direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C23⋊3D4, C14.1512+ 1+4, C23⋊4(C7×D4), C4⋊D4⋊5C14, C24⋊3(C2×C14), (C22×C14)⋊6D4, C22≀C2⋊2C14, (C22×D4)⋊6C14, C22.2(D4×C14), (D4×C14)⋊35C22, (C23×C14)⋊3C22, (C2×C14).354C24, (C2×C28).663C23, (C22×C28)⋊47C22, C22.D4⋊2C14, C14.189(C22×D4), C2.3(C7×2+ 1+4), C23.36(C22×C14), (C22×C14).90C23, C22.28(C23×C14), C4⋊C4⋊3(C2×C14), (D4×C2×C14)⋊21C2, C2.13(D4×C2×C14), (C2×D4)⋊3(C2×C14), (C7×C4⋊D4)⋊32C2, C22⋊C4⋊3(C2×C14), (C7×C4⋊C4)⋊37C22, (C22×C4)⋊7(C2×C14), (C2×C14).90(C2×D4), (C7×C22≀C2)⋊12C2, (C2×C22⋊C4)⋊12C14, (C14×C22⋊C4)⋊32C2, (C7×C22⋊C4)⋊38C22, (C2×C4).21(C22×C14), (C7×C22.D4)⋊21C2, SmallGroup(448,1317)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23⋊3D4
G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=e-1 >
Subgroups: 642 in 346 conjugacy classes, 162 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, C24, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C22≀C2, C4⋊D4, C22.D4, C22×D4, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C23⋊3D4, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, D4×C14, C23×C14, C23×C14, C14×C22⋊C4, C7×C22≀C2, C7×C4⋊D4, C7×C22.D4, D4×C2×C14, C7×C23⋊3D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C22×D4, 2+ 1+4, C7×D4, C22×C14, C23⋊3D4, D4×C14, C23×C14, D4×C2×C14, C7×2+ 1+4, C7×C23⋊3D4
►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)
(1 63)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 84)(9 78)(10 79)(11 80)(12 81)(13 82)(14 83)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 67)(30 68)(31 69)(32 70)(33 64)(34 65)(35 66)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 71)(43 53)(44 54)(45 55)(46 56)(47 50)(48 51)(49 52)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)(91 106)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 64)(7 65)(8 90)(9 91)(10 85)(11 86)(12 87)(13 88)(14 89)(15 97)(16 98)(17 92)(18 93)(19 94)(20 95)(21 96)(22 101)(23 102)(24 103)(25 104)(26 105)(27 99)(28 100)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 54)(37 55)(38 56)(39 50)(40 51)(41 52)(42 53)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(1 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 109)(16 110)(17 111)(18 112)(19 106)(20 107)(21 108)(29 48)(30 49)(31 43)(32 44)(33 45)(34 46)(35 47)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 76)(58 77)(59 71)(60 72)(61 73)(62 74)(63 75)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 104)(86 105)(87 99)(88 100)(89 101)(90 102)(91 103)
(1 94 35 91)(2 95 29 85)(3 96 30 86)(4 97 31 87)(5 98 32 88)(6 92 33 89)(7 93 34 90)(8 56 18 74)(9 50 19 75)(10 51 20 76)(11 52 21 77)(12 53 15 71)(13 54 16 72)(14 55 17 73)(22 64 111 61)(23 65 112 62)(24 66 106 63)(25 67 107 57)(26 68 108 58)(27 69 109 59)(28 70 110 60)(36 82 44 100)(37 83 45 101)(38 84 46 102)(39 78 47 103)(40 79 48 104)(41 80 49 105)(42 81 43 99)
(8 112)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 27)(16 28)(17 22)(18 23)(19 24)(20 25)(21 26)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 76)(58 77)(59 71)(60 72)(61 73)(62 74)(63 75)(78 103)(79 104)(80 105)(81 99)(82 100)(83 101)(84 102)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)
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), (1,63)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,84)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,67)(30,68)(31,69)(32,70)(33,64)(34,65)(35,66)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,71)(43,53)(44,54)(45,55)(46,56)(47,50)(48,51)(49,52)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,106), (1,66)(2,67)(3,68)(4,69)(5,70)(6,64)(7,65)(8,90)(9,91)(10,85)(11,86)(12,87)(13,88)(14,89)(15,97)(16,98)(17,92)(18,93)(19,94)(20,95)(21,96)(22,101)(23,102)(24,103)(25,104)(26,105)(27,99)(28,100)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,54)(37,55)(38,56)(39,50)(40,51)(41,52)(42,53)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103), (1,94,35,91)(2,95,29,85)(3,96,30,86)(4,97,31,87)(5,98,32,88)(6,92,33,89)(7,93,34,90)(8,56,18,74)(9,50,19,75)(10,51,20,76)(11,52,21,77)(12,53,15,71)(13,54,16,72)(14,55,17,73)(22,64,111,61)(23,65,112,62)(24,66,106,63)(25,67,107,57)(26,68,108,58)(27,69,109,59)(28,70,110,60)(36,82,44,100)(37,83,45,101)(38,84,46,102)(39,78,47,103)(40,79,48,104)(41,80,49,105)(42,81,43,99), (8,112)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,103)(79,104)(80,105)(81,99)(82,100)(83,101)(84,102)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)>;
