direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C16, C112⋊4C2, D14.2C8, C8.19D14, Dic7.2C8, C56.19C22, C7⋊C16⋊6C2, C7⋊1(C2×C16), C7⋊C8.3C4, C2.1(C8×D7), C14.1(C2×C8), (C4×D7).4C4, (C8×D7).3C2, C4.16(C4×D7), C28.21(C2×C4), SmallGroup(224,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C16 |
Generators and relations for D7×C16
G = < a,b,c | a16=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C16
►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 75 96 107 47 49 28)(2 76 81 108 48 50 29)(3 77 82 109 33 51 30)(4 78 83 110 34 52 31)(5 79 84 111 35 53 32)(6 80 85 112 36 54 17)(7 65 86 97 37 55 18)(8 66 87 98 38 56 19)(9 67 88 99 39 57 20)(10 68 89 100 40 58 21)(11 69 90 101 41 59 22)(12 70 91 102 42 60 23)(13 71 92 103 43 61 24)(14 72 93 104 44 62 25)(15 73 94 105 45 63 26)(16 74 95 106 46 64 27)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 17)(15 18)(16 19)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 81)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 89)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 79)(62 80)(63 65)(64 66)(97 105)(98 106)(99 107)(100 108)(101 109)(102 110)(103 111)(104 112)
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), (1,75,96,107,47,49,28)(2,76,81,108,48,50,29)(3,77,82,109,33,51,30)(4,78,83,110,34,52,31)(5,79,84,111,35,53,32)(6,80,85,112,36,54,17)(7,65,86,97,37,55,18)(8,66,87,98,38,56,19)(9,67,88,99,39,57,20)(10,68,89,100,40,58,21)(11,69,90,101,41,59,22)(12,70,91,102,42,60,23)(13,71,92,103,43,61,24)(14,72,93,104,44,62,25)(15,73,94,105,45,63,26)(16,74,95,106,46,64,27), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,17)(15,18)(16,19)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,65)(64,66)(97,105)(98,106)(99,107)(100,108)(101,109)(102,110)(103,111)(104,112)>;
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), (1,75,96,107,47,49,28)(2,76,81,108,48,50,29)(3,77,82,109,33,51,30)(4,78,83,110,34,52,31)(5,79,84,111,35,53,32)(6,80,85,112,36,54,17)(7,65,86,97,37,55,18)(8,66,87,98,38,56,19)(9,67,88,99,39,57,20)(10,68,89,100,40,58,21)(11,69,90,101,41,59,22)(12,70,91,102,42,60,23)(13,71,92,103,43,61,24)(14,72,93,104,44,62,25)(15,73,94,105,45,63,26)(16,74,95,106,46,64,27), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,17)(15,18)(16,19)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,65)(64,66)(97,105)(98,106)(99,107)(100,108)(101,109)(102,110)(103,111)(104,112) );
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)], [(1,75,96,107,47,49,28),(2,76,81,108,48,50,29),(3,77,82,109,33,51,30),(4,78,83,110,34,52,31),(5,79,84,111,35,53,32),(6,80,85,112,36,54,17),(7,65,86,97,37,55,18),(8,66,87,98,38,56,19),(9,67,88,99,39,57,20),(10,68,89,100,40,58,21),(11,69,90,101,41,59,22),(12,70,91,102,42,60,23),(13,71,92,103,43,61,24),(14,72,93,104,44,62,25),(15,73,94,105,45,63,26),(16,74,95,106,46,64,27)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,17),(15,18),(16,19),(33,90),(34,91),(35,92),(36,93),(37,94),(38,95),(39,96),(40,81),(41,82),(42,83),(43,84),(44,85),(45,86),(46,87),(47,88),(48,89),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78),(61,79),(62,80),(63,65),(64,66),(97,105),(98,106),(99,107),(100,108),(101,109),(102,110),(103,111),(104,112)]])
D7×C16 is a maximal subgroup of
C32⋊D7 D28.4C8 C16.12D14 D16⋊3D7 SD32⋊3D7 Q32⋊3D7
D7×C16 is a maximal quotient of C32⋊D7 Dic7⋊C16 D14⋊C16
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | 14B | 14C | 16A | ··· | 16H | 16I | ··· | 16P | 28A | ··· | 28F | 56A | ··· | 56L | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 16 | ··· | 16 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 7 | 7 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D7 | D14 | C4×D7 | C8×D7 | D7×C16 |
| kernel | D7×C16 | C7⋊C16 | C112 | C8×D7 | C7⋊C8 | C4×D7 | Dic7 | D14 | D7 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 3 | 3 | 6 | 12 | 24 |
Matrix representation of D7×C16 ►in GL2(𝔽113) generated by
| 35 | 0 |
| 0 | 35 |
| 0 | 1 |
| 112 | 9 |
| 0 | 112 |
| 112 | 0 |
G:=sub<GL(2,GF(113))| [35,0,0,35],[0,112,1,9],[0,112,112,0] >;
D7×C16 in GAP, Magma, Sage, TeX
D_7\times C_{16} % in TeX
G:=Group("D7xC16"); // GroupNames label
G:=SmallGroup(224,3);
// by ID
G=gap.SmallGroup(224,3);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,31,50,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^16=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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