direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×C4.4D4, C42⋊33D14, C4.31(D4×D7), (C2×Q8)⋊17D14, (C4×D7).24D4, C28.60(C2×D4), (D7×C42)⋊9C2, (C4×C28)⋊21C22, C22⋊C4⋊32D14, D14.60(C2×D4), Dic7.9(C2×D4), D14⋊C4⋊29C22, (C2×D4).170D14, C4.D28⋊23C2, (C2×C28).79C23, (Q8×C14)⋊11C22, C14.87(C22×D4), D14.40(C4○D4), C28.23D4⋊20C2, C28.17D4⋊23C2, (C2×C14).217C24, (C4×Dic7)⋊79C22, C23.D7⋊32C22, C23.39(C22×D7), Dic7.D4⋊39C2, (C2×Dic14)⋊32C22, (D4×C14).152C22, (C2×D28).162C22, (C22×C14).47C23, (C22×D7).95C23, (C23×D7).62C22, C22.238(C23×D7), (C2×Dic7).112C23, (C2×Q8×D7)⋊9C2, (C2×D4×D7).9C2, C2.60(C2×D4×D7), C7⋊3(C2×C4.4D4), C2.75(D7×C4○D4), (D7×C22⋊C4)⋊16C2, (C7×C4.4D4)⋊9C2, C14.186(C2×C4○D4), (C2×C4×D7).297C22, (C7×C22⋊C4)⋊27C22, (C2×C4).300(C22×D7), (C2×C7⋊D4).58C22, SmallGroup(448,1126)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×C4.4D4
G = < a,b,c,d,e | a7=b2=c4=d4=1, e2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=c2d-1 >
Subgroups: 1772 in 330 conjugacy classes, 111 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, D7, C14, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C4.4D4, C4.4D4, C22×D4, C22×Q8, Dic14, C4×D7, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×D7, C22×D7, C22×C14, C2×C4.4D4, C4×Dic7, C4×Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C2×Dic14, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, D4×D7, Q8×D7, C2×C7⋊D4, D4×C14, Q8×C14, C23×D7, D7×C42, C4.D28, D7×C22⋊C4, Dic7.D4, C28.17D4, C28.23D4, C7×C4.4D4, C2×D4×D7, C2×Q8×D7, D7×C4.4D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C4.4D4, C22×D4, C2×C4○D4, C22×D7, C2×C4.4D4, D4×D7, C23×D7, C2×D4×D7, D7×C4○D4, D7×C4.4D4
►On 112 points
Generators in S112
(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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)(64 66)(67 70)(68 69)(71 73)(74 77)(75 76)(78 80)(81 84)(82 83)(85 87)(88 91)(89 90)(92 94)(95 98)(96 97)(99 101)(102 105)(103 104)(106 108)(109 112)(110 111)
(1 111 27 90)(2 112 28 91)(3 106 22 85)(4 107 23 86)(5 108 24 87)(6 109 25 88)(7 110 26 89)(8 99 15 92)(9 100 16 93)(10 101 17 94)(11 102 18 95)(12 103 19 96)(13 104 20 97)(14 105 21 98)(29 71 50 64)(30 72 51 65)(31 73 52 66)(32 74 53 67)(33 75 54 68)(34 76 55 69)(35 77 56 70)(36 78 43 57)(37 79 44 58)(38 80 45 59)(39 81 46 60)(40 82 47 61)(41 83 48 62)(42 84 49 63)
(1 62 20 76)(2 63 21 77)(3 57 15 71)(4 58 16 72)(5 59 17 73)(6 60 18 74)(7 61 19 75)(8 64 22 78)(9 65 23 79)(10 66 24 80)(11 67 25 81)(12 68 26 82)(13 69 27 83)(14 70 28 84)(29 85 43 99)(30 86 44 100)(31 87 45 101)(32 88 46 102)(33 89 47 103)(34 90 48 104)(35 91 49 105)(36 92 50 106)(37 93 51 107)(38 94 52 108)(39 95 53 109)(40 96 54 110)(41 97 55 111)(42 98 56 112)
(1 76 27 69)(2 77 28 70)(3 71 22 64)(4 72 23 65)(5 73 24 66)(6 74 25 67)(7 75 26 68)(8 78 15 57)(9 79 16 58)(10 80 17 59)(11 81 18 60)(12 82 19 61)(13 83 20 62)(14 84 21 63)(29 85 50 106)(30 86 51 107)(31 87 52 108)(32 88 53 109)(33 89 54 110)(34 90 55 111)(35 91 56 112)(36 92 43 99)(37 93 44 100)(38 94 45 101)(39 95 46 102)(40 96 47 103)(41 97 48 104)(42 98 49 105)
G:=sub<Sym(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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)(43,45)(46,49)(47,48)(50,52)(53,56)(54,55)(57,59)(60,63)(61,62)(64,66)(67,70)(68,69)(71,73)(74,77)(75,76)(78,80)(81,84)(82,83)(85,87)(88,91)(89,90)(92,94)(95,98)(96,97)(99,101)(102,105)(103,104)(106,108)(109,112)(110,111), (1,111,27,90)(2,112,28,91)(3,106,22,85)(4,107,23,86)(5,108,24,87)(6,109,25,88)(7,110,26,89)(8,99,15,92)(9,100,16,93)(10,101,17,94)(11,102,18,95)(12,103,19,96)(13,104,20,97)(14,105,21,98)(29,71,50,64)(30,72,51,65)(31,73,52,66)(32,74,53,67)(33,75,54,68)(34,76,55,69)(35,77,56,70)(36,78,43,57)(37,79,44,58)(38,80,45,59)(39,81,46,60)(40,82,47,61)(41,83,48,62)(42,84,49,63), (1,62,20,76)(2,63,21,77)(3,57,15,71)(4,58,16,72)(5,59,17,73)(6,60,18,74)(7,61,19,75)(8,64,22,78)(9,65,23,79)(10,66,24,80)(11,67,25,81)(12,68,26,82)(13,69,27,83)(14,70,28,84)(29,85,43,99)(30,86,44,100)(31,87,45,101)(32,88,46,102)(33,89,47,103)(34,90,48,104)(35,91,49,105)(36,92,50,106)(37,93,51,107)(38,94,52,108)(39,95,53,109)(40,96,54,110)(41,97,55,111)(42,98,56,112), (1,76,27,69)(2,77,28,70)(3,71,22,64)(4,72,23,65)(5,73,24,66)(6,74,25,67)(7,75,26,68)(8,78,15,57)(9,79,16,58)(10,80,17,59)(11,81,18,60)(12,82,19,61)(13,83,20,62)(14,84,21,63)(29,85,50,106)(30,86,51,107)(31,87,52,108)(32,88,53,109)(33,89,54,110)(34,90,55,111)(35,91,56,112)(36,92,43,99)(37,93,44,100)(38,94,45,101)(39,95,46,102)(40,96,47,103)(41,97,48,104)(42,98,49,105)>;
G:=Group( (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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)(43,45)(46,49)(47,48)(50,52)(53,56)(54,55)(57,59)(60,63)(61,62)(64,66)(67,70)(68,69)(71,73)(74,77)(75,76)(78,80)(81,84)(82,83)(85,87)(88,91)(89,90)(92,94)(95,98)(96,97)(99,101)(102,105)(103,104)(106,108)(109,112)(110,111), (1,111,27,90)(2,112,28,91)(3,106,22,85)(4,107,23,86)(5,108,24,87)(6,109,25,88)(7,110,26,89)(8,99,15,92)(9,100,16,93)(10,101,17,94)(11,102,18,95)(12,103,19,96)(13,104,20,97)(14,105,21,98)(29,71,50,64)(30,72,51,65)(31,73,52,66)(32,74,53,67)(33,75,54,68)(34,76,55,69)(35,77,56,70)(36,78,43,57)(37,79,44,58)(38,80,45,59)(39,81,46,60)(40,82,47,61)(41,83,48,62)(42,84,49,63), (1,62,20,76)(2,63,21,77)(3,57,15,71)(4,58,16,72)(5,59,17,73)(6,60,18,74)(7,61,19,75)(8,64,22,78)(9,65,23,79)(10,66,24,80)(11,67,25,81)(12,68,26,82)(13,69,27,83)(14,70,28,84)(29,85,43,99)(30,86,44,100)(31,87,45,101)(32,88,46,102)(33,89,47,103)(34,90,48,104)(35,91,49,105)(36,92,50,106)(37,93,51,107)(38,94,52,108)(39,95,53,109)(40,96,54,110)(41,97,55,111)(42,98,56,112), (1,76,27,69)(2,77,28,70)(3,71,22,64)(4,72,23,65)(5,73,24,66)(6,74,25,67)(7,75,26,68)(8,78,15,57)(9,79,16,58)(10,80,17,59)(11,81,18,60)(12,82,19,61)(13,83,20,62)(14,84,21,63)(29,85,50,106)(30,86,51,107)(31,87,52,108)(32,88,53,109)(33,89,54,110)(34,90,55,111)(35,91,56,112)(36,92,43,99)(37,93,44,100)(38,94,45,101)(39,95,46,102)(40,96,47,103)(41,97,48,104)(42,98,49,105) );