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), (1,63)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,84)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,67)(30,68)(31,69)(32,70)(33,64)(34,65)(35,66)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,71)(43,53)(44,54)(45,55)(46,56)(47,50)(48,51)(49,52)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,106), (1,66)(2,67)(3,68)(4,69)(5,70)(6,64)(7,65)(8,90)(9,91)(10,85)(11,86)(12,87)(13,88)(14,89)(15,97)(16,98)(17,92)(18,93)(19,94)(20,95)(21,96)(22,101)(23,102)(24,103)(25,104)(26,105)(27,99)(28,100)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,54)(37,55)(38,56)(39,50)(40,51)(41,52)(42,53)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103), (1,94,35,91)(2,95,29,85)(3,96,30,86)(4,97,31,87)(5,98,32,88)(6,92,33,89)(7,93,34,90)(8,56,18,74)(9,50,19,75)(10,51,20,76)(11,52,21,77)(12,53,15,71)(13,54,16,72)(14,55,17,73)(22,64,111,61)(23,65,112,62)(24,66,106,63)(25,67,107,57)(26,68,108,58)(27,69,109,59)(28,70,110,60)(36,82,44,100)(37,83,45,101)(38,84,46,102)(39,78,47,103)(40,79,48,104)(41,80,49,105)(42,81,43,99), (8,112)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,103)(79,104)(80,105)(81,99)(82,100)(83,101)(84,102)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94) );
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)], [(1,63),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,84),(9,78),(10,79),(11,80),(12,81),(13,82),(14,83),(15,99),(16,100),(17,101),(18,102),(19,103),(20,104),(21,105),(22,92),(23,93),(24,94),(25,95),(26,96),(27,97),(28,98),(29,67),(30,68),(31,69),(32,70),(33,64),(34,65),(35,66),(36,72),(37,73),(38,74),(39,75),(40,76),(41,77),(42,71),(43,53),(44,54),(45,55),(46,56),(47,50),(48,51),(49,52),(85,107),(86,108),(87,109),(88,110),(89,111),(90,112),(91,106)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,64),(7,65),(8,90),(9,91),(10,85),(11,86),(12,87),(13,88),(14,89),(15,97),(16,98),(17,92),(18,93),(19,94),(20,95),(21,96),(22,101),(23,102),(24,103),(25,104),(26,105),(27,99),(28,100),(29,57),(30,58),(31,59),(32,60),(33,61),(34,62),(35,63),(36,54),(37,55),(38,56),(39,50),(40,51),(41,52),(42,53),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(1,39),(2,40),(3,41),(4,42),(5,36),(6,37),(7,38),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,22),(15,109),(16,110),(17,111),(18,112),(19,106),(20,107),(21,108),(29,48),(30,49),(31,43),(32,44),(33,45),(34,46),(35,47),(50,66),(51,67),(52,68),(53,69),(54,70),(55,64),(56,65),(57,76),(58,77),(59,71),(60,72),(61,73),(62,74),(63,75),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,104),(86,105),(87,99),(88,100),(89,101),(90,102),(91,103)], [(1,94,35,91),(2,95,29,85),(3,96,30,86),(4,97,31,87),(5,98,32,88),(6,92,33,89),(7,93,34,90),(8,56,18,74),(9,50,19,75),(10,51,20,76),(11,52,21,77),(12,53,15,71),(13,54,16,72),(14,55,17,73),(22,64,111,61),(23,65,112,62),(24,66,106,63),(25,67,107,57),(26,68,108,58),(27,69,109,59),(28,70,110,60),(36,82,44,100),(37,83,45,101),(38,84,46,102),(39,78,47,103),(40,79,48,104),(41,80,49,105),(42,81,43,99)], [(8,112),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,27),(16,28),(17,22),(18,23),(19,24),(20,25),(21,26),(50,66),(51,67),(52,68),(53,69),(54,70),(55,64),(56,65),(57,76),(58,77),(59,71),(60,72),(61,73),(62,74),(63,75),(78,103),(79,104),(80,105),(81,99),(82,100),(83,101),(84,102),(85,95),(86,96),(87,97),(88,98),(89,92),(90,93),(91,94)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14BB | 14BC | ··· | 14BZ | 28A | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | C7×D4 | 2+ 1+4 | C7×2+ 1+4 |
| kernel | C7×C23⋊3D4 | C14×C22⋊C4 | C7×C22≀C2 | C7×C4⋊D4 | C7×C22.D4 | D4×C2×C14 | C23⋊3D4 | C2×C22⋊C4 | C22≀C2 | C4⋊D4 | C22.D4 | C22×D4 | C22×C14 | C23 | C14 | C2 |
| # reps | 1 | 1 | 4 | 4 | 4 | 2 | 6 | 6 | 24 | 24 | 24 | 12 | 4 | 24 | 2 | 12 |
Matrix representation of C7×C23⋊3D4 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 27 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 28 | 0 | 28 |
| 0 | 0 | 1 | 1 | 28 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 27 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 1 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 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 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 27 | 0 |
| 0 | 0 | 28 | 0 | 1 | 28 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 28 | 28 | 1 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 28 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,28,1,0,0,27,1,28,1,0,0,0,0,0,28,0,0,0,0,28,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,27,1,28,1,0,0,0,0,0,1,0,0,0,0,1,0],[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],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,1,28,0,28,0,0,0,0,0,28,0,0,27,1,28,1,0,0,0,28,0,0],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,28,0,28,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28] >;
C7×C23⋊3D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3\rtimes_3D_4
% in TeX
G:=Group("C7xC2^3:3D4"); // GroupNames label
G:=SmallGroup(448,1317);
// by ID
G=gap.SmallGroup(448,1317);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,1227,3363]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations