direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C14×C8⋊C22, C56⋊8C23, C28.82C24, C8⋊(C22×C14), D8⋊3(C2×C14), (C2×D8)⋊11C14, (C14×D8)⋊25C2, C4.66(D4×C14), (C2×C56)⋊29C22, SD16⋊1(C2×C14), (C2×SD16)⋊4C14, C28.329(C2×D4), (C2×C28).525D4, (C7×D4)⋊13C23, (C7×D8)⋊19C22, D4⋊2(C22×C14), C4.5(C23×C14), (C7×Q8)⋊12C23, Q8⋊2(C22×C14), C23.50(C7×D4), (C14×SD16)⋊15C2, (C22×D4)⋊11C14, (D4×C14)⋊66C22, M4(2)⋊3(C2×C14), (C2×M4(2))⋊3C14, (Q8×C14)⋊54C22, C22.23(D4×C14), (C14×M4(2))⋊13C2, (C2×C28).975C23, (C7×SD16)⋊17C22, (C22×C14).172D4, C14.203(C22×D4), (C7×M4(2))⋊29C22, (C22×C28).465C22, (C2×C8)⋊2(C2×C14), (D4×C2×C14)⋊26C2, C2.27(D4×C2×C14), C4○D4⋊4(C2×C14), (C14×C4○D4)⋊27C2, (C2×C4○D4)⋊11C14, (C2×D4)⋊15(C2×C14), (C2×Q8)⋊14(C2×C14), (C2×C4).136(C7×D4), (C2×C14).419(C2×D4), (C7×C4○D4)⋊24C22, (C2×C4).45(C22×C14), (C22×C4).76(C2×C14), SmallGroup(448,1356)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×C8⋊C22
G = < a,b,c,d | a14=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >
Subgroups: 530 in 298 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C14, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C28, C28, C28, C2×C14, C2×C14, C2×C14, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, C2×C8⋊C22, C2×C56, C7×M4(2), C7×D8, C7×SD16, C22×C28, C22×C28, D4×C14, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C23×C14, C14×M4(2), C14×D8, C14×SD16, C7×C8⋊C22, D4×C2×C14, C14×C4○D4, C14×C8⋊C22
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C8⋊C22, C22×D4, C7×D4, C22×C14, C2×C8⋊C22, D4×C14, C23×C14, C7×C8⋊C22, D4×C2×C14, C14×C8⋊C22
►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 74 36 15 54 106 63 94)(2 75 37 16 55 107 64 95)(3 76 38 17 56 108 65 96)(4 77 39 18 43 109 66 97)(5 78 40 19 44 110 67 98)(6 79 41 20 45 111 68 85)(7 80 42 21 46 112 69 86)(8 81 29 22 47 99 70 87)(9 82 30 23 48 100 57 88)(10 83 31 24 49 101 58 89)(11 84 32 25 50 102 59 90)(12 71 33 26 51 103 60 91)(13 72 34 27 52 104 61 92)(14 73 35 28 53 105 62 93)
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 43)(12 44)(13 45)(14 46)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 98)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)
(15 94)(16 95)(17 96)(18 97)(19 98)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 91)(27 92)(28 93)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 99)(82 100)(83 101)(84 102)
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), (1,74,36,15,54,106,63,94)(2,75,37,16,55,107,64,95)(3,76,38,17,56,108,65,96)(4,77,39,18,43,109,66,97)(5,78,40,19,44,110,67,98)(6,79,41,20,45,111,68,85)(7,80,42,21,46,112,69,86)(8,81,29,22,47,99,70,87)(9,82,30,23,48,100,57,88)(10,83,31,24,49,101,58,89)(11,84,32,25,50,102,59,90)(12,71,33,26,51,103,60,91)(13,72,34,27,52,104,61,92)(14,73,35,28,53,105,62,93), (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,43)(12,44)(13,45)(14,46)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,98)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97), (15,94)(16,95)(17,96)(18,97)(19,98)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,99)(82,100)(83,101)(84,102)>;
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), (1,74,36,15,54,106,63,94)(2,75,37,16,55,107,64,95)(3,76,38,17,56,108,65,96)(4,77,39,18,43,109,66,97)(5,78,40,19,44,110,67,98)(6,79,41,20,45,111,68,85)(7,80,42,21,46,112,69,86)(8,81,29,22,47,99,70,87)(9,82,30,23,48,100,57,88)(10,83,31,24,49,101,58,89)(11,84,32,25,50,102,59,90)(12,71,33,26,51,103,60,91)(13,72,34,27,52,104,61,92)(14,73,35,28,53,105,62,93), (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,43)(12,44)(13,45)(14,46)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,98)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97), (15,94)(16,95)(17,96)(18,97)(19,98)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,99)(82,100)(83,101)(84,102) );
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)], [(1,74,36,15,54,106,63,94),(2,75,37,16,55,107,64,95),(3,76,38,17,56,108,65,96),(4,77,39,18,43,109,66,97),(5,78,40,19,44,110,67,98),(6,79,41,20,45,111,68,85),(7,80,42,21,46,112,69,86),(8,81,29,22,47,99,70,87),(9,82,30,23,48,100,57,88),(10,83,31,24,49,101,58,89),(11,84,32,25,50,102,59,90),(12,71,33,26,51,103,60,91),(13,72,34,27,52,104,61,92),(14,73,35,28,53,105,62,93)], [(1,47),(2,48),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,43),(12,44),(13,45),(14,46),(15,99),(16,100),(17,101),(18,102),(19,103),(20,104),(21,105),(22,106),(23,107),(24,108),(25,109),(26,110),(27,111),(28,112),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,98),(72,85),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97)], [(15,94),(16,95),(17,96),(18,97),(19,98),(20,85),(21,86),(22,87),(23,88),(24,89),(25,90),(26,91),(27,92),(28,93),(71,103),(72,104),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112),(81,99),(82,100),(83,101),(84,102)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14BN | 28A | ··· | 28X | 28Y | ··· | 28AJ | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | C7×D4 | C7×D4 | C8⋊C22 | C7×C8⋊C22 |
| kernel | C14×C8⋊C22 | C14×M4(2) | C14×D8 | C14×SD16 | C7×C8⋊C22 | D4×C2×C14 | C14×C4○D4 | C2×C8⋊C22 | C2×M4(2) | C2×D8 | C2×SD16 | C8⋊C22 | C22×D4 | C2×C4○D4 | C2×C28 | C22×C14 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 6 | 6 | 12 | 12 | 48 | 6 | 6 | 3 | 1 | 18 | 6 | 2 | 12 |
Matrix representation of C14×C8⋊C22 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 112 | 2 | 0 | 0 | 0 | 0 |
| 112 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 112 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[112,112,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,112,0,0],[112,112,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,112,0],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112] >;
C14×C8⋊C22 in GAP, Magma, Sage, TeX
C_{14}\times C_8\rtimes C_2^2 % in TeX
G:=Group("C14xC8:C2^2"); // GroupNames label
G:=SmallGroup(448,1356);
// by ID
G=gap.SmallGroup(448,1356);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations