metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C112⋊2C4, C16⋊2Dic7, C28.18C42, M5(2).3D7, C28.5M4(2), C7⋊C16⋊4C4, C8.33(C4×D7), C7⋊2(C16⋊C4), C56.77(C2×C4), (C2×C8).151D14, C4.9(C8⋊D7), (C4×Dic7).3C4, C56⋊C4.10C2, C8.20(C2×Dic7), C4.23(C4×Dic7), C14.5(C8⋊C4), C2.4(C56⋊C4), C28.C8.8C2, (C7×M5(2)).2C2, (C2×C14).4M4(2), (C2×C56).219C22, C22.6(C8⋊D7), (C2×C7⋊C8).1C4, (C2×C28).51(C2×C4), (C2×C4).136(C4×D7), SmallGroup(448,69)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C112⋊C4
G = < a,b | a112=b4=1, bab-1=a13 >
Smallest permutation representation of C112⋊C4
►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)
(2 70 58 14)(3 27)(4 96 60 40)(5 53)(6 10 62 66)(7 79)(8 36 64 92)(9 105)(11 19)(12 88 68 32)(13 45)(15 71)(16 28 72 84)(17 97)(18 54 74 110)(20 80 76 24)(21 37)(22 106 78 50)(23 63)(25 89)(26 46 82 102)(30 98 86 42)(31 55)(33 81)(34 38 90 94)(35 107)(39 47)(41 73)(43 99)(44 56 100 112)(48 108 104 52)(49 65)(51 91)(59 83)(61 109)(67 75)(69 101)(77 93)(87 111)(95 103)
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), (2,70,58,14)(3,27)(4,96,60,40)(5,53)(6,10,62,66)(7,79)(8,36,64,92)(9,105)(11,19)(12,88,68,32)(13,45)(15,71)(16,28,72,84)(17,97)(18,54,74,110)(20,80,76,24)(21,37)(22,106,78,50)(23,63)(25,89)(26,46,82,102)(30,98,86,42)(31,55)(33,81)(34,38,90,94)(35,107)(39,47)(41,73)(43,99)(44,56,100,112)(48,108,104,52)(49,65)(51,91)(59,83)(61,109)(67,75)(69,101)(77,93)(87,111)(95,103)>;
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), (2,70,58,14)(3,27)(4,96,60,40)(5,53)(6,10,62,66)(7,79)(8,36,64,92)(9,105)(11,19)(12,88,68,32)(13,45)(15,71)(16,28,72,84)(17,97)(18,54,74,110)(20,80,76,24)(21,37)(22,106,78,50)(23,63)(25,89)(26,46,82,102)(30,98,86,42)(31,55)(33,81)(34,38,90,94)(35,107)(39,47)(41,73)(43,99)(44,56,100,112)(48,108,104,52)(49,65)(51,91)(59,83)(61,109)(67,75)(69,101)(77,93)(87,111)(95,103) );
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)], [(2,70,58,14),(3,27),(4,96,60,40),(5,53),(6,10,62,66),(7,79),(8,36,64,92),(9,105),(11,19),(12,88,68,32),(13,45),(15,71),(16,28,72,84),(17,97),(18,54,74,110),(20,80,76,24),(21,37),(22,106,78,50),(23,63),(25,89),(26,46,82,102),(30,98,86,42),(31,55),(33,81),(34,38,90,94),(35,107),(39,47),(41,73),(43,99),(44,56,100,112),(48,108,104,52),(49,65),(51,91),(59,83),(61,109),(67,75),(69,101),(77,93),(87,111),(95,103)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 14A | 14B | 14C | 14D | 14E | 14F | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56R | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D7 | M4(2) | M4(2) | Dic7 | D14 | C4×D7 | C4×D7 | C8⋊D7 | C8⋊D7 | C16⋊C4 | C112⋊C4 |
| kernel | C112⋊C4 | C28.C8 | C56⋊C4 | C7×M5(2) | C7⋊C16 | C112 | C2×C7⋊C8 | C4×Dic7 | M5(2) | C28 | C2×C14 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 3 | 2 | 2 | 6 | 3 | 6 | 6 | 12 | 12 | 2 | 12 |
Matrix representation of C112⋊C4 ►in GL4(𝔽113) generated by
| 0 | 0 | 88 | 88 |
| 0 | 0 | 25 | 34 |
| 110 | 3 | 0 | 0 |
| 110 | 8 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 79 | 112 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 58 | 15 |
G:=sub<GL(4,GF(113))| [0,0,110,110,0,0,3,8,88,25,0,0,88,34,0,0],[1,79,0,0,0,112,0,0,0,0,98,58,0,0,0,15] >;
C112⋊C4 in GAP, Magma, Sage, TeX
C_{112}\rtimes C_4 % in TeX
G:=Group("C112:C4"); // GroupNames label
G:=SmallGroup(448,69);
// by ID
G=gap.SmallGroup(448,69);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,387,1123,80,102,18822]);
// Polycyclic
G:=Group<a,b|a^112=b^4=1,b*a*b^-1=a^13>;
// generators/relations
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