metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.32D14, C14.242+ (1+4), (D4×Dic7)⋊11C2, C22≀C2.2D7, C22⋊C4.1D14, (C2×D4).149D14, (C2×C28).26C23, Dic7⋊C4⋊7C22, C4⋊Dic7⋊24C22, C28.17D4⋊11C2, (C2×C14).131C24, (C4×Dic7)⋊13C22, C23.D7⋊12C22, C2.26(D4⋊6D14), C22⋊Dic14⋊12C2, C7⋊4(C22.45C24), (C2×Dic14)⋊19C22, (D4×C14).110C22, C23.18D14⋊3C2, C23.11D14⋊2C2, C23.D14⋊11C2, (C23×C14).67C22, C22.152(C23×D7), C23.176(C22×D7), C22.17(D4⋊2D7), (C22×C14).180C23, (C2×Dic7).220C23, (C22×Dic7)⋊11C22, C14.76(C2×C4○D4), (C7×C22≀C2).2C2, C2.27(C2×D4⋊2D7), (C2×C23.D7)⋊19C2, (C2×C4).26(C22×D7), (C2×C14).43(C4○D4), (C7×C22⋊C4).2C22, SmallGroup(448,1040)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 940 in 248 conjugacy classes, 99 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×11], C22, C22 [×4], C22 [×14], C7, C2×C4, C2×C4 [×2], C2×C4 [×15], D4 [×5], Q8, C23 [×2], C23 [×2], C23 [×5], C14, C14 [×2], C14 [×6], C42 [×3], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×11], C4⋊C4 [×8], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic7 [×8], C28 [×3], C2×C14, C2×C14 [×4], C2×C14 [×14], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C42⋊2C2 [×2], Dic14, C2×Dic7 [×8], C2×Dic7 [×7], C2×C28, C2×C28 [×2], C7×D4 [×5], C22×C14 [×2], C22×C14 [×2], C22×C14 [×5], C22.45C24, C4×Dic7, C4×Dic7 [×2], Dic7⋊C4 [×6], C4⋊Dic7 [×2], C23.D7, C23.D7 [×10], C7×C22⋊C4, C7×C22⋊C4 [×2], C2×Dic14, C22×Dic7, C22×Dic7 [×4], D4×C14, D4×C14 [×2], C23×C14, C23.11D14 [×2], C22⋊Dic14 [×2], C23.D14 [×2], D4×Dic7 [×2], C23.18D14, C23.18D14 [×2], C28.17D4, C2×C23.D7 [×2], C7×C22≀C2, C24.32D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×4], C24, D14 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D7 [×7], C22.45C24, D4⋊2D7 [×4], C23×D7, C2×D4⋊2D7 [×2], D4⋊6D14, C24.32D14
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=1, f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
►On 112 points
Generators in S112
(2 101)(4 103)(6 105)(8 107)(10 109)(12 111)(14 99)(16 51)(18 53)(20 55)(22 43)(24 45)(26 47)(28 49)(29 72)(31 74)(33 76)(35 78)(37 80)(39 82)(41 84)(57 98)(59 86)(61 88)(63 90)(65 92)(67 94)(69 96)
(2 72)(4 74)(6 76)(8 78)(10 80)(12 82)(14 84)(15 60)(16 51)(17 62)(18 53)(19 64)(20 55)(21 66)(22 43)(23 68)(24 45)(25 70)(26 47)(27 58)(28 49)(29 101)(31 103)(33 105)(35 107)(37 109)(39 111)(41 99)(44 95)(46 97)(48 85)(50 87)(52 89)(54 91)(56 93)(57 98)(59 86)(61 88)(63 90)(65 92)(67 94)(69 96)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 99)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 71)(57 98)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 85)(28 86)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 99)(42 100)(43 67)(44 68)(45 69)(46 70)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)
(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 86 71 28)(2 85 72 27)(3 98 73 26)(4 97 74 25)(5 96 75 24)(6 95 76 23)(7 94 77 22)(8 93 78 21)(9 92 79 20)(10 91 80 19)(11 90 81 18)(12 89 82 17)(13 88 83 16)(14 87 84 15)(29 48 101 58)(30 47 102 57)(31 46 103 70)(32 45 104 69)(33 44 105 68)(34 43 106 67)(35 56 107 66)(36 55 108 65)(37 54 109 64)(38 53 110 63)(39 52 111 62)(40 51 112 61)(41 50 99 60)(42 49 100 59)
G:=sub<Sym(112)| (2,101)(4,103)(6,105)(8,107)(10,109)(12,111)(14,99)(16,51)(18,53)(20,55)(22,43)(24,45)(26,47)(28,49)(29,72)(31,74)(33,76)(35,78)(37,80)(39,82)(41,84)(57,98)(59,86)(61,88)(63,90)(65,92)(67,94)(69,96), (2,72)(4,74)(6,76)(8,78)(10,80)(12,82)(14,84)(15,60)(16,51)(17,62)(18,53)(19,64)(20,55)(21,66)(22,43)(23,68)(24,45)(25,70)(26,47)(27,58)(28,49)(29,101)(31,103)(33,105)(35,107)(37,109)(39,111)(41,99)(44,95)(46,97)(48,85)(50,87)(52,89)(54,91)(56,93)(57,98)(59,86)(61,88)(63,90)(65,92)(67,94)(69,96), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,99)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,71)(57,98)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,85)(28,86)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,99)(42,100)(43,67)(44,68)(45,69)(46,70)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66), (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), (1,86,71,28)(2,85,72,27)(3,98,73,26)(4,97,74,25)(5,96,75,24)(6,95,76,23)(7,94,77,22)(8,93,78,21)(9,92,79,20)(10,91,80,19)(11,90,81,18)(12,89,82,17)(13,88,83,16)(14,87,84,15)(29,48,101,58)(30,47,102,57)(31,46,103,70)(32,45,104,69)(33,44,105,68)(34,43,106,67)(35,56,107,66)(36,55,108,65)(37,54,109,64)(38,53,110,63)(39,52,111,62)(40,51,112,61)(41,50,99,60)(42,49,100,59)>;
G:=Group( (2,101)(4,103)(6,105)(8,107)(10,109)(12,111)(14,99)(16,51)(18,53)(20,55)(22,43)(24,45)(26,47)(28,49)(29,72)(31,74)(33,76)(35,78)(37,80)(39,82)(41,84)(57,98)(59,86)(61,88)(63,90)(65,92)(67,94)(69,96), (2,72)(4,74)(6,76)(8,78)(10,80)(12,82)(14,84)(15,60)(16,51)(17,62)(18,53)(19,64)(20,55)(21,66)(22,43)(23,68)(24,45)(25,70)(26,47)(27,58)(28,49)(29,101)(31,103)(33,105)(35,107)(37,109)(39,111)(41,99)(44,95)(46,97)(48,85)(50,87)(52,89)(54,91)(56,93)(57,98)(59,86)(61,88)(63,90)(65,92)(67,94)(69,96), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,99)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,71)(57,98)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,85)(28,86)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,99)(42,100)(43,67)(44,68)(45,69)(46,70)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66), (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), (1,86,71,28)(2,85,72,27)(3,98,73,26)(4,97,74,25)(5,96,75,24)(6,95,76,23)(7,94,77,22)(8,93,78,21)(9,92,79,20)(10,91,80,19)(11,90,81,18)(12,89,82,17)(13,88,83,16)(14,87,84,15)(29,48,101,58)(30,47,102,57)(31,46,103,70)(32,45,104,69)(33,44,105,68)(34,43,106,67)(35,56,107,66)(36,55,108,65)(37,54,109,64)(38,53,110,63)(39,52,111,62)(40,51,112,61)(41,50,99,60)(42,49,100,59) );
G=PermutationGroup([(2,101),(4,103),(6,105),(8,107),(10,109),(12,111),(14,99),(16,51),(18,53),(20,55),(22,43),(24,45),(26,47),(28,49),(29,72),(31,74),(33,76),(35,78),(37,80),(39,82),(41,84),(57,98),(59,86),(61,88),(63,90),(65,92),(67,94),(69,96)], [(2,72),(4,74),(6,76),(8,78),(10,80),(12,82),(14,84),(15,60),(16,51),(17,62),(18,53),(19,64),(20,55),(21,66),(22,43),(23,68),(24,45),(25,70),(26,47),(27,58),(28,49),(29,101),(31,103),(33,105),(35,107),(37,109),(39,111),(41,99),(44,95),(46,97),(48,85),(50,87),(52,89),(54,91),(56,93),(57,98),(59,86),(61,88),(63,90),(65,92),(67,94),(69,96)], [(1,100),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,109),(11,110),(12,111),(13,112),(14,99),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,72),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(37,80),(38,81),(39,82),(40,83),(41,84),(42,71),(57,98),(58,85),(59,86),(60,87),(61,88),(62,89),(63,90),(64,91),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97)], [(1,71),(2,72),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,81),(12,82),(13,83),(14,84),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,85),(28,86),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,99),(42,100),(43,67),(44,68),(45,69),(46,70),(47,57),(48,58),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66)], [(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)], [(1,86,71,28),(2,85,72,27),(3,98,73,26),(4,97,74,25),(5,96,75,24),(6,95,76,23),(7,94,77,22),(8,93,78,21),(9,92,79,20),(10,91,80,19),(11,90,81,18),(12,89,82,17),(13,88,83,16),(14,87,84,15),(29,48,101,58),(30,47,102,57),(31,46,103,70),(32,45,104,69),(33,44,105,68),(34,43,106,67),(35,56,107,66),(36,55,108,65),(37,54,109,64),(38,53,110,63),(39,52,111,62),(40,51,112,61),(41,50,99,60),(42,49,100,59)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 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 | 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 | 0 | 0 |
| 0 | 0 | 6 | 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 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 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 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 25 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 3 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 0 | 0 | 0 | 17 | 0 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,6,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[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],[4,25,0,0,0,0,4,18,0,0,0,0,0,0,28,0,0,0,0,0,10,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,3,0,0,0,0,7,26,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,17,0,0,0,0,17,0] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | ··· | 4K | 4L | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 14AB | 14AC | 14AD | 28A | ··· | 28I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | 2+ (1+4) | D4⋊2D7 | D4⋊6D14 |
| kernel | C24.32D14 | C23.11D14 | C22⋊Dic14 | C23.D14 | D4×Dic7 | C23.18D14 | C28.17D4 | C2×C23.D7 | C7×C22≀C2 | C22≀C2 | C2×C14 | C22⋊C4 | C2×D4 | C24 | C14 | C22 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 3 | 1 | 2 | 1 | 3 | 8 | 9 | 9 | 3 | 1 | 12 | 6 |
In GAP, Magma, Sage, TeX
C_2^4._{32}D_{14} % in TeX
G:=Group("C2^4.32D14"); // GroupNames label
G:=SmallGroup(448,1040);
// by ID
G=gap.SmallGroup(448,1040);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,219,1571,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=1,f^2=d,a*b=b*a,e*a*e^-1=f*a*f^-1=a*c=c*a,a*d=d*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*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
○
≀