metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊5D14, D28.13D4, Dic14.13D4, (C2×Q8)⋊2D14, C4.51(D4×D7), C4.4D4⋊4D7, C28.28(C2×D4), (C4×C28)⋊13C22, (C2×D4).51D14, C28.D4⋊5C2, C7⋊3(D4.9D4), Dic14⋊C4⋊11C2, (Q8×C14)⋊2C22, C14.51C22≀C2, D4⋊6D14.4C2, C28.C23⋊2C2, (C22×C14).22D4, C4.Dic7⋊6C22, (C2×C28).379C23, C4○D28.19C22, (D4×C14).67C22, C23.10(C7⋊D4), C2.19(C23⋊D14), (C7×C4.4D4)⋊4C2, (C2×C14).510(C2×D4), C22.31(C2×C7⋊D4), (C2×C4).116(C22×D7), SmallGroup(448,595)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊5D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=a-1b2, dad=a-1b, cbc-1=b-1, bd=db, dcd=c-1 >
Subgroups: 876 in 152 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, M4(2), SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.D4, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C7⋊C8, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×C14, D4.9D4, C4.Dic7, Q8⋊D7, C7⋊Q16, C4×C28, C7×C22⋊C4, C4○D28, D4×D7, D4⋊2D7, C2×C7⋊D4, D4×C14, Q8×C14, Dic14⋊C4, C28.D4, C28.C23, C7×C4.4D4, D4⋊6D14, C42⋊5D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, D4.9D4, D4×D7, C2×C7⋊D4, C23⋊D14, C42⋊5D14
►On 112 points
Generators in S112
(2 50)(4 52)(6 54)(8 56)(10 44)(12 46)(14 48)(15 37)(17 39)(19 41)(21 29)(23 31)(25 33)(27 35)(57 81 89 108)(58 82 90 109)(59 83 91 110)(60 84 92 111)(61 71 93 112)(62 72 94 99)(63 73 95 100)(64 74 96 101)(65 75 97 102)(66 76 98 103)(67 77 85 104)(68 78 86 105)(69 79 87 106)(70 80 88 107)
(1 34 49 26)(2 27 50 35)(3 36 51 28)(4 15 52 37)(5 38 53 16)(6 17 54 39)(7 40 55 18)(8 19 56 41)(9 42 43 20)(10 21 44 29)(11 30 45 22)(12 23 46 31)(13 32 47 24)(14 25 48 33)(57 81 89 108)(58 109 90 82)(59 83 91 110)(60 111 92 84)(61 71 93 112)(62 99 94 72)(63 73 95 100)(64 101 96 74)(65 75 97 102)(66 103 98 76)(67 77 85 104)(68 105 86 78)(69 79 87 106)(70 107 88 80)
(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 89)(2 88)(3 87)(4 86)(5 85)(6 98)(7 97)(8 96)(9 95)(10 94)(11 93)(12 92)(13 91)(14 90)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 72)(22 71)(23 84)(24 83)(25 82)(26 81)(27 80)(28 79)(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 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)
G:=sub<Sym(112)| (2,50)(4,52)(6,54)(8,56)(10,44)(12,46)(14,48)(15,37)(17,39)(19,41)(21,29)(23,31)(25,33)(27,35)(57,81,89,108)(58,82,90,109)(59,83,91,110)(60,84,92,111)(61,71,93,112)(62,72,94,99)(63,73,95,100)(64,74,96,101)(65,75,97,102)(66,76,98,103)(67,77,85,104)(68,78,86,105)(69,79,87,106)(70,80,88,107), (1,34,49,26)(2,27,50,35)(3,36,51,28)(4,15,52,37)(5,38,53,16)(6,17,54,39)(7,40,55,18)(8,19,56,41)(9,42,43,20)(10,21,44,29)(11,30,45,22)(12,23,46,31)(13,32,47,24)(14,25,48,33)(57,81,89,108)(58,109,90,82)(59,83,91,110)(60,111,92,84)(61,71,93,112)(62,99,94,72)(63,73,95,100)(64,101,96,74)(65,75,97,102)(66,103,98,76)(67,77,85,104)(68,105,86,78)(69,79,87,106)(70,107,88,80), (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), (1,89)(2,88)(3,87)(4,86)(5,85)(6,98)(7,97)(8,96)(9,95)(10,94)(11,93)(12,92)(13,91)(14,90)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(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)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)>;
G:=Group( (2,50)(4,52)(6,54)(8,56)(10,44)(12,46)(14,48)(15,37)(17,39)(19,41)(21,29)(23,31)(25,33)(27,35)(57,81,89,108)(58,82,90,109)(59,83,91,110)(60,84,92,111)(61,71,93,112)(62,72,94,99)(63,73,95,100)(64,74,96,101)(65,75,97,102)(66,76,98,103)(67,77,85,104)(68,78,86,105)(69,79,87,106)(70,80,88,107), (1,34,49,26)(2,27,50,35)(3,36,51,28)(4,15,52,37)(5,38,53,16)(6,17,54,39)(7,40,55,18)(8,19,56,41)(9,42,43,20)(10,21,44,29)(11,30,45,22)(12,23,46,31)(13,32,47,24)(14,25,48,33)(57,81,89,108)(58,109,90,82)(59,83,91,110)(60,111,92,84)(61,71,93,112)(62,99,94,72)(63,73,95,100)(64,101,96,74)(65,75,97,102)(66,103,98,76)(67,77,85,104)(68,105,86,78)(69,79,87,106)(70,107,88,80), (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), (1,89)(2,88)(3,87)(4,86)(5,85)(6,98)(7,97)(8,96)(9,95)(10,94)(11,93)(12,92)(13,91)(14,90)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(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)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64) );
G=PermutationGroup([[(2,50),(4,52),(6,54),(8,56),(10,44),(12,46),(14,48),(15,37),(17,39),(19,41),(21,29),(23,31),(25,33),(27,35),(57,81,89,108),(58,82,90,109),(59,83,91,110),(60,84,92,111),(61,71,93,112),(62,72,94,99),(63,73,95,100),(64,74,96,101),(65,75,97,102),(66,76,98,103),(67,77,85,104),(68,78,86,105),(69,79,87,106),(70,80,88,107)], [(1,34,49,26),(2,27,50,35),(3,36,51,28),(4,15,52,37),(5,38,53,16),(6,17,54,39),(7,40,55,18),(8,19,56,41),(9,42,43,20),(10,21,44,29),(11,30,45,22),(12,23,46,31),(13,32,47,24),(14,25,48,33),(57,81,89,108),(58,109,90,82),(59,83,91,110),(60,111,92,84),(61,71,93,112),(62,99,94,72),(63,73,95,100),(64,101,96,74),(65,75,97,102),(66,103,98,76),(67,77,85,104),(68,105,86,78),(69,79,87,106),(70,107,88,80)], [(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)], [(1,89),(2,88),(3,87),(4,86),(5,85),(6,98),(7,97),(8,96),(9,95),(10,94),(11,93),(12,92),(13,91),(14,90),(15,78),(16,77),(17,76),(18,75),(19,74),(20,73),(21,72),(22,71),(23,84),(24,83),(25,82),(26,81),(27,80),(28,79),(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),(43,63),(44,62),(45,61),(46,60),(47,59),(48,58),(49,57),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 4 | 4 | 8 | 28 | 28 | 2 | 2 | 2 | 56 | 56 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | D4.9D4 | D4×D7 | C42⋊5D14 |
| kernel | C42⋊5D14 | Dic14⋊C4 | C28.D4 | C28.C23 | C7×C4.4D4 | D4⋊6D14 | Dic14 | D28 | C22×C14 | C4.4D4 | C42 | C2×D4 | C2×Q8 | C23 | C7 | C4 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 12 | 2 | 6 | 12 |
Matrix representation of C42⋊5D14 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 1 | 112 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 98 | 0 | 0 | 0 |
| 98 | 15 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 15 | 98 |
| 49 | 15 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 30 | 53 |
| 0 | 0 | 0 | 83 |
| 0 | 0 | 30 | 53 |
| 0 | 0 | 0 | 83 |
| 49 | 15 | 0 | 0 |
| 0 | 64 | 0 | 0 |
G:=sub<GL(4,GF(113))| [1,1,0,0,0,112,0,0,0,0,98,0,0,0,0,98],[98,98,0,0,0,15,0,0,0,0,15,15,0,0,0,98],[49,0,0,0,15,64,0,0,0,0,30,0,0,0,53,83],[0,0,49,0,0,0,15,64,30,0,0,0,53,83,0,0] >;
C42⋊5D14 in GAP, Magma, Sage, TeX
C_4^2\rtimes_5D_{14} % in TeX
G:=Group("C4^2:5D14"); // GroupNames label
G:=SmallGroup(448,595);
// by ID
G=gap.SmallGroup(448,595);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,184,1123,570,297,136,1684,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a^-1*b,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations