direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C14×C8⋊C22, C56⋊8C23, C28.82C24, C8⋊(C22×C14), D8⋊3(C2×C14), (C14×D8)⋊25C2, (C2×D8)⋊11C14, C4.66(D4×C14), (C2×C56)⋊29C22, SD16⋊1(C2×C14), (C2×SD16)⋊4C14, (C2×C28).525D4, C28.329(C2×D4), (C7×D4)⋊13C23, (C7×D8)⋊19C22, D4⋊2(C22×C14), C4.5(C23×C14), Q8⋊2(C22×C14), (C7×Q8)⋊12C23, C23.50(C7×D4), (C14×SD16)⋊15C2, (C22×D4)⋊11C14, (D4×C14)⋊66C22, (C2×M4(2))⋊3C14, M4(2)⋊3(C2×C14), (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), (C2×C4○D4)⋊11C14, (C14×C4○D4)⋊27C2, (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
Subgroups: 530 in 298 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], C7, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×6], D4 [×11], Q8 [×2], Q8, C23, C23 [×11], C14, C14 [×2], C14 [×8], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4, C2×D4 [×6], C2×D4 [×4], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, C28 [×2], C28 [×2], C28 [×2], C2×C14, C2×C14 [×2], C2×C14 [×22], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C56 [×4], C2×C28 [×2], C2×C28 [×4], C2×C28 [×5], C7×D4 [×6], C7×D4 [×11], C7×Q8 [×2], C7×Q8, C22×C14, C22×C14 [×11], C2×C8⋊C22, C2×C56 [×2], C7×M4(2) [×4], C7×D8 [×8], C7×SD16 [×8], C22×C28, C22×C28, D4×C14, D4×C14 [×6], D4×C14 [×4], Q8×C14, C7×C4○D4 [×4], C7×C4○D4 [×2], C23×C14, C14×M4(2), C14×D8 [×2], C14×SD16 [×2], C7×C8⋊C22 [×8], D4×C2×C14, C14×C4○D4, C14×C8⋊C22
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C24, C2×C14 [×35], C8⋊C22 [×2], C22×D4, C7×D4 [×4], C22×C14 [×15], C2×C8⋊C22, D4×C14 [×6], C23×C14, C7×C8⋊C22 [×2], D4×C2×C14, C14×C8⋊C22
Generators and relations
G = < a,b,c,d | a14=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >
►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)
(1 102 81 35 64 56 21 92)(2 103 82 36 65 43 22 93)(3 104 83 37 66 44 23 94)(4 105 84 38 67 45 24 95)(5 106 71 39 68 46 25 96)(6 107 72 40 69 47 26 97)(7 108 73 41 70 48 27 98)(8 109 74 42 57 49 28 85)(9 110 75 29 58 50 15 86)(10 111 76 30 59 51 16 87)(11 112 77 31 60 52 17 88)(12 99 78 32 61 53 18 89)(13 100 79 33 62 54 19 90)(14 101 80 34 63 55 20 91)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 102)(86 103)(87 104)(88 105)(89 106)(90 107)(91 108)(92 109)(93 110)(94 111)(95 112)(96 99)(97 100)(98 101)
(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 97)(41 98)(42 85)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 99)(54 100)(55 101)(56 102)
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), (1,102,81,35,64,56,21,92)(2,103,82,36,65,43,22,93)(3,104,83,37,66,44,23,94)(4,105,84,38,67,45,24,95)(5,106,71,39,68,46,25,96)(6,107,72,40,69,47,26,97)(7,108,73,41,70,48,27,98)(8,109,74,42,57,49,28,85)(9,110,75,29,58,50,15,86)(10,111,76,30,59,51,16,87)(11,112,77,31,60,52,17,88)(12,99,78,32,61,53,18,89)(13,100,79,33,62,54,19,90)(14,101,80,34,63,55,20,91), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,102)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,99)(97,100)(98,101), (29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,98)(42,85)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,99)(54,100)(55,101)(56,102)>;
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), (1,102,81,35,64,56,21,92)(2,103,82,36,65,43,22,93)(3,104,83,37,66,44,23,94)(4,105,84,38,67,45,24,95)(5,106,71,39,68,46,25,96)(6,107,72,40,69,47,26,97)(7,108,73,41,70,48,27,98)(8,109,74,42,57,49,28,85)(9,110,75,29,58,50,15,86)(10,111,76,30,59,51,16,87)(11,112,77,31,60,52,17,88)(12,99,78,32,61,53,18,89)(13,100,79,33,62,54,19,90)(14,101,80,34,63,55,20,91), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,102)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,99)(97,100)(98,101), (29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,98)(42,85)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,99)(54,100)(55,101)(56,102) );
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)], [(1,102,81,35,64,56,21,92),(2,103,82,36,65,43,22,93),(3,104,83,37,66,44,23,94),(4,105,84,38,67,45,24,95),(5,106,71,39,68,46,25,96),(6,107,72,40,69,47,26,97),(7,108,73,41,70,48,27,98),(8,109,74,42,57,49,28,85),(9,110,75,29,58,50,15,86),(10,111,76,30,59,51,16,87),(11,112,77,31,60,52,17,88),(12,99,78,32,61,53,18,89),(13,100,79,33,62,54,19,90),(14,101,80,34,63,55,20,91)], [(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,102),(86,103),(87,104),(88,105),(89,106),(90,107),(91,108),(92,109),(93,110),(94,111),(95,112),(96,99),(97,100),(98,101)], [(29,86),(30,87),(31,88),(32,89),(33,90),(34,91),(35,92),(36,93),(37,94),(38,95),(39,96),(40,97),(41,98),(42,85),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,99),(54,100),(55,101),(56,102)])
Matrix representation ►G ⊆ 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] >;
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 |
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