metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊26D14, C14.762+ (1+4), (C4×D7)⋊5D4, C4⋊1D4⋊5D7, C4.34(D4×D7), (C2×D4)⋊12D14, C28.65(C2×D4), C28⋊D4⋊26C2, (C4×C28)⋊26C22, D14.47(C2×D4), C23⋊D14⋊26C2, D14⋊C4⋊34C22, C4.D28⋊25C2, (D4×C14)⋊32C22, C42⋊D7⋊23C2, Dic7.52(C2×D4), C14.93(C22×D4), Dic7⋊D4⋊36C2, C28.17D4⋊26C2, (C2×C28).635C23, (C2×C14).259C24, Dic7⋊C4⋊71C22, C7⋊5(C22.29C24), (C4×Dic7)⋊39C22, C23.D7⋊36C22, C2.80(D4⋊6D14), C23.65(C22×D7), (C2×Dic14)⋊34C22, (C2×D28).170C22, (C22×C14).73C23, (C23×D7).72C22, C22.280(C23×D7), (C2×Dic7).134C23, (C22×Dic7)⋊29C22, (C22×D7).227C23, (C2×D4×D7)⋊19C2, C2.66(C2×D4×D7), (C7×C4⋊1D4)⋊6C2, (C2×D4⋊2D7)⋊20C2, (C2×C7⋊D4)⋊26C22, (C2×C4×D7).138C22, (C2×C4).213(C22×D7), SmallGroup(448,1168)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1900 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×30], C7, C2×C4, C2×C4 [×2], C2×C4 [×13], D4 [×22], Q8 [×2], C23 [×4], C23 [×11], D7 [×4], C14, C14 [×2], C14 [×4], C42, C42, C22⋊C4 [×10], C4⋊C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×4], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24 [×2], Dic7 [×2], Dic7 [×4], C28 [×2], C28 [×2], D14 [×2], D14 [×16], C2×C14, C2×C14 [×12], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊1D4, C22×D4, C2×C4○D4, Dic14 [×2], C4×D7 [×4], D28 [×2], C2×Dic7, C2×Dic7 [×4], C2×Dic7 [×4], C7⋊D4 [×12], C2×C28, C2×C28 [×2], C7×D4 [×8], C22×D7, C22×D7 [×2], C22×D7 [×8], C22×C14 [×4], C22.29C24, C4×Dic7, Dic7⋊C4 [×2], D14⋊C4 [×6], C23.D7 [×4], C4×C28, C2×Dic14, C2×C4×D7, C2×D28, D4×D7 [×4], D4⋊2D7 [×4], C22×Dic7 [×2], C2×C7⋊D4 [×8], D4×C14 [×2], D4×C14 [×4], C23×D7 [×2], C42⋊D7, C4.D28, C28.17D4, C23⋊D14 [×4], Dic7⋊D4 [×4], C28⋊D4, C7×C4⋊1D4, C2×D4×D7, C2×D4⋊2D7, C42⋊26D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, 2+ (1+4) [×2], C22×D7 [×7], C22.29C24, D4×D7 [×2], C23×D7, C2×D4×D7, D4⋊6D14 [×2], C42⋊26D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=b-1, dbd=a2b-1, dcd=c-1 >
►On 112 points
(1 42 19 35)(2 36 20 29)(3 30 21 37)(4 38 15 31)(5 32 16 39)(6 40 17 33)(7 34 18 41)(8 103 23 110)(9 111 24 104)(10 105 25 112)(11 99 26 106)(12 107 27 100)(13 101 28 108)(14 109 22 102)(43 79 91 61)(44 62 92 80)(45 81 93 63)(46 64 94 82)(47 83 95 65)(48 66 96 84)(49 71 97 67)(50 68 98 72)(51 73 85 69)(52 70 86 74)(53 75 87 57)(54 58 88 76)(55 77 89 59)(56 60 90 78)
(1 58 12 65)(2 66 13 59)(3 60 14 67)(4 68 8 61)(5 62 9 69)(6 70 10 63)(7 64 11 57)(15 72 23 79)(16 80 24 73)(17 74 25 81)(18 82 26 75)(19 76 27 83)(20 84 28 77)(21 78 22 71)(29 48 108 89)(30 90 109 49)(31 50 110 91)(32 92 111 51)(33 52 112 93)(34 94 99 53)(35 54 100 95)(36 96 101 55)(37 56 102 97)(38 98 103 43)(39 44 104 85)(40 86 105 45)(41 46 106 87)(42 88 107 47)
(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 11)(2 10)(3 9)(4 8)(5 14)(6 13)(7 12)(15 23)(16 22)(17 28)(18 27)(19 26)(20 25)(21 24)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 99)(36 112)(37 111)(38 110)(39 109)(40 108)(41 107)(42 106)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(57 83)(58 82)(59 81)(60 80)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(70 84)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)
G:=sub<Sym(112)| (1,42,19,35)(2,36,20,29)(3,30,21,37)(4,38,15,31)(5,32,16,39)(6,40,17,33)(7,34,18,41)(8,103,23,110)(9,111,24,104)(10,105,25,112)(11,99,26,106)(12,107,27,100)(13,101,28,108)(14,109,22,102)(43,79,91,61)(44,62,92,80)(45,81,93,63)(46,64,94,82)(47,83,95,65)(48,66,96,84)(49,71,97,67)(50,68,98,72)(51,73,85,69)(52,70,86,74)(53,75,87,57)(54,58,88,76)(55,77,89,59)(56,60,90,78), (1,58,12,65)(2,66,13,59)(3,60,14,67)(4,68,8,61)(5,62,9,69)(6,70,10,63)(7,64,11,57)(15,72,23,79)(16,80,24,73)(17,74,25,81)(18,82,26,75)(19,76,27,83)(20,84,28,77)(21,78,22,71)(29,48,108,89)(30,90,109,49)(31,50,110,91)(32,92,111,51)(33,52,112,93)(34,94,99,53)(35,54,100,95)(36,96,101,55)(37,56,102,97)(38,98,103,43)(39,44,104,85)(40,86,105,45)(41,46,106,87)(42,88,107,47), (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), (1,11)(2,10)(3,9)(4,8)(5,14)(6,13)(7,12)(15,23)(16,22)(17,28)(18,27)(19,26)(20,25)(21,24)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,112)(37,111)(38,110)(39,109)(40,108)(41,107)(42,106)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(70,84)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)>;
