(52,55),(53,54),(57,58),(59,64),(60,63),(61,62),(65,66),(67,72),(68,71),(69,70),(73,76),(74,75),(77,80),(78,79),(81,82),(83,88),(84,87),(85,86),(89,92),(90,91),(93,96),(94,95),(97,98),(99,104),(100,103),(101,102),(105,106),(107,112),(108,111),(109,110)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56L | 56M | ··· | 56AD | 56AE | ··· | 56AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 8 | 8 | 2 | 2 | 4 | 1 | ··· | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
112 irreducible representations
| dim | 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 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | D4 | C4○D4 | C7×D4 | C7×D4 | C7×D4 | C7×C4○D4 | D4.4D4 | C7×D4.4D4 |
| kernel | C7×D4.4D4 | C7×C4.D4 | C7×C8.C4 | C7×C8○D4 | C14×D8 | C7×C8⋊C22 | D4.4D4 | C4.D4 | C8.C4 | C8○D4 | C2×D8 | C8⋊C22 | C56 | C7×D4 | C7×Q8 | C2×C14 | C8 | D4 | Q8 | C22 | C7 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 6 | 12 | 6 | 6 | 6 | 12 | 2 | 1 | 1 | 2 | 12 | 6 | 6 | 12 | 2 | 12 |
Matrix representation of C7×D4.4D4 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 49 | 0 |
| 0 | 0 | 0 | 49 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 103 | 1 | 1 | 2 |
| 62 | 61 | 112 | 112 |
| 103 | 1 | 1 | 2 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 7 | 61 | 61 | 10 |
| 82 | 82 | 0 | 0 |
| 31 | 82 | 0 | 0 |
| 29 | 31 | 0 | 62 |
| 83 | 112 | 82 | 51 |
| 82 | 82 | 0 | 0 |
| 82 | 31 | 0 | 0 |
| 29 | 31 | 62 | 62 |
| 1 | 81 | 82 | 51 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,49,0,0,0,0,49],[0,112,103,62,1,0,1,61,0,0,1,112,0,0,2,112],[103,0,0,7,1,0,1,61,1,1,0,61,2,0,0,10],[82,31,29,83,82,82,31,112,0,0,0,82,0,0,62,51],[82,82,29,1,82,31,31,81,0,0,62,82,0,0,62,51] >;
C7×D4.4D4 in GAP, Magma, Sage, TeX
C_7\times D_4._4D_4
% in TeX
G:=Group("C7xD4.4D4"); // GroupNames label
G:=SmallGroup(448,880);
// by ID
G=gap.SmallGroup(448,880);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,1968,2438,9804,172,14117,3547,124]);
// 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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=b^2*d^3>;
// generators/relations