metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.31D14, C14.62+ 1+4, C24⋊D7.C2, C4⋊Dic7⋊7C22, D14⋊C4⋊49C22, D14.D4⋊3C2, (C2×C14).39C24, C22⋊C4.88D14, Dic7⋊4D4⋊41C2, C28.48D4⋊17C2, (C2×C28).575C23, Dic7⋊C4⋊50C22, C22⋊Dic14⋊3C2, Dic7.D4⋊3C2, (C2×Dic14)⋊4C22, (C4×Dic7)⋊49C22, C22.D28⋊3C2, C23.D14⋊2C2, (C22×C4).172D14, C2.10(D4⋊6D14), C22.78(C23×D7), C7⋊1(C22.45C24), C22.17(C4○D28), C23.21D14⋊3C2, (C23×C14).65C22, (C2×Dic7).12C23, (C22×D7).11C23, C23.222(C22×D7), C23.23D14⋊10C2, C23.11D14⋊25C2, C22.23(D4⋊2D7), (C22×C14).129C23, (C22×C28).101C22, C23.D7.140C22, (C22×Dic7).80C22, (C4×C7⋊D4)⋊2C2, (C2×C4×D7)⋊42C22, (C2×C22⋊C4)⋊18D7, C2.19(C2×C4○D28), C14.17(C2×C4○D4), (C14×C22⋊C4)⋊21C2, C2.12(C2×D4⋊2D7), (C2×C23.D7)⋊18C2, (C2×C7⋊D4).8C22, (C2×C14).40(C4○D4), (C2×C4).262(C22×D7), (C7×C22⋊C4).110C22, SmallGroup(448,948)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.31D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=f2=d, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, bc=cb, bd=db, be=eb, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >
Subgroups: 1012 in 248 conjugacy classes, 99 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C22×C14, C22.45C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C2×Dic14, C2×C4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, C23×C14, C23.11D14, C22⋊Dic14, C23.D14, Dic7⋊4D4, D14.D4, Dic7.D4, C22.D28, C28.48D4, C23.21D14, C4×C7⋊D4, C23.23D14, C2×C23.D7, C24⋊D7, C14×C22⋊C4, C24.31D14
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.45C24, C4○D28, D4⋊2D7, C23×D7, C2×C4○D28, C2×D4⋊2D7, D4⋊6D14, C24.31D14
►On 112 points
Generators in S112
(2 80)(4 82)(6 84)(8 58)(10 60)(12 62)(14 64)(16 66)(18 68)(20 70)(22 72)(24 74)(26 76)(28 78)(29 98)(30 44)(31 100)(32 46)(33 102)(34 48)(35 104)(36 50)(37 106)(38 52)(39 108)(40 54)(41 110)(42 56)(43 112)(45 86)(47 88)(49 90)(51 92)(53 94)(55 96)(85 99)(87 101)(89 103)(91 105)(93 107)(95 109)(97 111)
(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)(29 112)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(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)(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)
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 112)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(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 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)
(1 104 15 90)(2 89 16 103)(3 102 17 88)(4 87 18 101)(5 100 19 86)(6 85 20 99)(7 98 21 112)(8 111 22 97)(9 96 23 110)(10 109 24 95)(11 94 25 108)(12 107 26 93)(13 92 27 106)(14 105 28 91)(29 57 43 71)(30 70 44 84)(31 83 45 69)(32 68 46 82)(33 81 47 67)(34 66 48 80)(35 79 49 65)(36 64 50 78)(37 77 51 63)(38 62 52 76)(39 75 53 61)(40 60 54 74)(41 73 55 59)(42 58 56 72)
G:=sub<Sym(112)| (2,80)(4,82)(6,84)(8,58)(10,60)(12,62)(14,64)(16,66)(18,68)(20,70)(22,72)(24,74)(26,76)(28,78)(29,98)(30,44)(31,100)(32,46)(33,102)(34,48)(35,104)(36,50)(37,106)(38,52)(39,108)(40,54)(41,110)(42,56)(43,112)(45,86)(47,88)(49,90)(51,92)(53,94)(55,96)(85,99)(87,101)(89,103)(91,105)(93,107)(95,109)(97,111), (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)(29,112)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(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)(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), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,112)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(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), (1,15)(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), (1,104,15,90)(2,89,16,103)(3,102,17,88)(4,87,18,101)(5,100,19,86)(6,85,20,99)(7,98,21,112)(8,111,22,97)(9,96,23,110)(10,109,24,95)(11,94,25,108)(12,107,26,93)(13,92,27,106)(14,105,28,91)(29,57,43,71)(30,70,44,84)(31,83,45,69)(32,68,46,82)(33,81,47,67)(34,66,48,80)(35,79,49,65)(36,64,50,78)(37,77,51,63)(38,62,52,76)(39,75,53,61)(40,60,54,74)(41,73,55,59)(42,58,56,72)>;
