metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊Dic7, C28⋊1C4, C14.4D4, C2.1D28, C14.2Q8, C2.2Dic14, C22.5D14, C7⋊2(C4⋊C4), (C2×C4).3D7, C14.8(C2×C4), (C2×C28).3C2, C2.4(C2×Dic7), (C2×C14).5C22, (C2×Dic7).2C2, SmallGroup(112,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊Dic7
G = < a,b,c | a4=b14=1, c2=b7, ab=ba, cac-1=a-1, cbc-1=b-1 >
Smallest permutation representation of C4⋊Dic7
►Regular action on 112 points
Generators in S112
(1 106 91 48)(2 107 92 49)(3 108 93 50)(4 109 94 51)(5 110 95 52)(6 111 96 53)(7 112 97 54)(8 99 98 55)(9 100 85 56)(10 101 86 43)(11 102 87 44)(12 103 88 45)(13 104 89 46)(14 105 90 47)(15 34 67 72)(16 35 68 73)(17 36 69 74)(18 37 70 75)(19 38 57 76)(20 39 58 77)(21 40 59 78)(22 41 60 79)(23 42 61 80)(24 29 62 81)(25 30 63 82)(26 31 64 83)(27 32 65 84)(28 33 66 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 27 8 20)(2 26 9 19)(3 25 10 18)(4 24 11 17)(5 23 12 16)(6 22 13 15)(7 21 14 28)(29 44 36 51)(30 43 37 50)(31 56 38 49)(32 55 39 48)(33 54 40 47)(34 53 41 46)(35 52 42 45)(57 92 64 85)(58 91 65 98)(59 90 66 97)(60 89 67 96)(61 88 68 95)(62 87 69 94)(63 86 70 93)(71 112 78 105)(72 111 79 104)(73 110 80 103)(74 109 81 102)(75 108 82 101)(76 107 83 100)(77 106 84 99)
G:=sub<Sym(112)| (1,106,91,48)(2,107,92,49)(3,108,93,50)(4,109,94,51)(5,110,95,52)(6,111,96,53)(7,112,97,54)(8,99,98,55)(9,100,85,56)(10,101,86,43)(11,102,87,44)(12,103,88,45)(13,104,89,46)(14,105,90,47)(15,34,67,72)(16,35,68,73)(17,36,69,74)(18,37,70,75)(19,38,57,76)(20,39,58,77)(21,40,59,78)(22,41,60,79)(23,42,61,80)(24,29,62,81)(25,30,63,82)(26,31,64,83)(27,32,65,84)(28,33,66,71), (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), (1,27,8,20)(2,26,9,19)(3,25,10,18)(4,24,11,17)(5,23,12,16)(6,22,13,15)(7,21,14,28)(29,44,36,51)(30,43,37,50)(31,56,38,49)(32,55,39,48)(33,54,40,47)(34,53,41,46)(35,52,42,45)(57,92,64,85)(58,91,65,98)(59,90,66,97)(60,89,67,96)(61,88,68,95)(62,87,69,94)(63,86,70,93)(71,112,78,105)(72,111,79,104)(73,110,80,103)(74,109,81,102)(75,108,82,101)(76,107,83,100)(77,106,84,99)>;
G:=Group( (1,106,91,48)(2,107,92,49)(3,108,93,50)(4,109,94,51)(5,110,95,52)(6,111,96,53)(7,112,97,54)(8,99,98,55)(9,100,85,56)(10,101,86,43)(11,102,87,44)(12,103,88,45)(13,104,89,46)(14,105,90,47)(15,34,67,72)(16,35,68,73)(17,36,69,74)(18,37,70,75)(19,38,57,76)(20,39,58,77)(21,40,59,78)(22,41,60,79)(23,42,61,80)(24,29,62,81)(25,30,63,82)(26,31,64,83)(27,32,65,84)(28,33,66,71), (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), (1,27,8,20)(2,26,9,19)(3,25,10,18)(4,24,11,17)(5,23,12,16)(6,22,13,15)(7,21,14,28)(29,44,36,51)(30,43,37,50)(31,56,38,49)(32,55,39,48)(33,54,40,47)(34,53,41,46)(35,52,42,45)(57,92,64,85)(58,91,65,98)(59,90,66,97)(60,89,67,96)(61,88,68,95)(62,87,69,94)(63,86,70,93)(71,112,78,105)(72,111,79,104)(73,110,80,103)(74,109,81,102)(75,108,82,101)(76,107,83,100)(77,106,84,99) );
G=PermutationGroup([[(1,106,91,48),(2,107,92,49),(3,108,93,50),(4,109,94,51),(5,110,95,52),(6,111,96,53),(7,112,97,54),(8,99,98,55),(9,100,85,56),(10,101,86,43),(11,102,87,44),(12,103,88,45),(13,104,89,46),(14,105,90,47),(15,34,67,72),(16,35,68,73),(17,36,69,74),(18,37,70,75),(19,38,57,76),(20,39,58,77),(21,40,59,78),(22,41,60,79),(23,42,61,80),(24,29,62,81),(25,30,63,82),(26,31,64,83),(27,32,65,84),(28,33,66,71)], [(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)], [(1,27,8,20),(2,26,9,19),(3,25,10,18),(4,24,11,17),(5,23,12,16),(6,22,13,15),(7,21,14,28),(29,44,36,51),(30,43,37,50),(31,56,38,49),(32,55,39,48),(33,54,40,47),(34,53,41,46),(35,52,42,45),(57,92,64,85),(58,91,65,98),(59,90,66,97),(60,89,67,96),(61,88,68,95),(62,87,69,94),(63,86,70,93),(71,112,78,105),(72,111,79,104),(73,110,80,103),(74,109,81,102),(75,108,82,101),(76,107,83,100),(77,106,84,99)]])
C4⋊Dic7 is a maximal subgroup of
C28.Q8 C4.Dic14 C28.44D4 C8⋊Dic7 C56⋊1C4 C2.D56 D4⋊Dic7 Q8⋊Dic7 C4×Dic14 C28⋊2Q8 C28.6Q8 C4×D28 C22⋊Dic14 C23.D14 D14.D4 C22.D28 C28⋊Q8 Dic7.Q8 C28.3Q8 D7×C4⋊C4 C4⋊C4⋊7D7 D14⋊2Q8 C4⋊C4⋊D7 C28.48D4 C23.21D14 C28⋊7D4 D4×Dic7 C28⋊2D4 Q8×Dic7 D14⋊3Q8 C28⋊C12 C14.Dic6 C84⋊C4
C4⋊Dic7 is a maximal quotient of
C28⋊C8 C8⋊Dic7 C56⋊1C4 C56.C4 C14.C42 C14.Dic6 C84⋊C4
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | |
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D7 | Dic7 | D14 | Dic14 | D28 |
| kernel | C4⋊Dic7 | C2×Dic7 | C2×C28 | C28 | C14 | C14 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 3 | 6 | 3 | 6 | 6 |
Matrix representation of C4⋊Dic7 ►in GL3(𝔽29) generated by
| 28 | 0 | 0 |
| 0 | 2 | 6 |
| 0 | 4 | 27 |
| 28 | 0 | 0 |
| 0 | 28 | 1 |
| 0 | 20 | 8 |
| 12 | 0 | 0 |
| 0 | 24 | 25 |
| 0 | 6 | 5 |
G:=sub<GL(3,GF(29))| [28,0,0,0,2,4,0,6,27],[28,0,0,0,28,20,0,1,8],[12,0,0,0,24,6,0,25,5] >;
C4⋊Dic7 in GAP, Magma, Sage, TeX
C_4\rtimes {\rm Dic}_7 % in TeX
G:=Group("C4:Dic7"); // GroupNames label
G:=SmallGroup(112,12);
// by ID
G=gap.SmallGroup(112,12);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,20,101,46,2404]);
// Polycyclic
G:=Group<a,b,c|a^4=b^14=1,c^2=b^7,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
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