metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊20D14, C14.1262+ (1+4), (C2×Q8)⋊8D14, (C4×D28)⋊45C2, (C4×C28)⋊24C22, C22⋊C4⋊34D14, C4.4D4⋊12D7, C42⋊2D7⋊8C2, C22⋊D28⋊25C2, D14⋊D4⋊42C2, C23⋊D14⋊24C2, D14⋊C4⋊69C22, D14⋊3Q8⋊30C2, D14.8(C4○D4), (C2×D4).110D14, C4⋊Dic7⋊41C22, (Q8×C14)⋊14C22, Dic7⋊D4⋊34C2, Dic7⋊4D4⋊31C2, D14.D4⋊43C2, C28.23D4⋊22C2, (C2×C14).222C24, (C2×C28).631C23, Dic7⋊C4⋊36C22, C7⋊8(C22.32C24), (C4×Dic7)⋊36C22, C2.75(D4⋊6D14), C2.50(D4⋊8D14), C23.44(C22×D7), (D4×C14).210C22, (C2×D28).224C22, C22.D28⋊25C2, C23.D14⋊39C2, (C22×C14).52C23, (C22×D7).96C23, (C23×D7).65C22, C22.243(C23×D7), C23.D7.56C22, (C2×Dic7).254C23, (C22×Dic7)⋊27C22, C2.78(D7×C4○D4), (C2×C4×D7)⋊52C22, (D7×C22⋊C4)⋊18C2, C14.189(C2×C4○D4), (C7×C4.4D4)⋊14C2, (C2×C7⋊D4)⋊24C22, (C7×C22⋊C4)⋊30C22, (C2×C4).197(C22×D7), SmallGroup(448,1131)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1388 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×20], C7, C2×C4 [×5], C2×C4 [×9], D4 [×9], Q8, C23 [×2], C23 [×7], D7 [×4], C14 [×3], C14 [×2], C42, C42, C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×6], C2×Q8, C24, Dic7 [×5], C28 [×5], D14 [×2], D14 [×12], C2×C14, C2×C14 [×6], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C4.4D4, C42⋊2C2 [×2], C4×D7 [×3], D28 [×3], C2×Dic7 [×5], C2×Dic7, C7⋊D4 [×5], C2×C28 [×5], C7×D4, C7×Q8, C22×D7 [×3], C22×D7 [×4], C22×C14 [×2], C22.32C24, C4×Dic7, Dic7⋊C4 [×4], C4⋊Dic7 [×2], D14⋊C4 [×8], C23.D7 [×2], C4×C28, C7×C22⋊C4 [×4], C2×C4×D7 [×3], C2×D28 [×2], C22×Dic7, C2×C7⋊D4 [×4], D4×C14, Q8×C14, C23×D7, C4×D28, C42⋊2D7, C23.D14, D7×C22⋊C4, Dic7⋊4D4, C22⋊D28, D14.D4, D14⋊D4 [×2], C22.D28, C23⋊D14, Dic7⋊D4, D14⋊3Q8, C28.23D4, C7×C4.4D4, C42⋊20D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D7 [×7], C22.32C24, C23×D7, D4⋊6D14, D7×C4○D4, D4⋊8D14, C42⋊20D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >
►On 112 points
(1 48 12 82)(2 56 13 76)(3 50 14 84)(4 44 8 78)(5 52 9 72)(6 46 10 80)(7 54 11 74)(15 51 27 71)(16 45 28 79)(17 53 22 73)(18 47 23 81)(19 55 24 75)(20 49 25 83)(21 43 26 77)(29 87 36 59)(30 67 37 95)(31 89 38 61)(32 69 39 97)(33 91 40 63)(34 57 41 85)(35 93 42 65)(58 106 86 99)(60 108 88 101)(62 110 90 103)(64 112 92 105)(66 100 94 107)(68 102 96 109)(70 104 98 111)
(1 107 19 29)(2 101 20 37)(3 109 21 31)(4 103 15 39)(5 111 16 33)(6 105 17 41)(7 99 18 35)(8 110 27 32)(9 104 28 40)(10 112 22 34)(11 106 23 42)(12 100 24 36)(13 108 25 30)(14 102 26 38)(43 89 50 68)(44 62 51 97)(45 91 52 70)(46 64 53 85)(47 93 54 58)(48 66 55 87)(49 95 56 60)(57 80 92 73)(59 82 94 75)(61 84 96 77)(63 72 98 79)(65 74 86 81)(67 76 88 83)(69 78 90 71)
(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 27)(16 26)(17 25)(18 24)(19 23)(20 22)(21 28)(29 99)(30 112)(31 111)(32 110)(33 109)(34 108)(35 107)(36 106)(37 105)(38 104)(39 103)(40 102)(41 101)(42 100)(43 72)(44 71)(45 84)(46 83)(47 82)(48 81)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 67)(58 66)(59 65)(60 64)(61 63)(68 70)(85 95)(86 94)(87 93)(88 92)(89 91)(96 98)