G:=Group( (2,80)(4,82)(6,84)(8,58)(10,60)(12,62)(14,64)(16,66)(18,68)(20,70)(22,72)(24,74)(26,76)(28,78)(29,98)(30,44)(31,100)(32,46)(33,102)(34,48)(35,104)(36,50)(37,106)(38,52)(39,108)(40,54)(41,110)(42,56)(43,112)(45,86)(47,88)(49,90)(51,92)(53,94)(55,96)(85,99)(87,101)(89,103)(91,105)(93,107)(95,109)(97,111), (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)(29,112)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(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)(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), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,112)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(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), (1,15)(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), (1,104,15,90)(2,89,16,103)(3,102,17,88)(4,87,18,101)(5,100,19,86)(6,85,20,99)(7,98,21,112)(8,111,22,97)(9,96,23,110)(10,109,24,95)(11,94,25,108)(12,107,26,93)(13,92,27,106)(14,105,28,91)(29,57,43,71)(30,70,44,84)(31,83,45,69)(32,68,46,82)(33,81,47,67)(34,66,48,80)(35,79,49,65)(36,64,50,78)(37,77,51,63)(38,62,52,76)(39,75,53,61)(40,60,54,74)(41,73,55,59)(42,58,56,72) );
G=PermutationGroup([[(2,80),(4,82),(6,84),(8,58),(10,60),(12,62),(14,64),(16,66),(18,68),(20,70),(22,72),(24,74),(26,76),(28,78),(29,98),(30,44),(31,100),(32,46),(33,102),(34,48),(35,104),(36,50),(37,106),(38,52),(39,108),(40,54),(41,110),(42,56),(43,112),(45,86),(47,88),(49,90),(51,92),(53,94),(55,96),(85,99),(87,101),(89,103),(91,105),(93,107),(95,109),(97,111)], [(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),(29,112),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,98),(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),(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)], [(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,57),(8,58),(9,59),(10,60),(11,61),(12,62),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,112),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,98),(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)], [(1,15),(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)], [(1,104,15,90),(2,89,16,103),(3,102,17,88),(4,87,18,101),(5,100,19,86),(6,85,20,99),(7,98,21,112),(8,111,22,97),(9,96,23,110),(10,109,24,95),(11,94,25,108),(12,107,26,93),(13,92,27,106),(14,105,28,91),(29,57,43,71),(30,70,44,84),(31,83,45,69),(32,68,46,82),(33,81,47,67),(34,66,48,80),(35,79,49,65),(36,64,50,78),(37,77,51,63),(38,62,52,76),(39,75,53,61),(40,60,54,74),(41,73,55,59),(42,58,56,72)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 2 | 2 | 4 | 28 | 2 | 2 | 2 | 2 | 4 | 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 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | C4○D28 | 2+ 1+4 | D4⋊2D7 | D4⋊6D14 |
| kernel | C24.31D14 | C23.11D14 | C22⋊Dic14 | C23.D14 | Dic7⋊4D4 | D14.D4 | Dic7.D4 | C22.D28 | C28.48D4 | C23.21D14 | C4×C7⋊D4 | C23.23D14 | C2×C23.D7 | C24⋊D7 | C14×C22⋊C4 | C2×C22⋊C4 | C2×C14 | C22⋊C4 | C22×C4 | C24 | C22 | C14 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 8 | 12 | 6 | 3 | 24 | 1 | 6 | 6 |
Matrix representation of C24.31D14 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 21 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 1 | 27 |
| 0 | 0 | 1 | 28 |
| 0 | 11 | 0 | 0 |
| 21 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,28,0,0,0,0,1,1,0,0,0,28],[28,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[21,0,0,0,0,11,0,0,0,0,1,1,0,0,27,28],[0,21,0,0,11,0,0,0,0,0,12,0,0,0,0,12] >;
C24.31D14 in GAP, Magma, Sage, TeX
C_2^4._{31}D_{14} % in TeX
G:=Group("C2^4.31D14"); // GroupNames label
G:=SmallGroup(448,948);
// by ID
G=gap.SmallGroup(448,948);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,1571,18822]);
// 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,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d,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