metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊28D14, D14⋊5(C4○D4), C22⋊C4⋊31D14, (C2×Dic7)⋊21D4, D14⋊D4⋊31C2, C22⋊D28⋊20C2, D28⋊C4⋊32C2, C23⋊D14⋊16C2, C22.45(D4×D7), D14⋊C4⋊27C22, D14⋊Q8⋊29C2, (C2×D4).165D14, (C2×C28).73C23, Dic7.49(C2×D4), C14.85(C22×D4), Dic7⋊4D4⋊20C2, (C2×C14).200C24, Dic7⋊C4⋊23C22, C7⋊6(C22.19C24), (C4×Dic7)⋊32C22, (C22×C4).322D14, C22.D4⋊18D7, C23.27(C22×D7), (C2×Dic14)⋊28C22, (D4×C14).138C22, (C2×D28).156C22, (C22×C14).35C23, C22.221(C23×D7), C23.D7.43C22, C23.23D14⋊21C2, C23.11D14⋊13C2, (C22×C28).368C22, (C2×Dic7).104C23, (C22×Dic7)⋊25C22, (C23×D7).110C22, (C22×D7).208C23, C2.58(C2×D4×D7), C2.62(D7×C4○D4), (D7×C22×C4)⋊24C2, (C2×C4×D7)⋊22C22, (C7×C4⋊C4)⋊26C22, (C2×C14).61(C2×D4), (C2×D4⋊2D7)⋊17C2, C14.174(C2×C4○D4), (C2×C7⋊D4)⋊19C22, (C2×C4).63(C22×D7), (C7×C22⋊C4)⋊22C22, (C7×C22.D4)⋊8C2, SmallGroup(448,1109)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1644 in 330 conjugacy classes, 107 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×12], C22, C22 [×2], C22 [×24], C7, C2×C4, C2×C4 [×4], C2×C4 [×23], D4 [×14], Q8 [×2], C23 [×2], C23 [×9], D7 [×5], C14, C14 [×2], C14 [×3], C42 [×2], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4, C22×C4 [×11], C2×D4, C2×D4 [×6], C2×Q8, C4○D4 [×4], C24, Dic7 [×4], Dic7 [×3], C28 [×5], D14 [×4], D14 [×15], C2×C14, C2×C14 [×2], C2×C14 [×5], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C22.D4, C23×C4, C2×C4○D4, Dic14 [×2], C4×D7 [×10], D28 [×4], C2×Dic7 [×3], C2×Dic7 [×6], C2×Dic7 [×2], C7⋊D4 [×8], C2×C28, C2×C28 [×4], C2×C28 [×2], C7×D4 [×2], C22×D7, C22×D7 [×2], C22×D7 [×6], C22×C14 [×2], C22.19C24, C4×Dic7 [×2], Dic7⋊C4 [×4], D14⋊C4 [×6], C23.D7, C7×C22⋊C4, C7×C22⋊C4 [×2], C7×C4⋊C4 [×2], C2×Dic14, C2×C4×D7, C2×C4×D7 [×4], C2×C4×D7 [×4], C2×D28 [×2], D4⋊2D7 [×4], C22×Dic7 [×2], C2×C7⋊D4 [×2], C2×C7⋊D4 [×2], C22×C28, D4×C14, C23×D7, C23.11D14, Dic7⋊4D4 [×2], C22⋊D28, D14⋊D4 [×2], D28⋊C4 [×2], D14⋊Q8 [×2], C23.23D14, C23⋊D14, C7×C22.D4, D7×C22×C4, C2×D4⋊2D7, C4⋊C4⋊28D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×4], C24, D14 [×7], C22×D4, C2×C4○D4 [×2], C22×D7 [×7], C22.19C24, D4×D7 [×2], C23×D7, C2×D4×D7, D7×C4○D4 [×2], C4⋊C4⋊28D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=d2=1, bab-1=dad=a-1, cac-1=ab2, cbc-1=b-1, dbd=a2b, dcd=c-1 >
►On 112 points
(1 105 29 74)(2 99 30 82)(3 107 31 76)(4 101 32 84)(5 109 33 78)(6 103 34 72)(7 111 35 80)(8 66 15 98)(9 60 16 92)(10 68 17 86)(11 62 18 94)(12 70 19 88)(13 64 20 96)(14 58 21 90)(22 59 48 91)(23 67 49 85)(24 61 43 93)(25 69 44 87)(26 63 45 95)(27 57 46 89)(28 65 47 97)(36 108 53 77)(37 102 54 71)(38 110 55 79)(39 104 56 73)(40 112 50 81)(41 106 51 75)(42 100 52 83)
(1 85 40 92)(2 93 41 86)(3 87 42 94)(4 95 36 88)(5 89 37 96)(6 97 38 90)(7 91 39 98)(8 80 22 73)(9 74 23 81)(10 82 24 75)(11 76 25 83)(12 84 26 77)(13 78 27 71)(14 72 28 79)(15 111 48 104)(16 105 49 112)(17 99 43 106)(18 107 44 100)(19 101 45 108)(20 109 46 102)(21 103 47 110)(29 67 50 60)(30 61 51 68)(31 69 52 62)(32 63 53 70)(33 57 54 64)(34 65 55 58)(35 59 56 66)
(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 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 56)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 49)(15 22)(16 28)(17 27)(18 26)(19 25)(20 24)(21 23)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(69 70)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(83 84)(85 90)(86 89)(87 88)(91 98)(92 97)(93 96)(94 95)(99 102)(100 101)(103 112)(104 111)(105 110)(106 109)(107 108)
G:=sub<Sym(112)| (1,105,29,74)(2,99,30,82)(3,107,31,76)(4,101,32,84)(5,109,33,78)(6,103,34,72)(7,111,35,80)(8,66,15,98)(9,60,16,92)(10,68,17,86)(11,62,18,94)(12,70,19,88)(13,64,20,96)(14,58,21,90)(22,59,48,91)(23,67,49,85)(24,61,43,93)(25,69,44,87)(26,63,45,95)(27,57,46,89)(28,65,47,97)(36,108,53,77)(37,102,54,71)(38,110,55,79)(39,104,56,73)(40,112,50,81)(41,106,51,75)(42,100,52,83), (1,85,40,92)(2,93,41,86)(3,87,42,94)(4,95,36,88)(5,89,37,96)(6,97,38,90)(7,91,39,98)(8,80,22,73)(9,74,23,81)(10,82,24,75)(11,76,25,83)(12,84,26,77)(13,78,27,71)(14,72,28,79)(15,111,48,104)(16,105,49,112)(17,99,43,106)(18,107,44,100)(19,101,45,108)(20,109,46,102)(21,103,47,110)(29,67,50,60)(30,61,51,68)(31,69,52,62)(32,63,53,70)(33,57,54,64)(34,65,55,58)(35,59,56,66), (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,56)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,49)(15,22)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(69,70)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(83,84)(85,90)(86,89)(87,88)(91,98)(92,97)(93,96)(94,95)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108)>;
G:=Group( (1,105,29,74)(2,99,30,82)(3,107,31,76)(4,101,32,84)(5,109,33,78)(6,103,34,72)(7,111,35,80)(8,66,15,98)(9,60,16,92)(10,68,17,86)(11,62,18,94)(12,70,19,88)(13,64,20,96)(14,58,21,90)(22,59,48,91)(23,67,49,85)(24,61,43,93)(25,69,44,87)(26,63,45,95)(27,57,46,89)(28,65,47,97)(36,108,53,77)(37,102,54,71)(38,110,55,79)(39,104,56,73)(40,112,50,81)(41,106,51,75)(42,100,52,83), (1,85,40,92)(2,93,41,86)(3,87,42,94)(4,95,36,88)(5,89,37,96)(6,97,38,90)(7,91,39,98)(8,80,22,73)(9,74,23,81)(10,82,24,75)(11,76,25,83)(12,84,26,77)(13,78,27,71)(14,72,28,79)(15,111,48,104)(16,105,49,112)(17,99,43,106)(18,107,44,100)(19,101,45,108)(20,109,46,102)(21,103,47,110)(29,67,50,60)(30,61,51,68)(31,69,52,62)(32,63,53,70)(33,57,54,64)(34,65,55,58)(35,59,56,66), (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,56)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,49)(15,22)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(69,70)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(83,84)(85,90)(86,89)(87,88)(91,98)(92,97)(93,96)(94,95)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108) );
G=PermutationGroup([(1,105,29,74),(2,99,30,82),(3,107,31,76),(4,101,32,84),(5,109,33,78),(6,103,34,72),(7,111,35,80),(8,66,15,98),(9,60,16,92),(10,68,17,86),(11,62,18,94),(12,70,19,88),(13,64,20,96),(14,58,21,90),(22,59,48,91),(23,67,49,85),(24,61,43,93),(25,69,44,87),(26,63,45,95),(27,57,46,89),(28,65,47,97),(36,108,53,77),(37,102,54,71),(38,110,55,79),(39,104,56,73),(40,112,50,81),(41,106,51,75),(42,100,52,83)], [(1,85,40,92),(2,93,41,86),(3,87,42,94),(4,95,36,88),(5,89,37,96),(6,97,38,90),(7,91,39,98),(8,80,22,73),(9,74,23,81),(10,82,24,75),(11,76,25,83),(12,84,26,77),(13,78,27,71),(14,72,28,79),(15,111,48,104),(16,105,49,112),(17,99,43,106),(18,107,44,100),(19,101,45,108),(20,109,46,102),(21,103,47,110),(29,67,50,60),(30,61,51,68),(31,69,52,62),(32,63,53,70),(33,57,54,64),(34,65,55,58),(35,59,56,66)], [(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)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,56),(8,48),(9,47),(10,46),(11,45),(12,44),(13,43),(14,49),(15,22),(16,28),(17,27),(18,26),(19,25),(20,24),(21,23),(29,38),(30,37),(31,36),(32,42),(33,41),(34,40),(35,39),(57,68),(58,67),(59,66),(60,65),(61,64),(62,63),(69,70),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77),(83,84),(85,90),(86,89),(87,88),(91,98),(92,97),(93,96),(94,95),(99,102),(100,101),(103,112),(104,111),(105,110),(106,109),(107,108)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 12 | 17 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 7 |
| 0 | 0 | 0 | 0 | 8 | 28 |
| 28 | 1 | 0 | 0 | 0 | 0 |
| 27 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 7 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 26 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 20 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 21 | 1 |
G:=sub<GL(6,GF(29))| [12,0,0,0,0,0,17,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,7,28],[28,27,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,7,28],[1,2,0,0,0,0,0,28,0,0,0,0,0,0,3,26,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,28,0,0,0,0,0,0,1,20,0,0,0,0,0,28,0,0,0,0,0,0,28,21,0,0,0,0,0,1] >;
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 14 | 14 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
70 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | D4×D7 | D7×C4○D4 |
| kernel | C4⋊C4⋊28D14 | C23.11D14 | Dic7⋊4D4 | C22⋊D28 | D14⋊D4 | D28⋊C4 | D14⋊Q8 | C23.23D14 | C23⋊D14 | C7×C22.D4 | D7×C22×C4 | C2×D4⋊2D7 | C2×Dic7 | C22.D4 | D14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 8 | 9 | 6 | 3 | 3 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_{28}D_{14} % in TeX
G:=Group("C4:C4:28D14"); // GroupNames label
G:=SmallGroup(448,1109);
// by ID
G=gap.SmallGroup(448,1109);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,100,1123,346,297,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a*b^2,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations