metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.196D14, M4(2).23D14, C4≀C2⋊7D7, D4⋊D7⋊4C4, C7⋊2(C8○D8), Q8⋊D7⋊4C4, C7⋊C8.37D4, D4.D7⋊4C4, D4.4(C4×D7), C7⋊Q16⋊4C4, Q8.4(C4×D7), D28.C4⋊9C2, D28.7(C2×C4), C14.39(C4×D4), C4.203(D4×D7), Dic14⋊C4⋊7C2, C4○D4.21D14, C28.362(C2×D4), Q8.Dic7⋊2C2, C28.53D4⋊6C2, C28.20(C22×C4), (C4×C28).51C22, Dic14.7(C2×C4), (C2×C28).264C23, D4.8D14.2C2, C4○D28.13C22, C4.Dic7.9C22, C22.9(D4⋊2D7), C2.23(Dic7⋊4D4), (C7×M4(2)).17C22, (C4×C7⋊C8)⋊3C2, (C7×C4≀C2)⋊8C2, C7⋊C8.8(C2×C4), C4.20(C2×C4×D7), (C7×D4).7(C2×C4), (C7×Q8).7(C2×C4), (C2×C7⋊C8).222C22, (C7×C4○D4).5C22, (C2×C14).35(C4○D4), (C2×C4).370(C22×D7), SmallGroup(448,358)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.196D14
G = < a,b,c,d | a4=b4=c14=1, d2=cbc-1=b-1, ab=ba, cac-1=ab-1, ad=da, bd=db, dcd-1=b-1c-1 >
Subgroups: 412 in 106 conjugacy classes, 45 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D7, C14, C14, C42, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4×C8, C4≀C2, C4≀C2, C8.C4, C8○D4, C4○D8, C7⋊C8, C7⋊C8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C8○D8, C8×D7, C8⋊D7, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C4×C28, C7×M4(2), C4○D28, C7×C4○D4, C4×C7⋊C8, Dic14⋊C4, C28.53D4, C7×C4≀C2, D28.C4, Q8.Dic7, D4.8D14, C42.196D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, C22×D7, C8○D8, C2×C4×D7, D4×D7, D4⋊2D7, Dic7⋊4D4, C42.196D14
►On 112 points
Generators in S112
(1 26 70 44)(2 57)(3 28 58 46)(4 59)(5 16 60 48)(6 61)(7 18 62 50)(8 63)(9 20 64 52)(10 65)(11 22 66 54)(12 67)(13 24 68 56)(14 69)(15 47)(17 49)(19 51)(21 53)(23 55)(25 43)(27 45)(29 81)(30 107 82 98)(31 83)(32 109 84 86)(33 71)(34 111 72 88)(35 73)(36 99 74 90)(37 75)(38 101 76 92)(39 77)(40 103 78 94)(41 79)(42 105 80 96)(85 108)(87 110)(89 112)(91 100)(93 102)(95 104)(97 106)
(1 44 70 26)(2 27 57 45)(3 46 58 28)(4 15 59 47)(5 48 60 16)(6 17 61 49)(7 50 62 18)(8 19 63 51)(9 52 64 20)(10 21 65 53)(11 54 66 22)(12 23 67 55)(13 56 68 24)(14 25 69 43)(29 106 81 97)(30 98 82 107)(31 108 83 85)(32 86 84 109)(33 110 71 87)(34 88 72 111)(35 112 73 89)(36 90 74 99)(37 100 75 91)(38 92 76 101)(39 102 77 93)(40 94 78 103)(41 104 79 95)(42 96 80 105)
(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 92 26 38 70 101 44 76)(2 75 45 100 57 37 27 91)(3 90 28 36 58 99 46 74)(4 73 47 112 59 35 15 89)(5 88 16 34 60 111 48 72)(6 71 49 110 61 33 17 87)(7 86 18 32 62 109 50 84)(8 83 51 108 63 31 19 85)(9 98 20 30 64 107 52 82)(10 81 53 106 65 29 21 97)(11 96 22 42 66 105 54 80)(12 79 55 104 67 41 23 95)(13 94 24 40 68 103 56 78)(14 77 43 102 69 39 25 93)
