?
G = C14.1222+ (1+4) order 448 = 26·7
31st non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.1222+ (1+4), C4⋊C4⋊30D14, C22⋊C4⋊17D14, (D4×Dic7)⋊28C2, D14⋊2Q8⋊32C2, (C2×D4).166D14, C23⋊D14.2C2, (C2×C28).74C23, C4⋊Dic7⋊39C22, C22.D4⋊7D7, Dic7⋊4D4⋊21C2, D14.21(C4○D4), D14.D4⋊31C2, (C2×C14).202C24, Dic7⋊C4⋊24C22, D14⋊C4.33C22, (C4×Dic7)⋊55C22, (C22×C4).258D14, C2.43(D4⋊8D14), C23.D7⋊31C22, C22⋊Dic14⋊32C2, Dic7.D4⋊33C2, C7⋊6(C22.45C24), (C2×Dic14)⋊30C22, (D4×C14).140C22, C22.D28⋊21C2, (C22×D7).86C23, (C23×D7).59C22, C23.202(C22×D7), C22.223(C23×D7), C23.21D14⋊12C2, C22.19(D4⋊2D7), (C22×C14).222C23, (C22×C28).114C22, (C2×Dic7).245C23, (C22×Dic7)⋊26C22, C2.64(D7×C4○D4), (C2×D14⋊C4)⋊24C2, C4⋊C4⋊D7⋊28C2, C4⋊C4⋊7D7⋊32C2, (C7×C4⋊C4)⋊28C22, (D7×C22⋊C4)⋊14C2, C14.176(C2×C4○D4), C2.53(C2×D4⋊2D7), (C2×C4×D7).111C22, (C2×C14).47(C4○D4), (C7×C22⋊C4)⋊24C22, (C2×C4).297(C22×D7), (C2×C7⋊D4).47C22, (C7×C22.D4)⋊10C2, SmallGroup(448,1111)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1228 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×11], C22, C22 [×2], C22 [×16], C7, C2×C4 [×5], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], D7 [×3], C14 [×3], C14 [×3], C42 [×3], C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic7 [×6], C28 [×5], D14 [×2], D14 [×9], C2×C14, C2×C14 [×2], C2×C14 [×5], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C22.D4 [×2], C4.4D4, C42⋊2C2 [×2], Dic14, C4×D7 [×3], C2×Dic7 [×6], C2×Dic7 [×2], C7⋊D4 [×3], C2×C28 [×5], C2×C28 [×2], C7×D4 [×2], C22×D7 [×2], C22×D7 [×5], C22×C14 [×2], C22.45C24, C4×Dic7 [×3], Dic7⋊C4 [×2], C4⋊Dic7 [×4], D14⋊C4 [×8], C23.D7 [×3], C7×C22⋊C4 [×3], C7×C4⋊C4 [×2], C2×Dic14, C2×C4×D7 [×2], C22×Dic7 [×2], C2×C7⋊D4 [×2], C22×C28, D4×C14, C23×D7, C22⋊Dic14, D7×C22⋊C4, Dic7⋊4D4, D14.D4, Dic7.D4, C22.D28, C4⋊C4⋊7D7, D14⋊2Q8, C4⋊C4⋊D7 [×2], C23.21D14, C2×D14⋊C4, D4×Dic7, C23⋊D14, C7×C22.D4, C14.1222+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×4], C24, D14 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D7 [×7], C22.45C24, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, D7×C4○D4, D4⋊8D14, C14.1222+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7, bab-1=eae-1=a-1, ac=ca, ad=da, cbc=b-1, dbd-1=a7b, be=eb, dcd-1=a7c, ce=ec, ede-1=a7b2d >
►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 79 89 45)(2 78 90 44)(3 77 91 43)(4 76 92 56)(5 75 93 55)(6 74 94 54)(7 73 95 53)(8 72 96 52)(9 71 97 51)(10 84 98 50)(11 83 85 49)(12 82 86 48)(13 81 87 47)(14 80 88 46)(15 104 62 39)(16 103 63 38)(17 102 64 37)(18 101 65 36)(19 100 66 35)(20 99 67 34)(21 112 68 33)(22 111 69 32)(23 110 70 31)(24 109 57 30)(25 108 58 29)(26 107 59 42)(27 106 60 41)(28 105 61 40)
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 84)(14 71)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 94)(44 95)(45 96)(46 97)(47 98)(48 85)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)
(1 58 89 25)(2 59 90 26)(3 60 91 27)(4 61 92 28)(5 62 93 15)(6 63 94 16)(7 64 95 17)(8 65 96 18)(9 66 97 19)(10 67 98 20)(11 68 85 21)(12 69 86 22)(13 70 87 23)(14 57 88 24)(29 52 108 72)(30 53 109 73)(31 54 110 74)(32 55 111 75)(33 56 112 76)(34 43 99 77)(35 44 100 78)(36 45 101 79)(37 46 102 80)(38 47 103 81)(39 48 104 82)(40 49 105 83)(41 50 106 84)(42 51 107 71)
(1 65 8 58)(2 64 9 57)(3 63 10 70)(4 62 11 69)(5 61 12 68)(6 60 13 67)(7 59 14 66)(15 85 22 92)(16 98 23 91)(17 97 24 90)(18 96 25 89)(19 95 26 88)(20 94 27 87)(21 93 28 86)(29 79 36 72)(30 78 37 71)(31 77 38 84)(32 76 39 83)(33 75 40 82)(34 74 41 81)(35 73 42 80)(43 103 50 110)(44 102 51 109)(45 101 52 108)(46 100 53 107)(47 99 54 106)(48 112 55 105)(49 111 56 104)
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), (1,79,89,45)(2,78,90,44)(3,77,91,43)(4,76,92,56)(5,75,93,55)(6,74,94,54)(7,73,95,53)(8,72,96,52)(9,71,97,51)(10,84,98,50)(11,83,85,49)(12,82,86,48)(13,81,87,47)(14,80,88,46)(15,104,62,39)(16,103,63,38)(17,102,64,37)(18,101,65,36)(19,100,66,35)(20,99,67,34)(21,112,68,33)(22,111,69,32)(23,110,70,31)(24,109,57,30)(25,108,58,29)(26,107,59,42)(27,106,60,41)(28,105,61,40), (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,71)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,94)(44,95)(45,96)(46,97)(47,98)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93), (1,58,89,25)(2,59,90,26)(3,60,91,27)(4,61,92,28)(5,62,93,15)(6,63,94,16)(7,64,95,17)(8,65,96,18)(9,66,97,19)(10,67,98,20)(11,68,85,21)(12,69,86,22)(13,70,87,23)(14,57,88,24)(29,52,108,72)(30,53,109,73)(31,54,110,74)(32,55,111,75)(33,56,112,76)(34,43,99,77)(35,44,100,78)(36,45,101,79)(37,46,102,80)(38,47,103,81)(39,48,104,82)(40,49,105,83)(41,50,106,84)(42,51,107,71), (1,65,8,58)(2,64,9,57)(3,63,10,70)(4,62,11,69)(5,61,12,68)(6,60,13,67)(7,59,14,66)(15,85,22,92)(16,98,23,91)(17,97,24,90)(18,96,25,89)(19,95,26,88)(20,94,27,87)(21,93,28,86)(29,79,36,72)(30,78,37,71)(31,77,38,84)(32,76,39,83)(33,75,40,82)(34,74,41,81)(35,73,42,80)(43,103,50,110)(44,102,51,109)(45,101,52,108)(46,100,53,107)(47,99,54,106)(48,112,55,105)(49,111,56,104)>;
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), (1,79,89,45)(2,78,90,44)(3,77,91,43)(4,76,92,56)(5,75,93,55)(6,74,94,54)(7,73,95,53)(8,72,96,52)(9,71,97,51)(10,84,98,50)(11,83,85,49)(12,82,86,48)(13,81,87,47)(14,80,88,46)(15,104,62,39)(16,103,63,38)(17,102,64,37)(18,101,65,36)(19,100,66,35)(20,99,67,34)(21,112,68,33)(22,111,69,32)(23,110,70,31)(24,109,57,30)(25,108,58,29)(26,107,59,42)(27,106,60,41)(28,105,61,40), (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,71)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,94)(44,95)(45,96)(46,97)(47,98)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93), (1,58,89,25)(2,59,90,26)(3,60,91,27)(4,61,92,28)(5,62,93,15)(6,63,94,16)(7,64,95,17)(8,65,96,18)(9,66,97,19)(10,67,98,20)(11,68,85,21)(12,69,86,22)(13,70,87,23)(14,57,88,24)(29,52,108,72)(30,53,109,73)(31,54,110,74)(32,55,111,75)(33,56,112,76)(34,43,99,77)(35,44,100,78)(36,45,101,79)(37,46,102,80)(38,47,103,81)(39,48,104,82)(40,49,105,83)(41,50,106,84)(42,51,107,71), (1,65,8,58)(2,64,9,57)(3,63,10,70)(4,62,11,69)(5,61,12,68)(6,60,13,67)(7,59,14,66)(15,85,22,92)(16,98,23,91)(17,97,24,90)(18,96,25,89)(19,95,26,88)(20,94,27,87)(21,93,28,86)(29,79,36,72)(30,78,37,71)(31,77,38,84)(32,76,39,83)(33,75,40,82)(34,74,41,81)(35,73,42,80)(43,103,50,110)(44,102,51,109)(45,101,52,108)(46,100,53,107)(47,99,54,106)(48,112,55,105)(49,111,56,104) );
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)], [(1,79,89,45),(2,78,90,44),(3,77,91,43),(4,76,92,56),(5,75,93,55),(6,74,94,54),(7,73,95,53),(8,72,96,52),(9,71,97,51),(10,84,98,50),(11,83,85,49),(12,82,86,48),(13,81,87,47),(14,80,88,46),(15,104,62,39),(16,103,63,38),(17,102,64,37),(18,101,65,36),(19,100,66,35),(20,99,67,34),(21,112,68,33),(22,111,69,32),(23,110,70,31),(24,109,57,30),(25,108,58,29),(26,107,59,42),(27,106,60,41),(28,105,61,40)], [(1,72),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,81),(11,82),(12,83),(13,84),(14,71),(15,105),(16,106),(17,107),(18,108),(19,109),(20,110),(21,111),(22,112),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(41,63),(42,64),(43,94),(44,95),(45,96),(46,97),(47,98),(48,85),(49,86),(50,87),(51,88),(52,89),(53,90),(54,91),(55,92),(56,93)], [(1,58,89,25),(2,59,90,26),(3,60,91,27),(4,61,92,28),(5,62,93,15),(6,63,94,16),(7,64,95,17),(8,65,96,18),(9,66,97,19),(10,67,98,20),(11,68,85,21),(12,69,86,22),(13,70,87,23),(14,57,88,24),(29,52,108,72),(30,53,109,73),(31,54,110,74),(32,55,111,75),(33,56,112,76),(34,43,99,77),(35,44,100,78),(36,45,101,79),(37,46,102,80),(38,47,103,81),(39,48,104,82),(40,49,105,83),(41,50,106,84),(42,51,107,71)], [(1,65,8,58),(2,64,9,57),(3,63,10,70),(4,62,11,69),(5,61,12,68),(6,60,13,67),(7,59,14,66),(15,85,22,92),(16,98,23,91),(17,97,24,90),(18,96,25,89),(19,95,26,88),(20,94,27,87),(21,93,28,86),(29,79,36,72),(30,78,37,71),(31,77,38,84),(32,76,39,83),(33,75,40,82),(34,74,41,81),(35,73,42,80),(43,103,50,110),(44,102,51,109),(45,101,52,108),(46,100,53,107),(47,99,54,106),(48,112,55,105),(49,111,56,104)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 26 | 21 | 0 | 0 | 0 | 0 |
| 8 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 8 | 3 | 0 | 0 | 0 | 0 |
| 8 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 24 | 0 | 0 |
| 0 | 0 | 17 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 24 | 0 | 0 |
| 0 | 0 | 17 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 8 | 3 | 0 | 0 | 0 | 0 |
| 8 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 27 | 0 | 0 |
| 0 | 0 | 1 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [26,8,0,0,0,0,21,21,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[8,8,0,0,0,0,3,21,0,0,0,0,0,0,17,17,0,0,0,0,24,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,17,0,0,0,0,24,12,0,0,0,0,0,0,0,17,0,0,0,0,12,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,2,1,0,0,0,0,0,0,12,0,0,0,0,0,0,17],[8,8,0,0,0,0,3,21,0,0,0,0,0,0,1,1,0,0,0,0,27,28,0,0,0,0,0,0,17,0,0,0,0,0,0,17] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 14 | 14 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊2D7 | D7×C4○D4 | D4⋊8D14 |
| kernel | C14.1222+ (1+4) | C22⋊Dic14 | D7×C22⋊C4 | Dic7⋊4D4 | D14.D4 | Dic7.D4 | C22.D28 | C4⋊C4⋊7D7 | D14⋊2Q8 | C4⋊C4⋊D7 | C23.21D14 | C2×D14⋊C4 | D4×Dic7 | C23⋊D14 | C7×C22.D4 | C22.D4 | D14 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 9 | 6 | 3 | 3 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{122}2_+^{(1+4)} % in TeX
G:=Group("C14.122ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1111);
// by ID
G=gap.SmallGroup(448,1111);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,100,346,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=b^2,e^2=a^7,b*a*b^-1=e*a*e^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=a^7*b,b*e=e*b,d*c*d^-1=a^7*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations