metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7⋊C4, C14.5D4, C14.1Q8, C2.1Dic14, C22.4D14, C7⋊1(C4⋊C4), C2.4(C4×D7), (C2×C4).1D7, C14.4(C2×C4), (C2×C28).1C2, C2.1(C7⋊D4), (C2×C14).4C22, (C2×Dic7).1C2, SmallGroup(112,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7⋊C4
G = < a,b,c | a14=c4=1, b2=a7, bab-1=a-1, ac=ca, cbc-1=a7b >
Smallest permutation representation of Dic7⋊C4
►Regular action on 112 points
Generators in S112
(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 8 78)(2 84 9 77)(3 83 10 76)(4 82 11 75)(5 81 12 74)(6 80 13 73)(7 79 14 72)(15 58 22 65)(16 57 23 64)(17 70 24 63)(18 69 25 62)(19 68 26 61)(20 67 27 60)(21 66 28 59)(29 109 36 102)(30 108 37 101)(31 107 38 100)(32 106 39 99)(33 105 40 112)(34 104 41 111)(35 103 42 110)(43 98 50 91)(44 97 51 90)(45 96 52 89)(46 95 53 88)(47 94 54 87)(48 93 55 86)(49 92 56 85)
(1 56 23 32)(2 43 24 33)(3 44 25 34)(4 45 26 35)(5 46 27 36)(6 47 28 37)(7 48 15 38)(8 49 16 39)(9 50 17 40)(10 51 18 41)(11 52 19 42)(12 53 20 29)(13 54 21 30)(14 55 22 31)(57 106 78 85)(58 107 79 86)(59 108 80 87)(60 109 81 88)(61 110 82 89)(62 111 83 90)(63 112 84 91)(64 99 71 92)(65 100 72 93)(66 101 73 94)(67 102 74 95)(68 103 75 96)(69 104 76 97)(70 105 77 98)
G:=sub<Sym(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), (1,71,8,78)(2,84,9,77)(3,83,10,76)(4,82,11,75)(5,81,12,74)(6,80,13,73)(7,79,14,72)(15,58,22,65)(16,57,23,64)(17,70,24,63)(18,69,25,62)(19,68,26,61)(20,67,27,60)(21,66,28,59)(29,109,36,102)(30,108,37,101)(31,107,38,100)(32,106,39,99)(33,105,40,112)(34,104,41,111)(35,103,42,110)(43,98,50,91)(44,97,51,90)(45,96,52,89)(46,95,53,88)(47,94,54,87)(48,93,55,86)(49,92,56,85), (1,56,23,32)(2,43,24,33)(3,44,25,34)(4,45,26,35)(5,46,27,36)(6,47,28,37)(7,48,15,38)(8,49,16,39)(9,50,17,40)(10,51,18,41)(11,52,19,42)(12,53,20,29)(13,54,21,30)(14,55,22,31)(57,106,78,85)(58,107,79,86)(59,108,80,87)(60,109,81,88)(61,110,82,89)(62,111,83,90)(63,112,84,91)(64,99,71,92)(65,100,72,93)(66,101,73,94)(67,102,74,95)(68,103,75,96)(69,104,76,97)(70,105,77,98)>;
G:=Group( (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), (1,71,8,78)(2,84,9,77)(3,83,10,76)(4,82,11,75)(5,81,12,74)(6,80,13,73)(7,79,14,72)(15,58,22,65)(16,57,23,64)(17,70,24,63)(18,69,25,62)(19,68,26,61)(20,67,27,60)(21,66,28,59)(29,109,36,102)(30,108,37,101)(31,107,38,100)(32,106,39,99)(33,105,40,112)(34,104,41,111)(35,103,42,110)(43,98,50,91)(44,97,51,90)(45,96,52,89)(46,95,53,88)(47,94,54,87)(48,93,55,86)(49,92,56,85), (1,56,23,32)(2,43,24,33)(3,44,25,34)(4,45,26,35)(5,46,27,36)(6,47,28,37)(7,48,15,38)(8,49,16,39)(9,50,17,40)(10,51,18,41)(11,52,19,42)(12,53,20,29)(13,54,21,30)(14,55,22,31)(57,106,78,85)(58,107,79,86)(59,108,80,87)(60,109,81,88)(61,110,82,89)(62,111,83,90)(63,112,84,91)(64,99,71,92)(65,100,72,93)(66,101,73,94)(67,102,74,95)(68,103,75,96)(69,104,76,97)(70,105,77,98) );
G=PermutationGroup([[(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)], [(1,71,8,78),(2,84,9,77),(3,83,10,76),(4,82,11,75),(5,81,12,74),(6,80,13,73),(7,79,14,72),(15,58,22,65),(16,57,23,64),(17,70,24,63),(18,69,25,62),(19,68,26,61),(20,67,27,60),(21,66,28,59),(29,109,36,102),(30,108,37,101),(31,107,38,100),(32,106,39,99),(33,105,40,112),(34,104,41,111),(35,103,42,110),(43,98,50,91),(44,97,51,90),(45,96,52,89),(46,95,53,88),(47,94,54,87),(48,93,55,86),(49,92,56,85)], [(1,56,23,32),(2,43,24,33),(3,44,25,34),(4,45,26,35),(5,46,27,36),(6,47,28,37),(7,48,15,38),(8,49,16,39),(9,50,17,40),(10,51,18,41),(11,52,19,42),(12,53,20,29),(13,54,21,30),(14,55,22,31),(57,106,78,85),(58,107,79,86),(59,108,80,87),(60,109,81,88),(61,110,82,89),(62,111,83,90),(63,112,84,91),(64,99,71,92),(65,100,72,93),(66,101,73,94),(67,102,74,95),(68,103,75,96),(69,104,76,97),(70,105,77,98)]])
Dic7⋊C4 is a maximal subgroup of
C4×Dic14 C28.6Q8 C42⋊D7 C42⋊2D7 C23.11D14 C22⋊Dic14 C23.D14 Dic7⋊4D4 D14.D4 D14⋊D4 Dic7⋊3Q8 C28⋊Q8 Dic7.Q8 C28.3Q8 D7×C4⋊C4 D14.5D4 D14⋊Q8 C4⋊C4⋊D7 C28.48D4 C4×C7⋊D4 C23.23D14 C23.18D14 Dic7⋊D4 Dic7⋊Q8 D14⋊3Q8 Dic7⋊C12 C42.Q8 Dic21⋊C4 C42.4Q8
Dic7⋊C4 is a maximal quotient of
C28.Q8 C4.Dic14 Dic7⋊C8 C28.53D4 C14.C42 C42.Q8 Dic21⋊C4 C42.4Q8
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 | D14 | Dic14 | C4×D7 | C7⋊D4 |
| kernel | Dic7⋊C4 | C2×Dic7 | C2×C28 | Dic7 | C14 | C14 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 3 | 3 | 6 | 6 | 6 |
Matrix representation of Dic7⋊C4 ►in GL4(𝔽29) generated by
| 0 | 1 | 0 | 0 |
| 28 | 7 | 0 | 0 |
| 0 | 0 | 8 | 28 |
| 0 | 0 | 2 | 18 |
| 7 | 10 | 0 | 0 |
| 1 | 22 | 0 | 0 |
| 0 | 0 | 16 | 17 |
| 0 | 0 | 19 | 13 |
| 17 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 26 | 11 |
| 0 | 0 | 7 | 3 |
G:=sub<GL(4,GF(29))| [0,28,0,0,1,7,0,0,0,0,8,2,0,0,28,18],[7,1,0,0,10,22,0,0,0,0,16,19,0,0,17,13],[17,0,0,0,0,17,0,0,0,0,26,7,0,0,11,3] >;
Dic7⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_7\rtimes C_4 % in TeX
G:=Group("Dic7:C4"); // GroupNames label
G:=SmallGroup(112,11);
// by ID
G=gap.SmallGroup(112,11);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,40,101,26,2404]);
// Polycyclic
G:=Group<a,b,c|a^14=c^4=1,b^2=a^7,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;
// generators/relations
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