direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C16⋊C4, C16⋊2C28, C112⋊4C4, C42.1C28, C28.35C42, M5(2).2C14, C28.33M4(2), (C2×C56).5C4, (C2×C8).2C28, (C4×C28).4C4, C56.87(C2×C4), C4.11(C4×C28), C8.19(C2×C28), C8⋊C4.4C14, C14.9(C8⋊C4), C4.6(C7×M4(2)), (C7×M5(2)).6C2, (C2×C56).308C22, (C2×C14).16M4(2), C22.4(C7×M4(2)), C2.3(C7×C8⋊C4), (C7×C8⋊C4).9C2, (C2×C8).45(C2×C14), (C2×C4).66(C2×C28), (C2×C28).327(C2×C4), SmallGroup(448,151)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C16⋊C4
G = < a,b,c | a7=b16=c4=1, ab=ba, ac=ca, cbc-1=b13 >
Smallest permutation representation of C7×C16⋊C4
►On 112 points
Generators in S112
(1 72 83 39 23 106 58)(2 73 84 40 24 107 59)(3 74 85 41 25 108 60)(4 75 86 42 26 109 61)(5 76 87 43 27 110 62)(6 77 88 44 28 111 63)(7 78 89 45 29 112 64)(8 79 90 46 30 97 49)(9 80 91 47 31 98 50)(10 65 92 48 32 99 51)(11 66 93 33 17 100 52)(12 67 94 34 18 101 53)(13 68 95 35 19 102 54)(14 69 96 36 20 103 55)(15 70 81 37 21 104 56)(16 71 82 38 22 105 57)
(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)
(2 6 10 14)(3 11)(4 16 12 8)(7 15)(17 25)(18 30 26 22)(20 24 28 32)(21 29)(33 41)(34 46 42 38)(36 40 44 48)(37 45)(49 61 57 53)(51 55 59 63)(52 60)(56 64)(65 69 73 77)(66 74)(67 79 75 71)(70 78)(81 89)(82 94 90 86)(84 88 92 96)(85 93)(97 109 105 101)(99 103 107 111)(100 108)(104 112)
G:=sub<Sym(112)| (1,72,83,39,23,106,58)(2,73,84,40,24,107,59)(3,74,85,41,25,108,60)(4,75,86,42,26,109,61)(5,76,87,43,27,110,62)(6,77,88,44,28,111,63)(7,78,89,45,29,112,64)(8,79,90,46,30,97,49)(9,80,91,47,31,98,50)(10,65,92,48,32,99,51)(11,66,93,33,17,100,52)(12,67,94,34,18,101,53)(13,68,95,35,19,102,54)(14,69,96,36,20,103,55)(15,70,81,37,21,104,56)(16,71,82,38,22,105,57), (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), (2,6,10,14)(3,11)(4,16,12,8)(7,15)(17,25)(18,30,26,22)(20,24,28,32)(21,29)(33,41)(34,46,42,38)(36,40,44,48)(37,45)(49,61,57,53)(51,55,59,63)(52,60)(56,64)(65,69,73,77)(66,74)(67,79,75,71)(70,78)(81,89)(82,94,90,86)(84,88,92,96)(85,93)(97,109,105,101)(99,103,107,111)(100,108)(104,112)>;
G:=Group( (1,72,83,39,23,106,58)(2,73,84,40,24,107,59)(3,74,85,41,25,108,60)(4,75,86,42,26,109,61)(5,76,87,43,27,110,62)(6,77,88,44,28,111,63)(7,78,89,45,29,112,64)(8,79,90,46,30,97,49)(9,80,91,47,31,98,50)(10,65,92,48,32,99,51)(11,66,93,33,17,100,52)(12,67,94,34,18,101,53)(13,68,95,35,19,102,54)(14,69,96,36,20,103,55)(15,70,81,37,21,104,56)(16,71,82,38,22,105,57), (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), (2,6,10,14)(3,11)(4,16,12,8)(7,15)(17,25)(18,30,26,22)(20,24,28,32)(21,29)(33,41)(34,46,42,38)(36,40,44,48)(37,45)(49,61,57,53)(51,55,59,63)(52,60)(56,64)(65,69,73,77)(66,74)(67,79,75,71)(70,78)(81,89)(82,94,90,86)(84,88,92,96)(85,93)(97,109,105,101)(99,103,107,111)(100,108)(104,112) );
G=PermutationGroup([[(1,72,83,39,23,106,58),(2,73,84,40,24,107,59),(3,74,85,41,25,108,60),(4,75,86,42,26,109,61),(5,76,87,43,27,110,62),(6,77,88,44,28,111,63),(7,78,89,45,29,112,64),(8,79,90,46,30,97,49),(9,80,91,47,31,98,50),(10,65,92,48,32,99,51),(11,66,93,33,17,100,52),(12,67,94,34,18,101,53),(13,68,95,35,19,102,54),(14,69,96,36,20,103,55),(15,70,81,37,21,104,56),(16,71,82,38,22,105,57)], [(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)], [(2,6,10,14),(3,11),(4,16,12,8),(7,15),(17,25),(18,30,26,22),(20,24,28,32),(21,29),(33,41),(34,46,42,38),(36,40,44,48),(37,45),(49,61,57,53),(51,55,59,63),(52,60),(56,64),(65,69,73,77),(66,74),(67,79,75,71),(70,78),(81,89),(82,94,90,86),(84,88,92,96),(85,93),(97,109,105,101),(99,103,107,111),(100,108),(104,112)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 4E | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 8F | 14A | ··· | 14F | 14G | ··· | 14L | 16A | ··· | 16H | 28A | ··· | 28L | 28M | ··· | 28R | 28S | ··· | 28AD | 56A | ··· | 56X | 56Y | ··· | 56AJ | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C7 | C14 | C14 | C28 | C28 | C28 | M4(2) | M4(2) | C7×M4(2) | C7×M4(2) | C16⋊C4 | C7×C16⋊C4 |
| kernel | C7×C16⋊C4 | C7×C8⋊C4 | C7×M5(2) | C112 | C4×C28 | C2×C56 | C16⋊C4 | C8⋊C4 | M5(2) | C16 | C42 | C2×C8 | C28 | C2×C14 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 6 | 6 | 12 | 48 | 12 | 12 | 2 | 2 | 12 | 12 | 2 | 12 |
Matrix representation of C7×C16⋊C4 ►in GL4(𝔽113) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 1 | 0 |
| 112 | 0 | 15 | 2 |
| 0 | 15 | 0 | 0 |
| 57 | 57 | 57 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 106 | 0 | 112 | 15 |
G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[0,112,0,57,0,0,15,57,1,15,0,57,0,2,0,0],[1,0,0,106,0,112,0,0,0,0,98,112,0,0,0,15] >;
C7×C16⋊C4 in GAP, Magma, Sage, TeX
C_7\times C_{16}\rtimes C_4 % in TeX
G:=Group("C7xC16:C4"); // GroupNames label
G:=SmallGroup(448,151);
// by ID
G=gap.SmallGroup(448,151);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,196,1597,400,3538,136,9804,124]);
// Polycyclic
G:=Group<a,b,c|a^7=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^13>;
// generators/relations
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