G=PermutationGroup([[(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)], [(1,7),(2,6),(3,5),(8,10),(11,14),(12,13),(15,17),(18,21),(19,20),(22,24),(25,28),(26,27),(29,31),(32,35),(33,34),(36,38),(39,42),(40,41),(43,45),(46,49),(47,48),(50,52),(53,56),(54,55),(57,59),(60,63),(61,62),(64,66),(67,70),(68,69),(71,73),(74,77),(75,76),(78,80),(81,84),(82,83),(85,87),(88,91),(89,90),(92,94),(95,98),(96,97),(99,101),(102,105),(103,104),(106,108),(109,112),(110,111)], [(1,111,27,90),(2,112,28,91),(3,106,22,85),(4,107,23,86),(5,108,24,87),(6,109,25,88),(7,110,26,89),(8,99,15,92),(9,100,16,93),(10,101,17,94),(11,102,18,95),(12,103,19,96),(13,104,20,97),(14,105,21,98),(29,71,50,64),(30,72,51,65),(31,73,52,66),(32,74,53,67),(33,75,54,68),(34,76,55,69),(35,77,56,70),(36,78,43,57),(37,79,44,58),(38,80,45,59),(39,81,46,60),(40,82,47,61),(41,83,48,62),(42,84,49,63)], [(1,62,20,76),(2,63,21,77),(3,57,15,71),(4,58,16,72),(5,59,17,73),(6,60,18,74),(7,61,19,75),(8,64,22,78),(9,65,23,79),(10,66,24,80),(11,67,25,81),(12,68,26,82),(13,69,27,83),(14,70,28,84),(29,85,43,99),(30,86,44,100),(31,87,45,101),(32,88,46,102),(33,89,47,103),(34,90,48,104),(35,91,49,105),(36,92,50,106),(37,93,51,107),(38,94,52,108),(39,95,53,109),(40,96,54,110),(41,97,55,111),(42,98,56,112)], [(1,76,27,69),(2,77,28,70),(3,71,22,64),(4,72,23,65),(5,73,24,66),(6,74,25,67),(7,75,26,68),(8,78,15,57),(9,79,16,58),(10,80,17,59),(11,81,18,60),(12,82,19,61),(13,83,20,62),(14,84,21,63),(29,85,50,106),(30,86,51,107),(31,87,52,108),(32,88,53,109),(33,89,54,110),(34,90,55,111),(35,91,56,112),(36,92,43,99),(37,93,44,100),(38,94,45,101),(39,95,46,102),(40,96,47,103),(41,97,48,104),(42,98,49,105)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 7 | 7 | 7 | 7 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | D4×D7 | D7×C4○D4 |
| kernel | D7×C4.4D4 | D7×C42 | C4.D28 | D7×C22⋊C4 | Dic7.D4 | C28.17D4 | C28.23D4 | C7×C4.4D4 | C2×D4×D7 | C2×Q8×D7 | C4×D7 | C4.4D4 | D14 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 8 | 3 | 12 | 3 | 3 | 6 | 12 |
Matrix representation of D7×C4.4D4 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 1 | 0 | 0 |
| 0 | 0 | 13 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 7 | 0 | 0 |
| 0 | 0 | 16 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 18 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 17 | 24 | 0 | 0 | 0 | 0 |
| 17 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 13 | 12 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,25,13,0,0,0,0,1,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,16,0,0,0,0,7,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,18,12],[12,0,0,0,0,0,5,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[17,17,0,0,0,0,24,12,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,13,0,0,0,0,0,12] >;
D7×C4.4D4 in GAP, Magma, Sage, TeX
D_7\times C_4._4D_4
% in TeX
G:=Group("D7xC4.4D4"); // GroupNames label
G:=SmallGroup(448,1126);
// by ID
G=gap.SmallGroup(448,1126);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,100,346,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^4=d^4=1,e^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=c^2*d^-1>;
// generators/relations