metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.51C24, C14.16C25, D14.9C24, D28.37C23, 2+ (1+4)⋊5D7, Dic7.11C24, Dic14.38C23, C4○D4⋊11D14, (C2×D4)⋊32D14, (D4×D7)⋊14C22, (C2×C14).7C24, D4⋊6D14⋊10C2, (Q8×D7)⋊16C22, C2.17(D7×C24), C4.48(C23×D7), C7⋊D4.3C23, C4○D28⋊13C22, (D4×C14)⋊26C22, C7⋊2(C2.C25), D4.31(C22×D7), (C7×D4).31C23, (C4×D7).20C23, Q8.32(C22×D7), (C7×Q8).32C23, D4⋊2D7⋊16C22, C22.4(C23×D7), (C2×C28).122C23, Q8⋊2D7⋊19C22, (C7×2+ (1+4))⋊5C2, D4.10D14⋊11C2, C23.71(C22×D7), (C2×Dic14)⋊44C22, (C22×C14).79C23, (C2×Dic7).169C23, (C22×Dic7)⋊39C22, (C22×D7).143C23, (D7×C4○D4)⋊8C2, (C2×C4×D7)⋊37C22, (C2×D4⋊2D7)⋊31C2, (C7×C4○D4)⋊11C22, (C2×C7⋊D4)⋊33C22, (C2×C4).106(C22×D7), SmallGroup(448,1380)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 3060 in 810 conjugacy classes, 443 normal (8 characteristic)
C1, C2, C2 [×15], C4 [×6], C4 [×10], C22 [×9], C22 [×21], C7, C2×C4 [×9], C2×C4 [×51], D4 [×18], D4 [×42], Q8 [×2], Q8 [×18], C23 [×6], C23 [×9], D7 [×6], C14, C14 [×9], C22×C4 [×15], C2×D4 [×9], C2×D4 [×36], C2×Q8 [×15], C4○D4 [×6], C4○D4 [×74], Dic7, Dic7 [×9], C28 [×6], D14 [×6], D14 [×9], C2×C14 [×9], C2×C14 [×6], C2×C4○D4 [×15], 2+ (1+4), 2+ (1+4) [×9], 2- (1+4) [×6], Dic14 [×18], C4×D7 [×24], D28 [×6], C2×Dic7 [×27], C7⋊D4 [×36], C2×C28 [×9], C7×D4 [×18], C7×Q8 [×2], C22×D7 [×9], C22×C14 [×6], C2.C25, C2×Dic14 [×9], C2×C4×D7 [×9], C4○D28 [×18], D4×D7 [×18], D4⋊2D7 [×54], Q8×D7 [×6], Q8⋊2D7 [×2], C22×Dic7 [×6], C2×C7⋊D4 [×18], D4×C14 [×9], C7×C4○D4 [×6], C2×D4⋊2D7 [×9], D4⋊6D14 [×9], D7×C4○D4 [×6], D4.10D14 [×6], C7×2+ (1+4), D14.C24
Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], D7, C24 [×31], D14 [×15], C25, C22×D7 [×35], C2.C25, C23×D7 [×15], D7×C24, D14.C24
Generators and relations
G = < a,b,c,d,e,f | a14=b2=c2=e2=f2=1, d2=a7, bab=eae=a-1, ac=ca, ad=da, af=fa, cbc=fbf=a7b, bd=db, ebe=a12b, cd=dc, ce=ec, cf=fc, de=ed, fdf=a7d, fef=a7e >
►On 112 points
Generators in S112
(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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(29 38)(30 37)(31 36)(32 35)(33 34)(39 42)(40 41)(43 53)(44 52)(45 51)(46 50)(47 49)(54 56)(57 59)(60 70)(61 69)(62 68)(63 67)(64 66)(71 78)(72 77)(73 76)(74 75)(79 84)(80 83)(81 82)(85 87)(88 98)(89 97)(90 96)(91 95)(92 94)(99 100)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)
(1 69)(2 70)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 72)(16 73)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 81)(25 82)(26 83)(27 84)(28 71)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)
(1 34 8 41)(2 35 9 42)(3 36 10 29)(4 37 11 30)(5 38 12 31)(6 39 13 32)(7 40 14 33)(15 56 22 49)(16 43 23 50)(17 44 24 51)(18 45 25 52)(19 46 26 53)(20 47 27 54)(21 48 28 55)(57 92 64 85)(58 93 65 86)(59 94 66 87)(60 95 67 88)(61 96 68 89)(62 97 69 90)(63 98 70 91)(71 103 78 110)(72 104 79 111)(73 105 80 112)(74 106 81 99)(75 107 82 100)(76 108 83 101)(77 109 84 102)
(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(29 39)(30 38)(31 37)(32 36)(33 35)(40 42)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(55 56)(57 67)(58 66)(59 65)(60 64)(61 63)(68 70)(71 72)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 95)(86 94)(87 93)(88 92)(89 91)(96 98)(99 108)(100 107)(101 106)(102 105)(103 104)(109 112)(110 111)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 15)(13 16)(14 17)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 43)(40 44)(41 45)(42 46)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 71)(66 72)(67 73)(68 74)(69 75)(70 76)(85 109)(86 110)(87 111)(88 112)(89 99)(90 100)(91 101)(92 102)(93 103)(94 104)(95 105)(96 106)(97 107)(98 108)
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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(29,38)(30,37)(31,36)(32,35)(33,34)(39,42)(40,41)(43,53)(44,52)(45,51)(46,50)(47,49)(54,56)(57,59)(60,70)(61,69)(62,68)(63,67)(64,66)(71,78)(72,77)(73,76)(74,75)(79,84)(80,83)(81,82)(85,87)(88,98)(89,97)(90,96)(91,95)(92,94)(99,100)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107), (1,69)(2,70)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,72)(16,73)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,81)(25,82)(26,83)(27,84)(28,71)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104), (1,34,8,41)(2,35,9,42)(3,36,10,29)(4,37,11,30)(5,38,12,31)(6,39,13,32)(7,40,14,33)(15,56,22,49)(16,43,23,50)(17,44,24,51)(18,45,25,52)(19,46,26,53)(20,47,27,54)(21,48,28,55)(57,92,64,85)(58,93,65,86)(59,94,66,87)(60,95,67,88)(61,96,68,89)(62,97,69,90)(63,98,70,91)(71,103,78,110)(72,104,79,111)(73,105,80,112)(74,106,81,99)(75,107,82,100)(76,108,83,101)(77,109,84,102), (2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,39)(30,38)(31,37)(32,36)(33,35)(40,42)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(55,56)(57,67)(58,66)(59,65)(60,64)(61,63)(68,70)(71,72)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,95)(86,94)(87,93)(88,92)(89,91)(96,98)(99,108)(100,107)(101,106)(102,105)(103,104)(109,112)(110,111), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,15)(13,16)(14,17)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,43)(40,44)(41,45)(42,46)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76)(85,109)(86,110)(87,111)(88,112)(89,99)(90,100)(91,101)(92,102)(93,103)(94,104)(95,105)(96,106)(97,107)(98,108)>;
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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(29,38)(30,37)(31,36)(32,35)(33,34)(39,42)(40,41)(43,53)(44,52)(45,51)(46,50)(47,49)(54,56)(57,59)(60,70)(61,69)(62,68)(63,67)(64,66)(71,78)(72,77)(73,76)(74,75)(79,84)(80,83)(81,82)(85,87)(88,98)(89,97)(90,96)(91,95)(92,94)(99,100)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107), (1,69)(2,70)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,72)(16,73)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,81)(25,82)(26,83)(27,84)(28,71)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104), (1,34,8,41)(2,35,9,42)(3,36,10,29)(4,37,11,30)(5,38,12,31)(6,39,13,32)(7,40,14,33)(15,56,22,49)(16,43,23,50)(17,44,24,51)(18,45,25,52)(19,46,26,53)(20,47,27,54)(21,48,28,55)(57,92,64,85)(58,93,65,86)(59,94,66,87)(60,95,67,88)(61,96,68,89)(62,97,69,90)(63,98,70,91)(71,103,78,110)(72,104,79,111)(73,105,80,112)(74,106,81,99)(75,107,82,100)(76,108,83,101)(77,109,84,102), (2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,39)(30,38)(31,37)(32,36)(33,35)(40,42)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(55,56)(57,67)(58,66)(59,65)(60,64)(61,63)(68,70)(71,72)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,95)(86,94)(87,93)(88,92)(89,91)(96,98)(99,108)(100,107)(101,106)(102,105)(103,104)(109,112)(110,111), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,15)(13,16)(14,17)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,43)(40,44)(41,45)(42,46)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76)(85,109)(86,110)(87,111)(88,112)(89,99)(90,100)(91,101)(92,102)(93,103)(94,104)(95,105)(96,106)(97,107)(98,108) );
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)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22),(29,38),(30,37),(31,36),(32,35),(33,34),(39,42),(40,41),(43,53),(44,52),(45,51),(46,50),(47,49),(54,56),(57,59),(60,70),(61,69),(62,68),(63,67),(64,66),(71,78),(72,77),(73,76),(74,75),(79,84),(80,83),(81,82),(85,87),(88,98),(89,97),(90,96),(91,95),(92,94),(99,100),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107)], [(1,69),(2,70),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,67),(14,68),(15,72),(16,73),(17,74),(18,75),(19,76),(20,77),(21,78),(22,79),(23,80),(24,81),(25,82),(26,83),(27,84),(28,71),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,105),(44,106),(45,107),(46,108),(47,109),(48,110),(49,111),(50,112),(51,99),(52,100),(53,101),(54,102),(55,103),(56,104)], [(1,34,8,41),(2,35,9,42),(3,36,10,29),(4,37,11,30),(5,38,12,31),(6,39,13,32),(7,40,14,33),(15,56,22,49),(16,43,23,50),(17,44,24,51),(18,45,25,52),(19,46,26,53),(20,47,27,54),(21,48,28,55),(57,92,64,85),(58,93,65,86),(59,94,66,87),(60,95,67,88),(61,96,68,89),(62,97,69,90),(63,98,70,91),(71,103,78,110),(72,104,79,111),(73,105,80,112),(74,106,81,99),(75,107,82,100),(76,108,83,101),(77,109,84,102)], [(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22),(29,39),(30,38),(31,37),(32,36),(33,35),(40,42),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(55,56),(57,67),(58,66),(59,65),(60,64),(61,63),(68,70),(71,72),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(85,95),(86,94),(87,93),(88,92),(89,91),(96,98),(99,108),(100,107),(101,106),(102,105),(103,104),(109,112),(110,111)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,15),(13,16),(14,17),(29,47),(30,48),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,43),(40,44),(41,45),(42,46),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,71),(66,72),(67,73),(68,74),(69,75),(70,76),(85,109),(86,110),(87,111),(88,112),(89,99),(90,100),(91,101),(92,102),(93,103),(94,104),(95,105),(96,106),(97,107),(98,108)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 25 | 25 | 0 | 0 | 0 | 0 |
| 4 | 11 | 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 |
| 25 | 25 | 0 | 0 | 0 | 0 |
| 11 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 19 | 0 |
| 0 | 0 | 0 | 1 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 6 | 13 | 13 |
| 0 | 0 | 25 | 26 | 1 | 1 |
| 0 | 0 | 16 | 0 | 22 | 23 |
| 0 | 0 | 0 | 16 | 4 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 9 |
| 0 | 0 | 0 | 12 | 6 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 21 |
| 0 | 0 | 0 | 28 | 14 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 26 | 26 |
| 0 | 0 | 1 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(29))| [25,4,0,0,0,0,25,11,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[25,11,0,0,0,0,25,4,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,19,0,1,0,0,0,0,4,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,25,16,0,0,0,6,26,0,16,0,0,13,1,22,4,0,0,13,1,23,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,12,0,0,0,0,0,6,17,0,0,0,9,0,0,12],[28,11,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,14,1,0,0,0,21,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,26,3,0,1,0,0,26,3,1,0] >;
85 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2J | 2K | ··· | 2P | 4A | ··· | 4F | 4G | 4H | 4I | ··· | 4Q | 7A | 7B | 7C | 14A | 14B | 14C | 14D | ··· | 14AD | 28A | ··· | 28R |
| order | 1 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 7 | 7 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | C2.C25 | D14.C24 |
| kernel | D14.C24 | C2×D4⋊2D7 | D4⋊6D14 | D7×C4○D4 | D4.10D14 | C7×2+ (1+4) | 2+ (1+4) | C2×D4 | C4○D4 | C7 | C1 |
| # reps | 1 | 9 | 9 | 6 | 6 | 1 | 3 | 27 | 18 | 2 | 3 |
In GAP, Magma, Sage, TeX
D_{14}.C_2^4 % in TeX
G:=Group("D14.C2^4"); // GroupNames label
G:=SmallGroup(448,1380);
// by ID
G=gap.SmallGroup(448,1380);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,570,1684,438,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^14=b^2=c^2=e^2=f^2=1,d^2=a^7,b*a*b=e*a*e=a^-1,a*c=c*a,a*d=d*a,a*f=f*a,c*b*c=f*b*f=a^7*b,b*d=d*b,e*b*e=a^12*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=a^7*d,f*e*f=a^7*e>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