direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C14×M4(2), C56⋊14C22, C23.3C28, C28.53C23, (C2×C8)⋊6C14, C8⋊4(C2×C14), (C2×C56)⋊14C2, (C2×C4).6C28, C4.10(C2×C28), (C2×C28).15C4, C28.47(C2×C4), C2.6(C22×C28), (C22×C4).6C14, (C22×C14).4C4, C22.6(C2×C28), (C22×C28).16C2, C14.34(C22×C4), C4.11(C22×C14), (C2×C28).127C22, (C2×C4).33(C2×C14), (C2×C14).23(C2×C4), SmallGroup(224,165)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×M4(2)
G = < a,b,c | a14=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Subgroups: 76 in 68 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C14, C2×C8, M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C2×M4(2), C56, C2×C28, C2×C28, C22×C14, C2×C56, C7×M4(2), C22×C28, C14×M4(2)
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, M4(2), C22×C4, C28, C2×C14, C2×M4(2), C2×C28, C22×C14, C7×M4(2), C22×C28, C14×M4(2)
►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)
(1 112 35 77 62 87 52 15)(2 99 36 78 63 88 53 16)(3 100 37 79 64 89 54 17)(4 101 38 80 65 90 55 18)(5 102 39 81 66 91 56 19)(6 103 40 82 67 92 43 20)(7 104 41 83 68 93 44 21)(8 105 42 84 69 94 45 22)(9 106 29 71 70 95 46 23)(10 107 30 72 57 96 47 24)(11 108 31 73 58 97 48 25)(12 109 32 74 59 98 49 26)(13 110 33 75 60 85 50 27)(14 111 34 76 61 86 51 28)
(15 77)(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(85 110)(86 111)(87 112)(88 99)(89 100)(90 101)(91 102)(92 103)(93 104)(94 105)(95 106)(96 107)(97 108)(98 109)
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), (1,112,35,77,62,87,52,15)(2,99,36,78,63,88,53,16)(3,100,37,79,64,89,54,17)(4,101,38,80,65,90,55,18)(5,102,39,81,66,91,56,19)(6,103,40,82,67,92,43,20)(7,104,41,83,68,93,44,21)(8,105,42,84,69,94,45,22)(9,106,29,71,70,95,46,23)(10,107,30,72,57,96,47,24)(11,108,31,73,58,97,48,25)(12,109,32,74,59,98,49,26)(13,110,33,75,60,85,50,27)(14,111,34,76,61,86,51,28), (15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(85,110)(86,111)(87,112)(88,99)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109)>;
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), (1,112,35,77,62,87,52,15)(2,99,36,78,63,88,53,16)(3,100,37,79,64,89,54,17)(4,101,38,80,65,90,55,18)(5,102,39,81,66,91,56,19)(6,103,40,82,67,92,43,20)(7,104,41,83,68,93,44,21)(8,105,42,84,69,94,45,22)(9,106,29,71,70,95,46,23)(10,107,30,72,57,96,47,24)(11,108,31,73,58,97,48,25)(12,109,32,74,59,98,49,26)(13,110,33,75,60,85,50,27)(14,111,34,76,61,86,51,28), (15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(85,110)(86,111)(87,112)(88,99)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109) );
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)], [(1,112,35,77,62,87,52,15),(2,99,36,78,63,88,53,16),(3,100,37,79,64,89,54,17),(4,101,38,80,65,90,55,18),(5,102,39,81,66,91,56,19),(6,103,40,82,67,92,43,20),(7,104,41,83,68,93,44,21),(8,105,42,84,69,94,45,22),(9,106,29,71,70,95,46,23),(10,107,30,72,57,96,47,24),(11,108,31,73,58,97,48,25),(12,109,32,74,59,98,49,26),(13,110,33,75,60,85,50,27),(14,111,34,76,61,86,51,28)], [(15,77),(16,78),(17,79),(18,80),(19,81),(20,82),(21,83),(22,84),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(85,110),(86,111),(87,112),(88,99),(89,100),(90,101),(91,102),(92,103),(93,104),(94,105),(95,106),(96,107),(97,108),(98,109)]])
C14×M4(2) is a maximal subgroup of
C56.D4 M4(2)⋊Dic7 C28.3C42 (C2×C56)⋊C4 C23.9D28 C28.4C42 M4(2)⋊4Dic7 C28.21C42 Dic7⋊4M4(2) C28.439(C2×D4) C23.46D28 C23.47D28 C28.7C42 C23.Dic14 M4(2).Dic7 D14⋊6M4(2) C56⋊D4 C56⋊18D4 (C2×D28).14C4 C23.48D28 M4(2).31D14 C23.49D28 C56⋊2D4 C56⋊3D4 C56.4D4 C23.20D28 C28.70C24 C56.9C23
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C14 | C28 | C28 | M4(2) | C7×M4(2) |
| kernel | C14×M4(2) | C2×C56 | C7×M4(2) | C22×C28 | C2×C28 | C22×C14 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 6 | 12 | 24 | 6 | 36 | 12 | 4 | 24 |
Matrix representation of C14×M4(2) ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 106 | 0 |
| 0 | 0 | 106 |
| 1 | 0 | 0 |
| 0 | 100 | 2 |
| 0 | 36 | 13 |
| 112 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 13 | 112 |
G:=sub<GL(3,GF(113))| [112,0,0,0,106,0,0,0,106],[1,0,0,0,100,36,0,2,13],[112,0,0,0,1,13,0,0,112] >;
C14×M4(2) in GAP, Magma, Sage, TeX
C_{14}\times M_4(2) % in TeX
G:=Group("C14xM4(2)"); // GroupNames label
G:=SmallGroup(224,165);
// by ID
G=gap.SmallGroup(224,165);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,336,1369,88]);
// 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^5>;
// generators/relations