metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.38D14, C14.872+ (1+4), (D4×C14)⋊12C4, D4⋊6(C2×Dic7), (C2×D4)⋊11Dic7, (D4×Dic7)⋊37C2, C23⋊3(C2×Dic7), (C2×D4).251D14, C14.45(C23×C4), C28.94(C22×C4), C4⋊Dic7⋊76C22, (C22×D4).12D7, C2.5(D4⋊6D14), C2.7(C23×Dic7), (C2×C14).293C24, (C2×C28).541C23, C7⋊4(C22.11C24), (C4×Dic7)⋊40C22, (C22×C4).270D14, C23.D7⋊59C22, C4.17(C22×Dic7), C22.45(C23×D7), (D4×C14).270C22, (C23×C14).75C22, C23.204(C22×D7), C23.21D14⋊32C2, C22.1(C22×Dic7), (C22×C14).229C23, (C22×C28).274C22, (C2×Dic7).283C23, (C22×Dic7)⋊31C22, (D4×C2×C14).9C2, (C2×C28)⋊15(C2×C4), (C7×D4)⋊20(C2×C4), (C2×C4)⋊4(C2×Dic7), (C22×C14)⋊12(C2×C4), (C2×C23.D7)⋊26C2, (C2×C14).27(C22×C4), (C2×C4).624(C22×D7), SmallGroup(448,1251)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1108 in 338 conjugacy classes, 191 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×4], C4 [×8], C22, C22 [×10], C22 [×18], C7, C2×C4 [×6], C2×C4 [×16], D4 [×16], C23, C23 [×12], C23 [×4], C14, C14 [×2], C14 [×10], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×8], C2×D4 [×12], C24 [×2], Dic7 [×8], C28 [×4], C2×C14, C2×C14 [×10], C2×C14 [×18], C2×C22⋊C4 [×4], C42⋊C2 [×2], C4×D4 [×8], C22×D4, C2×Dic7 [×8], C2×Dic7 [×8], C2×C28 [×6], C7×D4 [×16], C22×C14, C22×C14 [×12], C22×C14 [×4], C22.11C24, C4×Dic7 [×4], C4⋊Dic7 [×4], C23.D7 [×12], C22×Dic7 [×8], C22×C28, D4×C14 [×12], C23×C14 [×2], C23.21D14 [×2], D4×Dic7 [×8], C2×C23.D7 [×4], D4×C2×C14, C24.38D14
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D7, C22×C4 [×14], C24, Dic7 [×8], D14 [×7], C23×C4, 2+ (1+4) [×2], C2×Dic7 [×28], C22×D7 [×7], C22.11C24, C22×Dic7 [×14], C23×D7, D4⋊6D14 [×2], C23×Dic7, C24.38D14
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=1, f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
►On 112 points
Generators in S112
(2 63)(4 65)(6 67)(8 69)(10 57)(12 59)(14 61)(15 103)(17 105)(19 107)(21 109)(23 111)(25 99)(27 101)(30 96)(32 98)(34 86)(36 88)(38 90)(40 92)(42 94)(44 78)(46 80)(48 82)(50 84)(52 72)(54 74)(56 76)
(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 99)(26 100)(27 101)(28 102)(43 77)(44 78)(45 79)(46 80)(47 81)(48 82)(49 83)(50 84)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)
(1 95)(2 96)(3 97)(4 98)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 82)(16 83)(17 84)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 57)(39 58)(40 59)(41 60)(42 61)(43 112)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 57)(11 58)(12 59)(13 60)(14 61)(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 99)(26 100)(27 101)(28 102)(29 95)(30 96)(31 97)(32 98)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 77)(44 78)(45 79)(46 80)(47 81)(48 82)(49 83)(50 84)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)
(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 54 95 109)(2 53 96 108)(3 52 97 107)(4 51 98 106)(5 50 85 105)(6 49 86 104)(7 48 87 103)(8 47 88 102)(9 46 89 101)(10 45 90 100)(11 44 91 99)(12 43 92 112)(13 56 93 111)(14 55 94 110)(15 68 82 35)(16 67 83 34)(17 66 84 33)(18 65 71 32)(19 64 72 31)(20 63 73 30)(21 62 74 29)(22 61 75 42)(23 60 76 41)(24 59 77 40)(25 58 78 39)(26 57 79 38)(27 70 80 37)(28 69 81 36)
G:=sub<Sym(112)| (2,63)(4,65)(6,67)(8,69)(10,57)(12,59)(14,61)(15,103)(17,105)(19,107)(21,109)(23,111)(25,99)(27,101)(30,96)(32,98)(34,86)(36,88)(38,90)(40,92)(42,94)(44,78)(46,80)(48,82)(50,84)(52,72)(54,74)(56,76), (15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,99)(26,100)(27,101)(28,102)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76), (1,95)(2,96)(3,97)(4,98)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,82)(16,83)(17,84)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,57)(39,58)(40,59)(41,60)(42,61)(43,112)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,57)(11,58)(12,59)(13,60)(14,61)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,99)(26,100)(27,101)(28,102)(29,95)(30,96)(31,97)(32,98)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76), (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), (1,54,95,109)(2,53,96,108)(3,52,97,107)(4,51,98,106)(5,50,85,105)(6,49,86,104)(7,48,87,103)(8,47,88,102)(9,46,89,101)(10,45,90,100)(11,44,91,99)(12,43,92,112)(13,56,93,111)(14,55,94,110)(15,68,82,35)(16,67,83,34)(17,66,84,33)(18,65,71,32)(19,64,72,31)(20,63,73,30)(21,62,74,29)(22,61,75,42)(23,60,76,41)(24,59,77,40)(25,58,78,39)(26,57,79,38)(27,70,80,37)(28,69,81,36)>;
G:=Group( (2,63)(4,65)(6,67)(8,69)(10,57)(12,59)(14,61)(15,103)(17,105)(19,107)(21,109)(23,111)(25,99)(27,101)(30,96)(32,98)(34,86)(36,88)(38,90)(40,92)(42,94)(44,78)(46,80)(48,82)(50,84)(52,72)(54,74)(56,76), (15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,99)(26,100)(27,101)(28,102)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76), (1,95)(2,96)(3,97)(4,98)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,82)(16,83)(17,84)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,57)(39,58)(40,59)(41,60)(42,61)(43,112)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,57)(11,58)(12,59)(13,60)(14,61)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,99)(26,100)(27,101)(28,102)(29,95)(30,96)(31,97)(32,98)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76), (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), (1,54,95,109)(2,53,96,108)(3,52,97,107)(4,51,98,106)(5,50,85,105)(6,49,86,104)(7,48,87,103)(8,47,88,102)(9,46,89,101)(10,45,90,100)(11,44,91,99)(12,43,92,112)(13,56,93,111)(14,55,94,110)(15,68,82,35)(16,67,83,34)(17,66,84,33)(18,65,71,32)(19,64,72,31)(20,63,73,30)(21,62,74,29)(22,61,75,42)(23,60,76,41)(24,59,77,40)(25,58,78,39)(26,57,79,38)(27,70,80,37)(28,69,81,36) );
G=PermutationGroup([(2,63),(4,65),(6,67),(8,69),(10,57),(12,59),(14,61),(15,103),(17,105),(19,107),(21,109),(23,111),(25,99),(27,101),(30,96),(32,98),(34,86),(36,88),(38,90),(40,92),(42,94),(44,78),(46,80),(48,82),(50,84),(52,72),(54,74),(56,76)], [(15,103),(16,104),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,99),(26,100),(27,101),(28,102),(43,77),(44,78),(45,79),(46,80),(47,81),(48,82),(49,83),(50,84),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76)], [(1,95),(2,96),(3,97),(4,98),(5,85),(6,86),(7,87),(8,88),(9,89),(10,90),(11,91),(12,92),(13,93),(14,94),(15,82),(16,83),(17,84),(18,71),(19,72),(20,73),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,81),(29,62),(30,63),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69),(37,70),(38,57),(39,58),(40,59),(41,60),(42,61),(43,112),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111)], [(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,57),(11,58),(12,59),(13,60),(14,61),(15,103),(16,104),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,99),(26,100),(27,101),(28,102),(29,95),(30,96),(31,97),(32,98),(33,85),(34,86),(35,87),(36,88),(37,89),(38,90),(39,91),(40,92),(41,93),(42,94),(43,77),(44,78),(45,79),(46,80),(47,81),(48,82),(49,83),(50,84),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76)], [(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)], [(1,54,95,109),(2,53,96,108),(3,52,97,107),(4,51,98,106),(5,50,85,105),(6,49,86,104),(7,48,87,103),(8,47,88,102),(9,46,89,101),(10,45,90,100),(11,44,91,99),(12,43,92,112),(13,56,93,111),(14,55,94,110),(15,68,82,35),(16,67,83,34),(17,66,84,33),(18,65,71,32),(19,64,72,31),(20,63,73,30),(21,62,74,29),(22,61,75,42),(23,60,76,41),(24,59,77,40),(25,58,78,39),(26,57,79,38),(27,70,80,37),(28,69,81,36)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 28 | 0 |
| 0 | 0 | 12 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 27 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 27 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 1 |
| 0 | 0 | 0 | 17 | 1 | 0 |
| 24 | 12 | 0 | 0 | 0 | 0 |
| 22 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 1 | 28 |
| 0 | 0 | 28 | 1 | 0 | 17 |
| 0 | 0 | 28 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,1,0,12,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,12,12,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[6,27,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,27,28,17,17,0,0,0,0,0,1,0,0,0,0,1,0],[24,22,0,0,0,0,12,5,0,0,0,0,0,0,12,0,28,28,0,0,0,0,1,0,0,0,0,1,0,0,0,0,27,28,17,17] >;
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
94 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D7 | D14 | Dic7 | D14 | D14 | 2+ (1+4) | D4⋊6D14 |
| kernel | C24.38D14 | C23.21D14 | D4×Dic7 | C2×C23.D7 | D4×C2×C14 | D4×C14 | C22×D4 | C22×C4 | C2×D4 | C2×D4 | C24 | C14 | C2 |
| # reps | 1 | 2 | 8 | 4 | 1 | 16 | 3 | 3 | 24 | 12 | 6 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_2^4._{38}D_{14} % in TeX
G:=Group("C2^4.38D14"); // GroupNames label
G:=SmallGroup(448,1251);
// by ID
G=gap.SmallGroup(448,1251);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,387,1123,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=1,f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,f*b*f^-1=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^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