direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C14×SD16, C28.42D4, C56⋊13C22, C28.45C23, (C2×C8)⋊5C14, C8⋊3(C2×C14), C4.7(C7×D4), (C2×C56)⋊13C2, (C2×Q8)⋊3C14, Q8⋊1(C2×C14), (Q8×C14)⋊10C2, D4.1(C2×C14), (C2×D4).6C14, C14.75(C2×D4), (C2×C14).53D4, C2.12(D4×C14), (C7×Q8)⋊9C22, (D4×C14).13C2, C4.2(C22×C14), C22.15(C7×D4), (C7×D4).11C22, (C2×C28).130C22, (C2×C4).26(C2×C14), SmallGroup(224,168)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×SD16
G = < a,b,c | a14=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Subgroups: 108 in 68 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C14, C14, C14, C2×C8, SD16, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, C2×SD16, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C2×C56, C7×SD16, D4×C14, Q8×C14, C14×SD16
Quotients: C1, C2, C22, C7, D4, C23, C14, SD16, C2×D4, C2×C14, C2×SD16, C7×D4, C22×C14, C7×SD16, D4×C14, C14×SD16
►On 112 points
(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)
(1 108 96 28 47 61 32 75)(2 109 97 15 48 62 33 76)(3 110 98 16 49 63 34 77)(4 111 85 17 50 64 35 78)(5 112 86 18 51 65 36 79)(6 99 87 19 52 66 37 80)(7 100 88 20 53 67 38 81)(8 101 89 21 54 68 39 82)(9 102 90 22 55 69 40 83)(10 103 91 23 56 70 41 84)(11 104 92 24 43 57 42 71)(12 105 93 25 44 58 29 72)(13 106 94 26 45 59 30 73)(14 107 95 27 46 60 31 74)
(15 109)(16 110)(17 111)(18 112)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)
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), (1,108,96,28,47,61,32,75)(2,109,97,15,48,62,33,76)(3,110,98,16,49,63,34,77)(4,111,85,17,50,64,35,78)(5,112,86,18,51,65,36,79)(6,99,87,19,52,66,37,80)(7,100,88,20,53,67,38,81)(8,101,89,21,54,68,39,82)(9,102,90,22,55,69,40,83)(10,103,91,23,56,70,41,84)(11,104,92,24,43,57,42,71)(12,105,93,25,44,58,29,72)(13,106,94,26,45,59,30,73)(14,107,95,27,46,60,31,74), (15,109)(16,110)(17,111)(18,112)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)>;
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), (1,108,96,28,47,61,32,75)(2,109,97,15,48,62,33,76)(3,110,98,16,49,63,34,77)(4,111,85,17,50,64,35,78)(5,112,86,18,51,65,36,79)(6,99,87,19,52,66,37,80)(7,100,88,20,53,67,38,81)(8,101,89,21,54,68,39,82)(9,102,90,22,55,69,40,83)(10,103,91,23,56,70,41,84)(11,104,92,24,43,57,42,71)(12,105,93,25,44,58,29,72)(13,106,94,26,45,59,30,73)(14,107,95,27,46,60,31,74), (15,109)(16,110)(17,111)(18,112)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84) );
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)], [(1,108,96,28,47,61,32,75),(2,109,97,15,48,62,33,76),(3,110,98,16,49,63,34,77),(4,111,85,17,50,64,35,78),(5,112,86,18,51,65,36,79),(6,99,87,19,52,66,37,80),(7,100,88,20,53,67,38,81),(8,101,89,21,54,68,39,82),(9,102,90,22,55,69,40,83),(10,103,91,23,56,70,41,84),(11,104,92,24,43,57,42,71),(12,105,93,25,44,58,29,72),(13,106,94,26,45,59,30,73),(14,107,95,27,46,60,31,74)], [(15,109),(16,110),(17,111),(18,112),(19,99),(20,100),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,93),(30,94),(31,95),(32,96),(33,97),(34,98),(35,85),(36,86),(37,87),(38,88),(39,89),(40,90),(41,91),(42,92),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84)]])
C14×SD16 is a maximal subgroup of
Dic7⋊3SD16 Dic7⋊5SD16 SD16⋊Dic7 (C7×D4).D4 (C7×Q8).D4 C56.31D4 C56.43D4 D14⋊6SD16 Dic14⋊7D4 C56⋊14D4 D28⋊7D4 Dic14.16D4 C56⋊8D4 C56⋊15D4 C56⋊9D4 C56.44D4 D28.29D4
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | D4 | D4 | SD16 | C7×D4 | C7×D4 | C7×SD16 |
| kernel | C14×SD16 | C2×C56 | C7×SD16 | D4×C14 | Q8×C14 | C2×SD16 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C28 | C2×C14 | C14 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 6 | 6 | 24 | 6 | 6 | 1 | 1 | 4 | 6 | 6 | 24 |
Matrix representation of C14×SD16 ►in GL4(𝔽113) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 109 | 0 |
| 0 | 0 | 0 | 109 |
| 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 87 | 87 |
| 0 | 0 | 13 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 112 | 112 |
G:=sub<GL(4,GF(113))| [4,0,0,0,0,4,0,0,0,0,109,0,0,0,0,109],[0,1,0,0,112,0,0,0,0,0,87,13,0,0,87,0],[1,0,0,0,0,112,0,0,0,0,1,112,0,0,0,112] >;
C14×SD16 in GAP, Magma, Sage, TeX
C_{14}\times {\rm SD}_{16} % in TeX
G:=Group("C14xSD16"); // GroupNames label
G:=SmallGroup(224,168);
// by ID
G=gap.SmallGroup(224,168);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,672,697,5044,2530,88]);
// Polycyclic
G:=Group<a,b,c|a^14=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations