metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.14C25, C28.49C24, D14.8C24, D28.41C23, Dic7.9C24, Dic14.41C23, C4○D4⋊19D14, (C2×D4)⋊47D14, (C2×Q8)⋊36D14, (D4×D7)⋊13C22, (C22×C4)⋊34D14, (C2×C14).5C24, D4⋊6D14⋊11C2, D4⋊8D14⋊13C2, (Q8×D7)⋊15C22, C2.15(D7×C24), C4.46(C23×D7), C7⋊D4.2C23, C4○D28⋊27C22, (D4×C14)⋊54C22, (C2×D28)⋊65C22, C7⋊1(C2.C25), (Q8×C14)⋊47C22, (C7×D4).30C23, D4.30(C22×D7), (C4×D7).38C23, (C7×Q8).31C23, Q8.31(C22×D7), D4⋊2D7⋊14C22, (C2×C28).568C23, Q8.10D14⋊9C2, (C22×C28)⋊29C22, Q8⋊2D7⋊14C22, D4.10D14⋊13C2, C22.10(C23×D7), (C2×Dic14)⋊76C22, C23.142(C22×D7), (C22×C14).250C23, (C2×Dic7).168C23, (C22×D7).142C23, (D7×C4○D4)⋊6C2, (C2×C4○D4)⋊15D7, (C2×C4×D7)⋊35C22, (C14×C4○D4)⋊16C2, (C2×C4○D28)⋊39C2, (C7×C4○D4)⋊22C22, (C2×C7⋊D4)⋊55C22, (C2×C4).646(C22×D7), SmallGroup(448,1378)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 3156 in 810 conjugacy classes, 443 normal (17 characteristic)
C1, C2, C2 [×15], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×6], C22 [×23], C7, C2×C4, C2×C4 [×15], C2×C4 [×44], D4 [×12], D4 [×48], Q8 [×4], Q8 [×16], C23 [×3], C23 [×12], D7 [×8], C14, C14 [×7], C22×C4 [×3], C22×C4 [×12], C2×D4 [×3], C2×D4 [×42], C2×Q8, C2×Q8 [×14], C4○D4 [×8], C4○D4 [×72], Dic7 [×8], C28 [×2], C28 [×6], D14 [×8], D14 [×12], C2×C14, C2×C14 [×6], C2×C14 [×3], C2×C4○D4, C2×C4○D4 [×14], 2+ (1+4) [×10], 2- (1+4) [×6], Dic14 [×16], C4×D7 [×32], D28 [×16], C2×Dic7 [×12], C7⋊D4 [×32], C2×C28, C2×C28 [×15], C7×D4 [×12], C7×Q8 [×4], C22×D7 [×12], C22×C14 [×3], C2.C25, C2×Dic14 [×6], C2×C4×D7 [×12], C2×D28 [×6], C4○D28 [×40], D4×D7 [×24], D4⋊2D7 [×24], Q8×D7 [×8], Q8⋊2D7 [×8], C2×C7⋊D4 [×12], C22×C28 [×3], D4×C14 [×3], Q8×C14, C7×C4○D4 [×8], C2×C4○D28 [×6], D4⋊6D14 [×6], Q8.10D14 [×2], D7×C4○D4 [×8], D4⋊8D14 [×4], D4.10D14 [×4], C14×C4○D4, C14.C25
Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], D7, C24 [×31], D14 [×15], C25, C22×D7 [×35], C2.C25, C23×D7 [×15], D7×C24, C14.C25
Generators and relations
G = < a,b,c,d,e,f | a14=b2=c2=e2=f2=1, d2=a7, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a7b, cd=dc, ece=a7c, cf=fc, de=ed, df=fd, ef=fe >
►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)
(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(15 16)(17 28)(18 27)(19 26)(20 25)(21 24)(22 23)(29 33)(30 32)(34 42)(35 41)(36 40)(37 39)(43 52)(44 51)(45 50)(46 49)(47 48)(53 56)(54 55)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(71 76)(72 75)(73 74)(77 84)(78 83)(79 82)(80 81)(85 91)(86 90)(87 89)(92 98)(93 97)(94 96)(99 108)(100 107)(101 106)(102 105)(103 104)(109 112)(110 111)
(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 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 81)(24 82)(25 83)(26 84)(27 71)(28 72)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)
(1 31 8 38)(2 32 9 39)(3 33 10 40)(4 34 11 41)(5 35 12 42)(6 36 13 29)(7 37 14 30)(15 47 22 54)(16 48 23 55)(17 49 24 56)(18 50 25 43)(19 51 26 44)(20 52 27 45)(21 53 28 46)(57 95 64 88)(58 96 65 89)(59 97 66 90)(60 98 67 91)(61 85 68 92)(62 86 69 93)(63 87 70 94)(71 101 78 108)(72 102 79 109)(73 103 80 110)(74 104 81 111)(75 105 82 112)(76 106 83 99)(77 107 84 100)
(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 15)(12 16)(13 17)(14 18)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(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)(85 111)(86 112)(87 99)(88 100)(89 101)(90 102)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)
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), (2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(15,16)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(29,33)(30,32)(34,42)(35,41)(36,40)(37,39)(43,52)(44,51)(45,50)(46,49)(47,48)(53,56)(54,55)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(71,76)(72,75)(73,74)(77,84)(78,83)(79,82)(80,81)(85,91)(86,90)(87,89)(92,98)(93,97)(94,96)(99,108)(100,107)(101,106)(102,105)(103,104)(109,112)(110,111), (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)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,81)(24,82)(25,83)(26,84)(27,71)(28,72)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112), (1,31,8,38)(2,32,9,39)(3,33,10,40)(4,34,11,41)(5,35,12,42)(6,36,13,29)(7,37,14,30)(15,47,22,54)(16,48,23,55)(17,49,24,56)(18,50,25,43)(19,51,26,44)(20,52,27,45)(21,53,28,46)(57,95,64,88)(58,96,65,89)(59,97,66,90)(60,98,67,91)(61,85,68,92)(62,86,69,93)(63,87,70,94)(71,101,78,108)(72,102,79,109)(73,103,80,110)(74,104,81,111)(75,105,82,112)(76,106,83,99)(77,107,84,100), (57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,15)(12,16)(13,17)(14,18)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(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)(85,111)(86,112)(87,99)(88,100)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)>;
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), (2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(15,16)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(29,33)(30,32)(34,42)(35,41)(36,40)(37,39)(43,52)(44,51)(45,50)(46,49)(47,48)(53,56)(54,55)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(71,76)(72,75)(73,74)(77,84)(78,83)(79,82)(80,81)(85,91)(86,90)(87,89)(92,98)(93,97)(94,96)(99,108)(100,107)(101,106)(102,105)(103,104)(109,112)(110,111), (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)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,81)(24,82)(25,83)(26,84)(27,71)(28,72)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112), (1,31,8,38)(2,32,9,39)(3,33,10,40)(4,34,11,41)(5,35,12,42)(6,36,13,29)(7,37,14,30)(15,47,22,54)(16,48,23,55)(17,49,24,56)(18,50,25,43)(19,51,26,44)(20,52,27,45)(21,53,28,46)(57,95,64,88)(58,96,65,89)(59,97,66,90)(60,98,67,91)(61,85,68,92)(62,86,69,93)(63,87,70,94)(71,101,78,108)(72,102,79,109)(73,103,80,110)(74,104,81,111)(75,105,82,112)(76,106,83,99)(77,107,84,100), (57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,15)(12,16)(13,17)(14,18)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(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)(85,111)(86,112)(87,99)(88,100)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110) );
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)], [(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(15,16),(17,28),(18,27),(19,26),(20,25),(21,24),(22,23),(29,33),(30,32),(34,42),(35,41),(36,40),(37,39),(43,52),(44,51),(45,50),(46,49),(47,48),(53,56),(54,55),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(71,76),(72,75),(73,74),(77,84),(78,83),(79,82),(80,81),(85,91),(86,90),(87,89),(92,98),(93,97),(94,96),(99,108),(100,107),(101,106),(102,105),(103,104),(109,112),(110,111)], [(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),(15,73),(16,74),(17,75),(18,76),(19,77),(20,78),(21,79),(22,80),(23,81),(24,82),(25,83),(26,84),(27,71),(28,72),(29,93),(30,94),(31,95),(32,96),(33,97),(34,98),(35,85),(36,86),(37,87),(38,88),(39,89),(40,90),(41,91),(42,92),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112)], [(1,31,8,38),(2,32,9,39),(3,33,10,40),(4,34,11,41),(5,35,12,42),(6,36,13,29),(7,37,14,30),(15,47,22,54),(16,48,23,55),(17,49,24,56),(18,50,25,43),(19,51,26,44),(20,52,27,45),(21,53,28,46),(57,95,64,88),(58,96,65,89),(59,97,66,90),(60,98,67,91),(61,85,68,92),(62,86,69,93),(63,87,70,94),(71,101,78,108),(72,102,79,109),(73,103,80,110),(74,104,81,111),(75,105,82,112),(76,106,83,99),(77,107,84,100)], [(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,15),(12,16),(13,17),(14,18),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(36,56),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(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),(85,111),(86,112),(87,99),(88,100),(89,101),(90,102),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108),(97,109),(98,110)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 4 | 4 | 0 | 0 |
| 25 | 18 | 0 | 0 |
| 0 | 0 | 4 | 4 |
| 0 | 0 | 25 | 18 |
| 1 | 0 | 0 | 0 |
| 18 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 18 | 28 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 18 | 27 | 0 | 0 |
| 2 | 11 | 0 | 0 |
| 0 | 0 | 18 | 27 |
| 0 | 0 | 2 | 11 |
G:=sub<GL(4,GF(29))| [4,25,0,0,4,18,0,0,0,0,4,25,0,0,4,18],[1,18,0,0,0,28,0,0,0,0,1,18,0,0,0,28],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[18,2,0,0,27,11,0,0,0,0,18,2,0,0,27,11] >;
94 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2H | 2I | ··· | 2P | 4A | 4B | 4C | ··· | 4I | 4J | ··· | 4Q | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | ··· | 2 | 14 | ··· | 14 | 1 | 1 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
94 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | D14 | C2.C25 | C14.C25 |
| kernel | C14.C25 | C2×C4○D28 | D4⋊6D14 | Q8.10D14 | D7×C4○D4 | D4⋊8D14 | D4.10D14 | C14×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C7 | C1 |
| # reps | 1 | 6 | 6 | 2 | 8 | 4 | 4 | 1 | 3 | 9 | 9 | 3 | 24 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_{14}.C_2^5 % in TeX
G:=Group("C14.C2^5"); // GroupNames label
G:=SmallGroup(448,1378);
// by ID
G=gap.SmallGroup(448,1378);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,570,1684,18822]);
// 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=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a^7*b,c*d=d*c,e*c*e=a^7*c,c*f=f*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