G:=Group( (1,42,19,35)(2,36,20,29)(3,30,21,37)(4,38,15,31)(5,32,16,39)(6,40,17,33)(7,34,18,41)(8,103,23,110)(9,111,24,104)(10,105,25,112)(11,99,26,106)(12,107,27,100)(13,101,28,108)(14,109,22,102)(43,79,91,61)(44,62,92,80)(45,81,93,63)(46,64,94,82)(47,83,95,65)(48,66,96,84)(49,71,97,67)(50,68,98,72)(51,73,85,69)(52,70,86,74)(53,75,87,57)(54,58,88,76)(55,77,89,59)(56,60,90,78), (1,58,12,65)(2,66,13,59)(3,60,14,67)(4,68,8,61)(5,62,9,69)(6,70,10,63)(7,64,11,57)(15,72,23,79)(16,80,24,73)(17,74,25,81)(18,82,26,75)(19,76,27,83)(20,84,28,77)(21,78,22,71)(29,48,108,89)(30,90,109,49)(31,50,110,91)(32,92,111,51)(33,52,112,93)(34,94,99,53)(35,54,100,95)(36,96,101,55)(37,56,102,97)(38,98,103,43)(39,44,104,85)(40,86,105,45)(41,46,106,87)(42,88,107,47), (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), (1,11)(2,10)(3,9)(4,8)(5,14)(6,13)(7,12)(15,23)(16,22)(17,28)(18,27)(19,26)(20,25)(21,24)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,112)(37,111)(38,110)(39,109)(40,108)(41,107)(42,106)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(70,84)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92) );
G=PermutationGroup([(1,42,19,35),(2,36,20,29),(3,30,21,37),(4,38,15,31),(5,32,16,39),(6,40,17,33),(7,34,18,41),(8,103,23,110),(9,111,24,104),(10,105,25,112),(11,99,26,106),(12,107,27,100),(13,101,28,108),(14,109,22,102),(43,79,91,61),(44,62,92,80),(45,81,93,63),(46,64,94,82),(47,83,95,65),(48,66,96,84),(49,71,97,67),(50,68,98,72),(51,73,85,69),(52,70,86,74),(53,75,87,57),(54,58,88,76),(55,77,89,59),(56,60,90,78)], [(1,58,12,65),(2,66,13,59),(3,60,14,67),(4,68,8,61),(5,62,9,69),(6,70,10,63),(7,64,11,57),(15,72,23,79),(16,80,24,73),(17,74,25,81),(18,82,26,75),(19,76,27,83),(20,84,28,77),(21,78,22,71),(29,48,108,89),(30,90,109,49),(31,50,110,91),(32,92,111,51),(33,52,112,93),(34,94,99,53),(35,54,100,95),(36,96,101,55),(37,56,102,97),(38,98,103,43),(39,44,104,85),(40,86,105,45),(41,46,106,87),(42,88,107,47)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,14),(6,13),(7,12),(15,23),(16,22),(17,28),(18,27),(19,26),(20,25),(21,24),(29,105),(30,104),(31,103),(32,102),(33,101),(34,100),(35,99),(36,112),(37,111),(38,110),(39,109),(40,108),(41,107),(42,106),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(57,83),(58,82),(59,81),(60,80),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(70,84),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 2 |
| 0 | 0 | 0 | 0 | 27 | 18 |
| 0 | 0 | 18 | 27 | 0 | 0 |
| 0 | 0 | 2 | 11 | 0 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 28 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 28 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 25 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 25 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 28 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 18 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 18 | 25 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,18,2,0,0,0,0,27,11,0,0,11,27,0,0,0,0,2,18,0,0],[1,28,0,0,0,0,2,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,0,1,0,0],[1,28,0,0,0,0,0,28,0,0,0,0,0,0,25,4,0,0,0,0,25,11,0,0,0,0,0,0,4,25,0,0,0,0,4,18],[1,28,0,0,0,0,0,28,0,0,0,0,0,0,4,18,0,0,0,0,4,25,0,0,0,0,0,0,4,18,0,0,0,0,4,25] >;
64 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 | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | 2+ (1+4) | D4×D7 | D4⋊6D14 |
| kernel | C42⋊26D14 | C42⋊D7 | C4.D28 | C28.17D4 | C23⋊D14 | Dic7⋊D4 | C28⋊D4 | C7×C4⋊1D4 | C2×D4×D7 | C2×D4⋊2D7 | C4×D7 | C4⋊1D4 | C42 | C2×D4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 18 | 2 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{26}D_{14} % in TeX
G:=Group("C4^2:26D14"); // GroupNames label
G:=SmallGroup(448,1168);
// by ID
G=gap.SmallGroup(448,1168);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,675,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=b^-1,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations