metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.42D14, C14.912+ (1+4), (C7×D4)⋊17D4, D4⋊8(C7⋊D4), C28⋊2D4⋊41C2, C7⋊10(D4⋊5D4), (D4×Dic7)⋊39C2, C28.253(C2×D4), (C22×D4)⋊12D7, D14⋊C4⋊73C22, (C2×D4).231D14, C24⋊D7⋊14C2, C4⋊Dic7⋊45C22, Dic7⋊D4⋊42C2, C28.17D4⋊29C2, C28.48D4⋊37C2, C22⋊5(D4⋊2D7), (C2×C14).301C24, (C2×C28).546C23, Dic7⋊C4⋊39C22, (C4×Dic7)⋊43C22, (C22×C4).273D14, C14.148(C22×D4), C23.D7⋊40C22, C2.94(D4⋊6D14), (C2×Dic14)⋊42C22, (D4×C14).312C22, (C23×C14).80C22, C22.314(C23×D7), C23.236(C22×D7), C23.18D14⋊30C2, (C22×C28).278C22, (C22×C14).235C23, (C2×Dic7).286C23, (C22×Dic7)⋊35C22, (C22×D7).132C23, (D4×C2×C14)⋊8C2, (C4×C7⋊D4)⋊26C2, (C2×C4×D7)⋊32C22, C4.68(C2×C7⋊D4), (C2×C14).74(C2×D4), (C2×D4⋊2D7)⋊27C2, (C2×C14)⋊15(C4○D4), C22.3(C2×C7⋊D4), C14.107(C2×C4○D4), C2.71(C2×D4⋊2D7), (C2×C7⋊D4)⋊29C22, (C2×C23.D7)⋊31C2, C2.21(C22×C7⋊D4), (C2×C4).239(C22×D7), SmallGroup(448,1259)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1332 in 334 conjugacy classes, 115 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×23], C7, C2×C4 [×2], C2×C4 [×17], D4 [×4], D4 [×14], Q8 [×2], C23, C23 [×4], C23 [×11], D7, C14 [×3], C14 [×8], C42, C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×D4 [×9], C2×Q8, C4○D4 [×4], C24 [×2], Dic7 [×7], C28 [×2], C28, D14 [×3], C2×C14, C2×C14 [×6], C2×C14 [×20], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic14 [×2], C4×D7 [×2], C2×Dic7 [×3], C2×Dic7 [×4], C2×Dic7 [×6], C7⋊D4 [×8], C2×C28 [×2], C2×C28 [×2], C7×D4 [×4], C7×D4 [×6], C22×D7, C22×C14, C22×C14 [×4], C22×C14 [×10], D4⋊5D4, C4×Dic7, Dic7⋊C4, Dic7⋊C4 [×2], C4⋊Dic7, D14⋊C4, C23.D7, C23.D7 [×10], C2×Dic14, C2×C4×D7, D4⋊2D7 [×4], C22×Dic7 [×4], C2×C7⋊D4, C2×C7⋊D4 [×4], C22×C28, D4×C14 [×2], D4×C14 [×2], D4×C14 [×4], C23×C14 [×2], C28.48D4, C4×C7⋊D4, D4×Dic7, C23.18D14 [×2], C28.17D4, C28⋊2D4, Dic7⋊D4 [×2], C2×C23.D7 [×2], C24⋊D7 [×2], C2×D4⋊2D7, D4×C2×C14, C24.42D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×2], C24, D14 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C7⋊D4 [×4], C22×D7 [×7], D4⋊5D4, D4⋊2D7 [×2], C2×C7⋊D4 [×6], C23×D7, C2×D4⋊2D7, D4⋊6D14, C22×C7⋊D4, C24.42D14
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=f2=d, ab=ba, ac=ca, eae-1=ad=da, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >
►On 112 points
(1 59)(2 74)(3 61)(4 76)(5 63)(6 78)(7 65)(8 80)(9 67)(10 82)(11 69)(12 84)(13 71)(14 58)(15 73)(16 60)(17 75)(18 62)(19 77)(20 64)(21 79)(22 66)(23 81)(24 68)(25 83)(26 70)(27 57)(28 72)(29 111)(30 98)(31 85)(32 100)(33 87)(34 102)(35 89)(36 104)(37 91)(38 106)(39 93)(40 108)(41 95)(42 110)(43 97)(44 112)(45 99)(46 86)(47 101)(48 88)(49 103)(50 90)(51 105)(52 92)(53 107)(54 94)(55 109)(56 96)
(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)(53 100)(54 101)(55 102)(56 103)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 81)(17 82)(18 83)(19 84)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)(53 100)(54 101)(55 102)(56 103)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 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)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(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 50 15 36)(2 35 16 49)(3 48 17 34)(4 33 18 47)(5 46 19 32)(6 31 20 45)(7 44 21 30)(8 29 22 43)(9 42 23 56)(10 55 24 41)(11 40 25 54)(12 53 26 39)(13 38 27 52)(14 51 28 37)(57 92 71 106)(58 105 72 91)(59 90 73 104)(60 103 74 89)(61 88 75 102)(62 101 76 87)(63 86 77 100)(64 99 78 85)(65 112 79 98)(66 97 80 111)(67 110 81 96)(68 95 82 109)(69 108 83 94)(70 93 84 107)
G:=sub<Sym(112)| (1,59)(2,74)(3,61)(4,76)(5,63)(6,78)(7,65)(8,80)(9,67)(10,82)(11,69)(12,84)(13,71)(14,58)(15,73)(16,60)(17,75)(18,62)(19,77)(20,64)(21,79)(22,66)(23,81)(24,68)(25,83)(26,70)(27,57)(28,72)(29,111)(30,98)(31,85)(32,100)(33,87)(34,102)(35,89)(36,104)(37,91)(38,106)(39,93)(40,108)(41,95)(42,110)(43,97)(44,112)(45,99)(46,86)(47,101)(48,88)(49,103)(50,90)(51,105)(52,92)(53,107)(54,94)(55,109)(56,96), (29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(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)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,50,15,36)(2,35,16,49)(3,48,17,34)(4,33,18,47)(5,46,19,32)(6,31,20,45)(7,44,21,30)(8,29,22,43)(9,42,23,56)(10,55,24,41)(11,40,25,54)(12,53,26,39)(13,38,27,52)(14,51,28,37)(57,92,71,106)(58,105,72,91)(59,90,73,104)(60,103,74,89)(61,88,75,102)(62,101,76,87)(63,86,77,100)(64,99,78,85)(65,112,79,98)(66,97,80,111)(67,110,81,96)(68,95,82,109)(69,108,83,94)(70,93,84,107)>;
G:=Group( (1,59)(2,74)(3,61)(4,76)(5,63)(6,78)(7,65)(8,80)(9,67)(10,82)(11,69)(12,84)(13,71)(14,58)(15,73)(16,60)(17,75)(18,62)(19,77)(20,64)(21,79)(22,66)(23,81)(24,68)(25,83)(26,70)(27,57)(28,72)(29,111)(30,98)(31,85)(32,100)(33,87)(34,102)(35,89)(36,104)(37,91)(38,106)(39,93)(40,108)(41,95)(42,110)(43,97)(44,112)(45,99)(46,86)(47,101)(48,88)(49,103)(50,90)(51,105)(52,92)(53,107)(54,94)(55,109)(56,96), (29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(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)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,50,15,36)(2,35,16,49)(3,48,17,34)(4,33,18,47)(5,46,19,32)(6,31,20,45)(7,44,21,30)(8,29,22,43)(9,42,23,56)(10,55,24,41)(11,40,25,54)(12,53,26,39)(13,38,27,52)(14,51,28,37)(57,92,71,106)(58,105,72,91)(59,90,73,104)(60,103,74,89)(61,88,75,102)(62,101,76,87)(63,86,77,100)(64,99,78,85)(65,112,79,98)(66,97,80,111)(67,110,81,96)(68,95,82,109)(69,108,83,94)(70,93,84,107) );
G=PermutationGroup([(1,59),(2,74),(3,61),(4,76),(5,63),(6,78),(7,65),(8,80),(9,67),(10,82),(11,69),(12,84),(13,71),(14,58),(15,73),(16,60),(17,75),(18,62),(19,77),(20,64),(21,79),(22,66),(23,81),(24,68),(25,83),(26,70),(27,57),(28,72),(29,111),(30,98),(31,85),(32,100),(33,87),(34,102),(35,89),(36,104),(37,91),(38,106),(39,93),(40,108),(41,95),(42,110),(43,97),(44,112),(45,99),(46,86),(47,101),(48,88),(49,103),(50,90),(51,105),(52,92),(53,107),(54,94),(55,109),(56,96)], [(29,104),(30,105),(31,106),(32,107),(33,108),(34,109),(35,110),(36,111),(37,112),(38,85),(39,86),(40,87),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,96),(50,97),(51,98),(52,99),(53,100),(54,101),(55,102),(56,103)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,73),(9,74),(10,75),(11,76),(12,77),(13,78),(14,79),(15,80),(16,81),(17,82),(18,83),(19,84),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,65),(29,104),(30,105),(31,106),(32,107),(33,108),(34,109),(35,110),(36,111),(37,112),(38,85),(39,86),(40,87),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,96),(50,97),(51,98),(52,99),(53,100),(54,101),(55,102),(56,103)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(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),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)], [(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)], [(1,50,15,36),(2,35,16,49),(3,48,17,34),(4,33,18,47),(5,46,19,32),(6,31,20,45),(7,44,21,30),(8,29,22,43),(9,42,23,56),(10,55,24,41),(11,40,25,54),(12,53,26,39),(13,38,27,52),(14,51,28,37),(57,92,71,106),(58,105,72,91),(59,90,73,104),(60,103,74,89),(61,88,75,102),(62,101,76,87),(63,86,77,100),(64,99,78,85),(65,112,79,98),(66,97,80,111),(67,110,81,96),(68,95,82,109),(69,108,83,94),(70,93,84,107)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 22 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 0 | 1 | 0 |
| 0 | 4 | 0 | 0 |
| 22 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[22,0,0,0,0,4,0,0,0,0,0,1,0,0,28,0],[0,22,0,0,4,0,0,0,0,0,12,0,0,0,0,12] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 28 | 2 | 2 | 4 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | 2+ (1+4) | D4⋊2D7 | D4⋊6D14 |
| kernel | C24.42D14 | C28.48D4 | C4×C7⋊D4 | D4×Dic7 | C23.18D14 | C28.17D4 | C28⋊2D4 | Dic7⋊D4 | C2×C23.D7 | C24⋊D7 | C2×D4⋊2D7 | D4×C2×C14 | C7×D4 | C22×D4 | C2×C14 | C22×C4 | C2×D4 | C24 | D4 | C14 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 4 | 3 | 4 | 3 | 12 | 6 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_2^4._{42}D_{14} % in TeX
G:=Group("C2^4.42D14"); // GroupNames label
G:=SmallGroup(448,1259);
// by ID
G=gap.SmallGroup(448,1259);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,387,675,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,a*f=f*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^13>;
// generators/relations