direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×Dic7, C7⋊C42, C28⋊2C4, C22.3D14, (C2×C4).6D7, C2.2(C4×D7), C14.3(C2×C4), (C2×C28).7C2, C2.2(C2×Dic7), (C2×C14).3C22, (C2×Dic7).4C2, SmallGroup(112,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C4×Dic7 |
Generators and relations for C4×Dic7
G = < a,b,c | a4=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4×Dic7
►Regular action on 112 points
Generators in S112
(1 105 43 90)(2 106 44 91)(3 107 45 92)(4 108 46 93)(5 109 47 94)(6 110 48 95)(7 111 49 96)(8 112 50 97)(9 99 51 98)(10 100 52 85)(11 101 53 86)(12 102 54 87)(13 103 55 88)(14 104 56 89)(15 69 32 72)(16 70 33 73)(17 57 34 74)(18 58 35 75)(19 59 36 76)(20 60 37 77)(21 61 38 78)(22 62 39 79)(23 63 40 80)(24 64 41 81)(25 65 42 82)(26 66 29 83)(27 67 30 84)(28 68 31 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 57 8 64)(2 70 9 63)(3 69 10 62)(4 68 11 61)(5 67 12 60)(6 66 13 59)(7 65 14 58)(15 85 22 92)(16 98 23 91)(17 97 24 90)(18 96 25 89)(19 95 26 88)(20 94 27 87)(21 93 28 86)(29 103 36 110)(30 102 37 109)(31 101 38 108)(32 100 39 107)(33 99 40 106)(34 112 41 105)(35 111 42 104)(43 74 50 81)(44 73 51 80)(45 72 52 79)(46 71 53 78)(47 84 54 77)(48 83 55 76)(49 82 56 75)
G:=sub<Sym(112)| (1,105,43,90)(2,106,44,91)(3,107,45,92)(4,108,46,93)(5,109,47,94)(6,110,48,95)(7,111,49,96)(8,112,50,97)(9,99,51,98)(10,100,52,85)(11,101,53,86)(12,102,54,87)(13,103,55,88)(14,104,56,89)(15,69,32,72)(16,70,33,73)(17,57,34,74)(18,58,35,75)(19,59,36,76)(20,60,37,77)(21,61,38,78)(22,62,39,79)(23,63,40,80)(24,64,41,81)(25,65,42,82)(26,66,29,83)(27,67,30,84)(28,68,31,71), (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), (1,57,8,64)(2,70,9,63)(3,69,10,62)(4,68,11,61)(5,67,12,60)(6,66,13,59)(7,65,14,58)(15,85,22,92)(16,98,23,91)(17,97,24,90)(18,96,25,89)(19,95,26,88)(20,94,27,87)(21,93,28,86)(29,103,36,110)(30,102,37,109)(31,101,38,108)(32,100,39,107)(33,99,40,106)(34,112,41,105)(35,111,42,104)(43,74,50,81)(44,73,51,80)(45,72,52,79)(46,71,53,78)(47,84,54,77)(48,83,55,76)(49,82,56,75)>;
G:=Group( (1,105,43,90)(2,106,44,91)(3,107,45,92)(4,108,46,93)(5,109,47,94)(6,110,48,95)(7,111,49,96)(8,112,50,97)(9,99,51,98)(10,100,52,85)(11,101,53,86)(12,102,54,87)(13,103,55,88)(14,104,56,89)(15,69,32,72)(16,70,33,73)(17,57,34,74)(18,58,35,75)(19,59,36,76)(20,60,37,77)(21,61,38,78)(22,62,39,79)(23,63,40,80)(24,64,41,81)(25,65,42,82)(26,66,29,83)(27,67,30,84)(28,68,31,71), (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), (1,57,8,64)(2,70,9,63)(3,69,10,62)(4,68,11,61)(5,67,12,60)(6,66,13,59)(7,65,14,58)(15,85,22,92)(16,98,23,91)(17,97,24,90)(18,96,25,89)(19,95,26,88)(20,94,27,87)(21,93,28,86)(29,103,36,110)(30,102,37,109)(31,101,38,108)(32,100,39,107)(33,99,40,106)(34,112,41,105)(35,111,42,104)(43,74,50,81)(44,73,51,80)(45,72,52,79)(46,71,53,78)(47,84,54,77)(48,83,55,76)(49,82,56,75) );
G=PermutationGroup([[(1,105,43,90),(2,106,44,91),(3,107,45,92),(4,108,46,93),(5,109,47,94),(6,110,48,95),(7,111,49,96),(8,112,50,97),(9,99,51,98),(10,100,52,85),(11,101,53,86),(12,102,54,87),(13,103,55,88),(14,104,56,89),(15,69,32,72),(16,70,33,73),(17,57,34,74),(18,58,35,75),(19,59,36,76),(20,60,37,77),(21,61,38,78),(22,62,39,79),(23,63,40,80),(24,64,41,81),(25,65,42,82),(26,66,29,83),(27,67,30,84),(28,68,31,71)], [(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)], [(1,57,8,64),(2,70,9,63),(3,69,10,62),(4,68,11,61),(5,67,12,60),(6,66,13,59),(7,65,14,58),(15,85,22,92),(16,98,23,91),(17,97,24,90),(18,96,25,89),(19,95,26,88),(20,94,27,87),(21,93,28,86),(29,103,36,110),(30,102,37,109),(31,101,38,108),(32,100,39,107),(33,99,40,106),(34,112,41,105),(35,111,42,104),(43,74,50,81),(44,73,51,80),(45,72,52,79),(46,71,53,78),(47,84,54,77),(48,83,55,76),(49,82,56,75)]])
C4×Dic7 is a maximal subgroup of
Dic7⋊C8 C56⋊C4 D28⋊4C4 D4⋊2Dic7 D7×C42 C42⋊D7 C23.11D14 C23.D14 Dic7⋊4D4 Dic7.D4 Dic7⋊3Q8 C28⋊Q8 Dic7.Q8 C28.3Q8 C4⋊C4⋊7D7 D28⋊C4 C4⋊C4⋊D7 C23.21D14 C28.17D4 C28⋊D4 Dic7⋊Q8 C28.23D4
C4×Dic7 is a maximal quotient of
C42.D7 C56⋊C4 C14.C42
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 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 | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | C4 | D7 | Dic7 | D14 | C4×D7 |
| kernel | C4×Dic7 | C2×Dic7 | C2×C28 | Dic7 | C28 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 3 | 6 | 3 | 12 |
Matrix representation of C4×Dic7 ►in GL3(𝔽29) generated by
| 1 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 28 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 28 | 3 |
| 12 | 0 | 0 |
| 0 | 5 | 2 |
| 0 | 17 | 24 |
G:=sub<GL(3,GF(29))| [1,0,0,0,12,0,0,0,12],[28,0,0,0,0,28,0,1,3],[12,0,0,0,5,17,0,2,24] >;
C4×Dic7 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_7 % in TeX
G:=Group("C4xDic7"); // GroupNames label
G:=SmallGroup(112,10);
// by ID
G=gap.SmallGroup(112,10);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,20,46,2404]);
// Polycyclic
G:=Group<a,b,c|a^4=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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