G:=sub<Sym(112)| (1,48,12,82)(2,56,13,76)(3,50,14,84)(4,44,8,78)(5,52,9,72)(6,46,10,80)(7,54,11,74)(15,51,27,71)(16,45,28,79)(17,53,22,73)(18,47,23,81)(19,55,24,75)(20,49,25,83)(21,43,26,77)(29,87,36,59)(30,67,37,95)(31,89,38,61)(32,69,39,97)(33,91,40,63)(34,57,41,85)(35,93,42,65)(58,106,86,99)(60,108,88,101)(62,110,90,103)(64,112,92,105)(66,100,94,107)(68,102,96,109)(70,104,98,111), (1,107,19,29)(2,101,20,37)(3,109,21,31)(4,103,15,39)(5,111,16,33)(6,105,17,41)(7,99,18,35)(8,110,27,32)(9,104,28,40)(10,112,22,34)(11,106,23,42)(12,100,24,36)(13,108,25,30)(14,102,26,38)(43,89,50,68)(44,62,51,97)(45,91,52,70)(46,64,53,85)(47,93,54,58)(48,66,55,87)(49,95,56,60)(57,80,92,73)(59,82,94,75)(61,84,96,77)(63,72,98,79)(65,74,86,81)(67,76,88,83)(69,78,90,71), (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)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(21,28)(29,99)(30,112)(31,111)(32,110)(33,109)(34,108)(35,107)(36,106)(37,105)(38,104)(39,103)(40,102)(41,101)(42,100)(43,72)(44,71)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,67)(58,66)(59,65)(60,64)(61,63)(68,70)(85,95)(86,94)(87,93)(88,92)(89,91)(96,98)>;
G:=Group( (1,48,12,82)(2,56,13,76)(3,50,14,84)(4,44,8,78)(5,52,9,72)(6,46,10,80)(7,54,11,74)(15,51,27,71)(16,45,28,79)(17,53,22,73)(18,47,23,81)(19,55,24,75)(20,49,25,83)(21,43,26,77)(29,87,36,59)(30,67,37,95)(31,89,38,61)(32,69,39,97)(33,91,40,63)(34,57,41,85)(35,93,42,65)(58,106,86,99)(60,108,88,101)(62,110,90,103)(64,112,92,105)(66,100,94,107)(68,102,96,109)(70,104,98,111), (1,107,19,29)(2,101,20,37)(3,109,21,31)(4,103,15,39)(5,111,16,33)(6,105,17,41)(7,99,18,35)(8,110,27,32)(9,104,28,40)(10,112,22,34)(11,106,23,42)(12,100,24,36)(13,108,25,30)(14,102,26,38)(43,89,50,68)(44,62,51,97)(45,91,52,70)(46,64,53,85)(47,93,54,58)(48,66,55,87)(49,95,56,60)(57,80,92,73)(59,82,94,75)(61,84,96,77)(63,72,98,79)(65,74,86,81)(67,76,88,83)(69,78,90,71), (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)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(21,28)(29,99)(30,112)(31,111)(32,110)(33,109)(34,108)(35,107)(36,106)(37,105)(38,104)(39,103)(40,102)(41,101)(42,100)(43,72)(44,71)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,67)(58,66)(59,65)(60,64)(61,63)(68,70)(85,95)(86,94)(87,93)(88,92)(89,91)(96,98) );
G=PermutationGroup([(1,48,12,82),(2,56,13,76),(3,50,14,84),(4,44,8,78),(5,52,9,72),(6,46,10,80),(7,54,11,74),(15,51,27,71),(16,45,28,79),(17,53,22,73),(18,47,23,81),(19,55,24,75),(20,49,25,83),(21,43,26,77),(29,87,36,59),(30,67,37,95),(31,89,38,61),(32,69,39,97),(33,91,40,63),(34,57,41,85),(35,93,42,65),(58,106,86,99),(60,108,88,101),(62,110,90,103),(64,112,92,105),(66,100,94,107),(68,102,96,109),(70,104,98,111)], [(1,107,19,29),(2,101,20,37),(3,109,21,31),(4,103,15,39),(5,111,16,33),(6,105,17,41),(7,99,18,35),(8,110,27,32),(9,104,28,40),(10,112,22,34),(11,106,23,42),(12,100,24,36),(13,108,25,30),(14,102,26,38),(43,89,50,68),(44,62,51,97),(45,91,52,70),(46,64,53,85),(47,93,54,58),(48,66,55,87),(49,95,56,60),(57,80,92,73),(59,82,94,75),(61,84,96,77),(63,72,98,79),(65,74,86,81),(67,76,88,83),(69,78,90,71)], [(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),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22),(21,28),(29,99),(30,112),(31,111),(32,110),(33,109),(34,108),(35,107),(36,106),(37,105),(38,104),(39,103),(40,102),(41,101),(42,100),(43,72),(44,71),(45,84),(46,83),(47,82),(48,81),(49,80),(50,79),(51,78),(52,77),(53,76),(54,75),(55,74),(56,73),(57,67),(58,66),(59,65),(60,64),(61,63),(68,70),(85,95),(86,94),(87,93),(88,92),(89,91),(96,98)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 23 | 2 |
| 0 | 0 | 0 | 0 | 0 | 17 | 25 | 2 |
| 0 | 0 | 0 | 0 | 28 | 1 | 12 | 0 |
| 0 | 0 | 0 | 0 | 27 | 3 | 0 | 12 |
| 1 | 0 | 0 | 14 | 0 | 0 | 0 | 0 |
| 0 | 1 | 15 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 18 |
| 0 | 0 | 0 | 0 | 0 | 0 | 26 | 16 |
| 20 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 3 | 8 | 8 | 0 | 0 | 0 | 0 |
| 22 | 26 | 21 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 2 | 0 | 8 |
| 0 | 0 | 0 | 0 | 16 | 26 | 18 | 11 |
| 26 | 26 | 0 | 0 | 0 | 0 | 0 | 0 |
| 22 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 26 | 21 | 21 | 0 | 0 | 0 | 0 |
| 17 | 17 | 26 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 18 | 18 | 8 |
| 0 | 0 | 0 | 0 | 26 | 16 | 14 | 11 |
G:=sub<GL(8,GF(29))| [17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,28,27,0,0,0,0,0,17,1,3,0,0,0,0,23,25,12,0,0,0,0,0,2,2,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,15,28,0,0,0,0,0,14,4,0,28,0,0,0,0,0,0,0,0,2,11,0,0,0,0,0,0,18,27,0,0,0,0,0,0,0,0,13,26,0,0,0,0,0,0,18,16],[20,16,21,22,0,0,0,0,21,27,3,26,0,0,0,0,0,0,8,21,0,0,0,0,0,0,8,3,0,0,0,0,0,0,0,0,21,8,18,16,0,0,0,0,21,26,2,26,0,0,0,0,0,0,0,18,0,0,0,0,0,0,8,11],[26,22,8,17,0,0,0,0,26,3,26,17,0,0,0,0,0,0,21,26,0,0,0,0,0,0,21,8,0,0,0,0,0,0,0,0,21,26,2,26,0,0,0,0,21,8,18,16,0,0,0,0,0,0,18,14,0,0,0,0,0,0,8,11] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊6D14 | D7×C4○D4 | D4⋊8D14 |
| kernel | C42⋊20D14 | C4×D28 | C42⋊2D7 | C23.D14 | D7×C22⋊C4 | Dic7⋊4D4 | C22⋊D28 | D14.D4 | D14⋊D4 | C22.D28 | C23⋊D14 | Dic7⋊D4 | D14⋊3Q8 | C28.23D4 | C7×C4.4D4 | C4.4D4 | D14 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C14 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 3 | 12 | 3 | 3 | 2 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{20}D_{14} % in TeX
G:=Group("C4^2:20D14"); // GroupNames label
G:=SmallGroup(448,1131);
// by ID
G=gap.SmallGroup(448,1131);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,675,570,80,18822]);
// 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*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations