metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊28D14, D14⋊5(C4○D4), C22⋊C4⋊31D14, (C2×Dic7)⋊21D4, D28⋊C4⋊32C2, C23⋊D14⋊16C2, D14⋊D4⋊31C2, C22⋊D28⋊20C2, 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), (C2×C4×D7)⋊22C22, (D7×C22×C4)⋊24C2, (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
Generators and relations for C4⋊C4⋊28D14
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 >
Subgroups: 1644 in 330 conjugacy classes, 107 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C23×C4, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C22.19C24, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×C4×D7, C2×D28, D4⋊2D7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, 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, C4⋊C4⋊28D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, C22×D7, C22.19C24, D4×D7, C23×D7, C2×D4×D7, D7×C4○D4, C4⋊C4⋊28D14
►On 112 points
(1 100 22 82)(2 108 23 76)(3 102 24 84)(4 110 25 78)(5 104 26 72)(6 112 27 80)(7 106 28 74)(8 87 46 68)(9 95 47 62)(10 89 48 70)(11 97 49 64)(12 91 43 58)(13 85 44 66)(14 93 45 60)(15 111 35 79)(16 105 29 73)(17 99 30 81)(18 107 31 75)(19 101 32 83)(20 109 33 77)(21 103 34 71)(36 94 53 61)(37 88 54 69)(38 96 55 63)(39 90 56 57)(40 98 50 65)(41 92 51 59)(42 86 52 67)
(1 88 18 95)(2 96 19 89)(3 90 20 97)(4 98 21 91)(5 92 15 85)(6 86 16 93)(7 94 17 87)(8 106 36 99)(9 100 37 107)(10 108 38 101)(11 102 39 109)(12 110 40 103)(13 104 41 111)(14 112 42 105)(22 69 31 62)(23 63 32 70)(24 57 33 64)(25 65 34 58)(26 59 35 66)(27 67 29 60)(28 61 30 68)(43 78 50 71)(44 72 51 79)(45 80 52 73)(46 74 53 81)(47 82 54 75)(48 76 55 83)(49 84 56 77)
(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 30)(2 29)(3 35)(4 34)(5 33)(6 32)(7 31)(8 54)(9 53)(10 52)(11 51)(12 50)(13 56)(14 55)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(36 47)(37 46)(38 45)(39 44)(40 43)(41 49)(42 48)(57 66)(58 65)(59 64)(60 63)(61 62)(67 70)(68 69)(71 78)(72 77)(73 76)(74 75)(79 84)(80 83)(81 82)(85 90)(86 89)(87 88)(91 98)(92 97)(93 96)(94 95)(99 100)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)
G:=sub<Sym(112)| (1,100,22,82)(2,108,23,76)(3,102,24,84)(4,110,25,78)(5,104,26,72)(6,112,27,80)(7,106,28,74)(8,87,46,68)(9,95,47,62)(10,89,48,70)(11,97,49,64)(12,91,43,58)(13,85,44,66)(14,93,45,60)(15,111,35,79)(16,105,29,73)(17,99,30,81)(18,107,31,75)(19,101,32,83)(20,109,33,77)(21,103,34,71)(36,94,53,61)(37,88,54,69)(38,96,55,63)(39,90,56,57)(40,98,50,65)(41,92,51,59)(42,86,52,67), (1,88,18,95)(2,96,19,89)(3,90,20,97)(4,98,21,91)(5,92,15,85)(6,86,16,93)(7,94,17,87)(8,106,36,99)(9,100,37,107)(10,108,38,101)(11,102,39,109)(12,110,40,103)(13,104,41,111)(14,112,42,105)(22,69,31,62)(23,63,32,70)(24,57,33,64)(25,65,34,58)(26,59,35,66)(27,67,29,60)(28,61,30,68)(43,78,50,71)(44,72,51,79)(45,80,52,73)(46,74,53,81)(47,82,54,75)(48,76,55,83)(49,84,56,77), (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), (1,30)(2,29)(3,35)(4,34)(5,33)(6,32)(7,31)(8,54)(9,53)(10,52)(11,51)(12,50)(13,56)(14,55)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(36,47)(37,46)(38,45)(39,44)(40,43)(41,49)(42,48)(57,66)(58,65)(59,64)(60,63)(61,62)(67,70)(68,69)(71,78)(72,77)(73,76)(74,75)(79,84)(80,83)(81,82)(85,90)(86,89)(87,88)(91,98)(92,97)(93,96)(94,95)(99,100)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)>;
G:=Group( (1,100,22,82)(2,108,23,76)(3,102,24,84)(4,110,25,78)(5,104,26,72)(6,112,27,80)(7,106,28,74)(8,87,46,68)(9,95,47,62)(10,89,48,70)(11,97,49,64)(12,91,43,58)(13,85,44,66)(14,93,45,60)(15,111,35,79)(16,105,29,73)(17,99,30,81)(18,107,31,75)(19,101,32,83)(20,109,33,77)(21,103,34,71)(36,94,53,61)(37,88,54,69)(38,96,55,63)(39,90,56,57)(40,98,50,65)(41,92,51,59)(42,86,52,67), (1,88,18,95)(2,96,19,89)(3,90,20,97)(4,98,21,91)(5,92,15,85)(6,86,16,93)(7,94,17,87)(8,106,36,99)(9,100,37,107)(10,108,38,101)(11,102,39,109)(12,110,40,103)(13,104,41,111)(14,112,42,105)(22,69,31,62)(23,63,32,70)(24,57,33,64)(25,65,34,58)(26,59,35,66)(27,67,29,60)(28,61,30,68)(43,78,50,71)(44,72,51,79)(45,80,52,73)(46,74,53,81)(47,82,54,75)(48,76,55,83)(49,84,56,77), (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), (1,30)(2,29)(3,35)(4,34)(5,33)(6,32)(7,31)(8,54)(9,53)(10,52)(11,51)(12,50)(13,56)(14,55)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(36,47)(37,46)(38,45)(39,44)(40,43)(41,49)(42,48)(57,66)(58,65)(59,64)(60,63)(61,62)(67,70)(68,69)(71,78)(72,77)(73,76)(74,75)(79,84)(80,83)(81,82)(85,90)(86,89)(87,88)(91,98)(92,97)(93,96)(94,95)(99,100)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107) );
G=PermutationGroup([[(1,100,22,82),(2,108,23,76),(3,102,24,84),(4,110,25,78),(5,104,26,72),(6,112,27,80),(7,106,28,74),(8,87,46,68),(9,95,47,62),(10,89,48,70),(11,97,49,64),(12,91,43,58),(13,85,44,66),(14,93,45,60),(15,111,35,79),(16,105,29,73),(17,99,30,81),(18,107,31,75),(19,101,32,83),(20,109,33,77),(21,103,34,71),(36,94,53,61),(37,88,54,69),(38,96,55,63),(39,90,56,57),(40,98,50,65),(41,92,51,59),(42,86,52,67)], [(1,88,18,95),(2,96,19,89),(3,90,20,97),(4,98,21,91),(5,92,15,85),(6,86,16,93),(7,94,17,87),(8,106,36,99),(9,100,37,107),(10,108,38,101),(11,102,39,109),(12,110,40,103),(13,104,41,111),(14,112,42,105),(22,69,31,62),(23,63,32,70),(24,57,33,64),(25,65,34,58),(26,59,35,66),(27,67,29,60),(28,61,30,68),(43,78,50,71),(44,72,51,79),(45,80,52,73),(46,74,53,81),(47,82,54,75),(48,76,55,83),(49,84,56,77)], [(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)], [(1,30),(2,29),(3,35),(4,34),(5,33),(6,32),(7,31),(8,54),(9,53),(10,52),(11,51),(12,50),(13,56),(14,55),(15,24),(16,23),(17,22),(18,28),(19,27),(20,26),(21,25),(36,47),(37,46),(38,45),(39,44),(40,43),(41,49),(42,48),(57,66),(58,65),(59,64),(60,63),(61,62),(67,70),(68,69),(71,78),(72,77),(73,76),(74,75),(79,84),(80,83),(81,82),(85,90),(86,89),(87,88),(91,98),(92,97),(93,96),(94,95),(99,100),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107)]])
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 |
Matrix representation of C4⋊C4⋊28D14 ►in 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] >;
C4⋊C4⋊28D14 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