direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic14, C14⋊Q8, C4.11D14, C14.1C23, C22.8D14, C28.11C22, Dic7.1C22, C7⋊1(C2×Q8), (C2×C4).4D7, (C2×C28).4C2, C2.3(C22×D7), (C2×C14).8C22, (C2×Dic7).3C2, SmallGroup(112,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic14
G = < a,b,c | a2=b28=1, c2=b14, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic14
►Regular action on 112 points
Generators in S112
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 29)(26 30)(27 31)(28 32)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(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 71 15 57)(2 70 16 84)(3 69 17 83)(4 68 18 82)(5 67 19 81)(6 66 20 80)(7 65 21 79)(8 64 22 78)(9 63 23 77)(10 62 24 76)(11 61 25 75)(12 60 26 74)(13 59 27 73)(14 58 28 72)(29 103 43 89)(30 102 44 88)(31 101 45 87)(32 100 46 86)(33 99 47 85)(34 98 48 112)(35 97 49 111)(36 96 50 110)(37 95 51 109)(38 94 52 108)(39 93 53 107)(40 92 54 106)(41 91 55 105)(42 90 56 104)
G:=sub<Sym(112)| (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,29)(26,30)(27,31)(28,32)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (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), (1,71,15,57)(2,70,16,84)(3,69,17,83)(4,68,18,82)(5,67,19,81)(6,66,20,80)(7,65,21,79)(8,64,22,78)(9,63,23,77)(10,62,24,76)(11,61,25,75)(12,60,26,74)(13,59,27,73)(14,58,28,72)(29,103,43,89)(30,102,44,88)(31,101,45,87)(32,100,46,86)(33,99,47,85)(34,98,48,112)(35,97,49,111)(36,96,50,110)(37,95,51,109)(38,94,52,108)(39,93,53,107)(40,92,54,106)(41,91,55,105)(42,90,56,104)>;
G:=Group( (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,29)(26,30)(27,31)(28,32)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (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), (1,71,15,57)(2,70,16,84)(3,69,17,83)(4,68,18,82)(5,67,19,81)(6,66,20,80)(7,65,21,79)(8,64,22,78)(9,63,23,77)(10,62,24,76)(11,61,25,75)(12,60,26,74)(13,59,27,73)(14,58,28,72)(29,103,43,89)(30,102,44,88)(31,101,45,87)(32,100,46,86)(33,99,47,85)(34,98,48,112)(35,97,49,111)(36,96,50,110)(37,95,51,109)(38,94,52,108)(39,93,53,107)(40,92,54,106)(41,91,55,105)(42,90,56,104) );
G=PermutationGroup([[(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,29),(26,30),(27,31),(28,32),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(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)], [(1,71,15,57),(2,70,16,84),(3,69,17,83),(4,68,18,82),(5,67,19,81),(6,66,20,80),(7,65,21,79),(8,64,22,78),(9,63,23,77),(10,62,24,76),(11,61,25,75),(12,60,26,74),(13,59,27,73),(14,58,28,72),(29,103,43,89),(30,102,44,88),(31,101,45,87),(32,100,46,86),(33,99,47,85),(34,98,48,112),(35,97,49,111),(36,96,50,110),(37,95,51,109),(38,94,52,108),(39,93,53,107),(40,92,54,106),(41,91,55,105),(42,90,56,104)]])
C2×Dic14 is a maximal subgroup of
C14.Q16 C28.44D4 C4.12D28 C28⋊2Q8 C4.D28 C22⋊Dic14 Dic7.D4 Dic7⋊3Q8 C28⋊Q8 D14⋊Q8 D14⋊2Q8 C8.D14 C28.48D4 C28.17D4 Dic7⋊Q8 D4.9D14 C2×Q8×D7 D4.10D14
C2×Dic14 is a maximal quotient of
C28⋊2Q8 C28.6Q8 C22⋊Dic14 C28⋊Q8 C28.3Q8 C28.48D4
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 |
| type | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D7 | D14 | D14 | Dic14 |
| kernel | C2×Dic14 | Dic14 | C2×Dic7 | C2×C28 | C14 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 3 | 6 | 3 | 12 |
Matrix representation of C2×Dic14 ►in GL3(𝔽29) generated by
| 28 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 28 | 0 | 0 |
| 0 | 6 | 8 |
| 0 | 21 | 4 |
| 28 | 0 | 0 |
| 0 | 0 | 12 |
| 0 | 12 | 0 |
G:=sub<GL(3,GF(29))| [28,0,0,0,1,0,0,0,1],[28,0,0,0,6,21,0,8,4],[28,0,0,0,0,12,0,12,0] >;
C2×Dic14 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{14} % in TeX
G:=Group("C2xDic14"); // GroupNames label
G:=SmallGroup(112,27);
// by ID
G=gap.SmallGroup(112,27);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,40,182,42,2404]);
// Polycyclic
G:=Group<a,b,c|a^2=b^28=1,c^2=b^14,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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