G:=sub<Sym(112)| (1,26,70,44)(2,57)(3,28,58,46)(4,59)(5,16,60,48)(6,61)(7,18,62,50)(8,63)(9,20,64,52)(10,65)(11,22,66,54)(12,67)(13,24,68,56)(14,69)(15,47)(17,49)(19,51)(21,53)(23,55)(25,43)(27,45)(29,81)(30,107,82,98)(31,83)(32,109,84,86)(33,71)(34,111,72,88)(35,73)(36,99,74,90)(37,75)(38,101,76,92)(39,77)(40,103,78,94)(41,79)(42,105,80,96)(85,108)(87,110)(89,112)(91,100)(93,102)(95,104)(97,106), (1,44,70,26)(2,27,57,45)(3,46,58,28)(4,15,59,47)(5,48,60,16)(6,17,61,49)(7,50,62,18)(8,19,63,51)(9,52,64,20)(10,21,65,53)(11,54,66,22)(12,23,67,55)(13,56,68,24)(14,25,69,43)(29,106,81,97)(30,98,82,107)(31,108,83,85)(32,86,84,109)(33,110,71,87)(34,88,72,111)(35,112,73,89)(36,90,74,99)(37,100,75,91)(38,92,76,101)(39,102,77,93)(40,94,78,103)(41,104,79,95)(42,96,80,105), (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), (1,92,26,38,70,101,44,76)(2,75,45,100,57,37,27,91)(3,90,28,36,58,99,46,74)(4,73,47,112,59,35,15,89)(5,88,16,34,60,111,48,72)(6,71,49,110,61,33,17,87)(7,86,18,32,62,109,50,84)(8,83,51,108,63,31,19,85)(9,98,20,30,64,107,52,82)(10,81,53,106,65,29,21,97)(11,96,22,42,66,105,54,80)(12,79,55,104,67,41,23,95)(13,94,24,40,68,103,56,78)(14,77,43,102,69,39,25,93)>;
G:=Group( (1,26,70,44)(2,57)(3,28,58,46)(4,59)(5,16,60,48)(6,61)(7,18,62,50)(8,63)(9,20,64,52)(10,65)(11,22,66,54)(12,67)(13,24,68,56)(14,69)(15,47)(17,49)(19,51)(21,53)(23,55)(25,43)(27,45)(29,81)(30,107,82,98)(31,83)(32,109,84,86)(33,71)(34,111,72,88)(35,73)(36,99,74,90)(37,75)(38,101,76,92)(39,77)(40,103,78,94)(41,79)(42,105,80,96)(85,108)(87,110)(89,112)(91,100)(93,102)(95,104)(97,106), (1,44,70,26)(2,27,57,45)(3,46,58,28)(4,15,59,47)(5,48,60,16)(6,17,61,49)(7,50,62,18)(8,19,63,51)(9,52,64,20)(10,21,65,53)(11,54,66,22)(12,23,67,55)(13,56,68,24)(14,25,69,43)(29,106,81,97)(30,98,82,107)(31,108,83,85)(32,86,84,109)(33,110,71,87)(34,88,72,111)(35,112,73,89)(36,90,74,99)(37,100,75,91)(38,92,76,101)(39,102,77,93)(40,94,78,103)(41,104,79,95)(42,96,80,105), (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), (1,92,26,38,70,101,44,76)(2,75,45,100,57,37,27,91)(3,90,28,36,58,99,46,74)(4,73,47,112,59,35,15,89)(5,88,16,34,60,111,48,72)(6,71,49,110,61,33,17,87)(7,86,18,32,62,109,50,84)(8,83,51,108,63,31,19,85)(9,98,20,30,64,107,52,82)(10,81,53,106,65,29,21,97)(11,96,22,42,66,105,54,80)(12,79,55,104,67,41,23,95)(13,94,24,40,68,103,56,78)(14,77,43,102,69,39,25,93) );
G=PermutationGroup([[(1,26,70,44),(2,57),(3,28,58,46),(4,59),(5,16,60,48),(6,61),(7,18,62,50),(8,63),(9,20,64,52),(10,65),(11,22,66,54),(12,67),(13,24,68,56),(14,69),(15,47),(17,49),(19,51),(21,53),(23,55),(25,43),(27,45),(29,81),(30,107,82,98),(31,83),(32,109,84,86),(33,71),(34,111,72,88),(35,73),(36,99,74,90),(37,75),(38,101,76,92),(39,77),(40,103,78,94),(41,79),(42,105,80,96),(85,108),(87,110),(89,112),(91,100),(93,102),(95,104),(97,106)], [(1,44,70,26),(2,27,57,45),(3,46,58,28),(4,15,59,47),(5,48,60,16),(6,17,61,49),(7,50,62,18),(8,19,63,51),(9,52,64,20),(10,21,65,53),(11,54,66,22),(12,23,67,55),(13,56,68,24),(14,25,69,43),(29,106,81,97),(30,98,82,107),(31,108,83,85),(32,86,84,109),(33,110,71,87),(34,88,72,111),(35,112,73,89),(36,90,74,99),(37,100,75,91),(38,92,76,101),(39,102,77,93),(40,94,78,103),(41,104,79,95),(42,96,80,105)], [(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)], [(1,92,26,38,70,101,44,76),(2,75,45,100,57,37,27,91),(3,90,28,36,58,99,46,74),(4,73,47,112,59,35,15,89),(5,88,16,34,60,111,48,72),(6,71,49,110,61,33,17,87),(7,86,18,32,62,109,50,84),(8,83,51,108,63,31,19,85),(9,98,20,30,64,107,52,82),(10,81,53,106,65,29,21,97),(11,96,22,42,66,105,54,80),(12,79,55,104,67,41,23,95),(13,94,24,40,68,103,56,78),(14,77,43,102,69,39,25,93)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | ··· | 8L | 8M | 8N | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28U | 28V | 28W | 28X | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 28 | 1 | 1 | 2 | ··· | 2 | 4 | 28 | 2 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 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 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4×D7 | C4×D7 | C8○D8 | D4×D7 | D4⋊2D7 | C42.196D14 |
| kernel | C42.196D14 | C4×C7⋊C8 | Dic14⋊C4 | C28.53D4 | C7×C4≀C2 | D28.C4 | Q8.Dic7 | D4.8D14 | D4⋊D7 | D4.D7 | Q8⋊D7 | C7⋊Q16 | C7⋊C8 | C4≀C2 | C2×C14 | C42 | M4(2) | C4○D4 | D4 | Q8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 2 | 3 | 3 | 3 | 6 | 6 | 8 | 3 | 3 | 12 |
Matrix representation of C42.196D14 ►in GL4(𝔽113) generated by
| 98 | 111 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 15 | 85 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 98 | 28 | 0 | 0 |
| 105 | 15 | 0 | 0 |
| 0 | 0 | 97 | 0 |
| 0 | 0 | 29 | 7 |
| 95 | 36 | 0 | 0 |
| 0 | 69 | 0 | 0 |
| 0 | 0 | 93 | 27 |
| 0 | 0 | 48 | 20 |
G:=sub<GL(4,GF(113))| [98,0,0,0,111,112,0,0,0,0,1,0,0,0,0,1],[15,0,0,0,85,98,0,0,0,0,1,0,0,0,0,1],[98,105,0,0,28,15,0,0,0,0,97,29,0,0,0,7],[95,0,0,0,36,69,0,0,0,0,93,48,0,0,27,20] >;
C42.196D14 in GAP, Magma, Sage, TeX
C_4^2._{196}D_{14} % in TeX
G:=Group("C4^2.196D14"); // GroupNames label
G:=SmallGroup(448,358);
// by ID
G=gap.SmallGroup(448,358);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,555,58,136,1684,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations