?
G = C14.382+ (1+4) order 448 = 26·7
38th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.382+ (1+4), (C4×D7)⋊1D4, C4⋊C4⋊22D14, (C2×D4)⋊7D14, C4⋊D4⋊10D7, C4.183(D4×D7), C23⋊D14⋊9C2, C28⋊2D4⋊18C2, C28⋊7D4⋊33C2, C28⋊D4⋊16C2, C28⋊1D4⋊21C2, C22⋊C4⋊10D14, D14.41(C2×D4), C28.227(C2×D4), (C22×C4)⋊16D14, Dic7.6(C2×D4), C22⋊D28⋊12C2, D14⋊C4⋊17C22, (D4×C14)⋊12C22, (C2×D28)⋊23C22, C4⋊Dic7⋊31C22, C14.66(C22×D4), (C2×C28).173C23, (C2×C14).151C24, (C22×C28)⋊20C22, C7⋊3(C22.29C24), (C4×Dic7)⋊21C22, C23.D7⋊23C22, C2.27(D4⋊8D14), C2.40(D4⋊6D14), Dic7.D4⋊18C2, (C2×Dic14)⋊54C22, (C22×C14).20C23, (C2×Dic7).72C23, (C23×D7).46C22, C23.112(C22×D7), C22.172(C23×D7), (C22×D7).186C23, (C2×D4×D7)⋊10C2, C2.39(C2×D4×D7), (C2×C4×D7)⋊13C22, C4⋊C4⋊7D7⋊20C2, (C2×C4○D28)⋊20C2, (C7×C4⋊D4)⋊13C2, (C7×C4⋊C4)⋊10C22, (C2×C7⋊D4)⋊14C22, (C2×C4).38(C22×D7), (C7×C22⋊C4)⋊12C22, SmallGroup(448,1060)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1996 in 334 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×30], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×12], D4 [×22], Q8 [×2], C23, C23 [×2], C23 [×12], D7 [×5], C14 [×3], C14 [×3], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4, C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×16], C2×Q8, C4○D4 [×4], C24 [×2], Dic7 [×2], Dic7 [×3], C28 [×2], C28 [×3], D14 [×2], D14 [×19], C2×C14, C2×C14 [×9], C42⋊C2, C22≀C2 [×4], C4⋊D4, C4⋊D4 [×3], C4.4D4 [×2], C4⋊1D4 [×2], C22×D4, C2×C4○D4, Dic14 [×2], C4×D7 [×4], C4×D7 [×2], D28 [×6], C2×Dic7 [×2], C2×Dic7 [×2], C7⋊D4 [×12], C2×C28 [×2], C2×C28 [×2], C2×C28 [×2], C7×D4 [×4], C22×D7 [×2], C22×D7 [×2], C22×D7 [×8], C22×C14, C22×C14 [×2], C22.29C24, C4×Dic7 [×2], C4⋊Dic7, D14⋊C4 [×6], C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14, C2×C4×D7 [×2], C2×D28 [×2], C2×D28 [×2], C4○D28 [×4], D4×D7 [×4], C2×C7⋊D4 [×2], C2×C7⋊D4 [×6], C22×C28, D4×C14, D4×C14 [×2], C23×D7 [×2], C22⋊D28 [×2], Dic7.D4 [×2], C4⋊C4⋊7D7, C28⋊1D4, C28⋊7D4, C23⋊D14 [×2], C28⋊2D4, C28⋊D4 [×2], C7×C4⋊D4, C2×C4○D28, C2×D4×D7, C14.382+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, 2+ (1+4) [×2], C22×D7 [×7], C22.29C24, D4×D7 [×2], C23×D7, C2×D4×D7, D4⋊6D14, D4⋊8D14, C14.382+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece=a7c, ede=b2d >
►On 112 points
(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 74 16 58)(2 75 17 59)(3 76 18 60)(4 77 19 61)(5 78 20 62)(6 79 21 63)(7 80 22 64)(8 81 23 65)(9 82 24 66)(10 83 25 67)(11 84 26 68)(12 71 27 69)(13 72 28 70)(14 73 15 57)(29 101 44 87)(30 102 45 88)(31 103 46 89)(32 104 47 90)(33 105 48 91)(34 106 49 92)(35 107 50 93)(36 108 51 94)(37 109 52 95)(38 110 53 96)(39 111 54 97)(40 112 55 98)(41 99 56 85)(42 100 43 86)
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 70)(14 57)(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 94)(30 95)(31 96)(32 97)(33 98)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)
(1 54 16 39)(2 53 17 38)(3 52 18 37)(4 51 19 36)(5 50 20 35)(6 49 21 34)(7 48 22 33)(8 47 23 32)(9 46 24 31)(10 45 25 30)(11 44 26 29)(12 43 27 42)(13 56 28 41)(14 55 15 40)(57 112 73 98)(58 111 74 97)(59 110 75 96)(60 109 76 95)(61 108 77 94)(62 107 78 93)(63 106 79 92)(64 105 80 91)(65 104 81 90)(66 103 82 89)(67 102 83 88)(68 101 84 87)(69 100 71 86)(70 99 72 85)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 29)(13 30)(14 31)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 43)(27 44)(28 45)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 85)(68 86)(69 87)(70 88)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 99)(84 100)
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), (1,74,16,58)(2,75,17,59)(3,76,18,60)(4,77,19,61)(5,78,20,62)(6,79,21,63)(7,80,22,64)(8,81,23,65)(9,82,24,66)(10,83,25,67)(11,84,26,68)(12,71,27,69)(13,72,28,70)(14,73,15,57)(29,101,44,87)(30,102,45,88)(31,103,46,89)(32,104,47,90)(33,105,48,91)(34,106,49,92)(35,107,50,93)(36,108,51,94)(37,109,52,95)(38,110,53,96)(39,111,54,97)(40,112,55,98)(41,99,56,85)(42,100,43,86), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,57)(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)(29,94)(30,95)(31,96)(32,97)(33,98)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106), (1,54,16,39)(2,53,17,38)(3,52,18,37)(4,51,19,36)(5,50,20,35)(6,49,21,34)(7,48,22,33)(8,47,23,32)(9,46,24,31)(10,45,25,30)(11,44,26,29)(12,43,27,42)(13,56,28,41)(14,55,15,40)(57,112,73,98)(58,111,74,97)(59,110,75,96)(60,109,76,95)(61,108,77,94)(62,107,78,93)(63,106,79,92)(64,105,80,91)(65,104,81,90)(66,103,82,89)(67,102,83,88)(68,101,84,87)(69,100,71,86)(70,99,72,85), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,29)(13,30)(14,31)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,43)(27,44)(28,45)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,85)(68,86)(69,87)(70,88)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,99)(84,100)>;
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), (1,74,16,58)(2,75,17,59)(3,76,18,60)(4,77,19,61)(5,78,20,62)(6,79,21,63)(7,80,22,64)(8,81,23,65)(9,82,24,66)(10,83,25,67)(11,84,26,68)(12,71,27,69)(13,72,28,70)(14,73,15,57)(29,101,44,87)(30,102,45,88)(31,103,46,89)(32,104,47,90)(33,105,48,91)(34,106,49,92)(35,107,50,93)(36,108,51,94)(37,109,52,95)(38,110,53,96)(39,111,54,97)(40,112,55,98)(41,99,56,85)(42,100,43,86), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,57)(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)(29,94)(30,95)(31,96)(32,97)(33,98)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106), (1,54,16,39)(2,53,17,38)(3,52,18,37)(4,51,19,36)(5,50,20,35)(6,49,21,34)(7,48,22,33)(8,47,23,32)(9,46,24,31)(10,45,25,30)(11,44,26,29)(12,43,27,42)(13,56,28,41)(14,55,15,40)(57,112,73,98)(58,111,74,97)(59,110,75,96)(60,109,76,95)(61,108,77,94)(62,107,78,93)(63,106,79,92)(64,105,80,91)(65,104,81,90)(66,103,82,89)(67,102,83,88)(68,101,84,87)(69,100,71,86)(70,99,72,85), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,29)(13,30)(14,31)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,43)(27,44)(28,45)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,85)(68,86)(69,87)(70,88)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,99)(84,100) );
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)], [(1,74,16,58),(2,75,17,59),(3,76,18,60),(4,77,19,61),(5,78,20,62),(6,79,21,63),(7,80,22,64),(8,81,23,65),(9,82,24,66),(10,83,25,67),(11,84,26,68),(12,71,27,69),(13,72,28,70),(14,73,15,57),(29,101,44,87),(30,102,45,88),(31,103,46,89),(32,104,47,90),(33,105,48,91),(34,106,49,92),(35,107,50,93),(36,108,51,94),(37,109,52,95),(38,110,53,96),(39,111,54,97),(40,112,55,98),(41,99,56,85),(42,100,43,86)], [(1,58),(2,59),(3,60),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,68),(12,69),(13,70),(14,57),(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),(29,94),(30,95),(31,96),(32,97),(33,98),(34,85),(35,86),(36,87),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,107),(44,108),(45,109),(46,110),(47,111),(48,112),(49,99),(50,100),(51,101),(52,102),(53,103),(54,104),(55,105),(56,106)], [(1,54,16,39),(2,53,17,38),(3,52,18,37),(4,51,19,36),(5,50,20,35),(6,49,21,34),(7,48,22,33),(8,47,23,32),(9,46,24,31),(10,45,25,30),(11,44,26,29),(12,43,27,42),(13,56,28,41),(14,55,15,40),(57,112,73,98),(58,111,74,97),(59,110,75,96),(60,109,76,95),(61,108,77,94),(62,107,78,93),(63,106,79,92),(64,105,80,91),(65,104,81,90),(66,103,82,89),(67,102,83,88),(68,101,84,87),(69,100,71,86),(70,99,72,85)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,29),(13,30),(14,31),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,56),(26,43),(27,44),(28,45),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,85),(68,86),(69,87),(70,88),(71,101),(72,102),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,109),(80,110),(81,111),(82,112),(83,99),(84,100)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 25 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 25 |
| 0 | 0 | 0 | 0 | 4 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 27 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 27 | 0 | 0 |
| 0 | 0 | 3 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 27 |
| 0 | 0 | 0 | 0 | 3 | 11 |
| 1 | 27 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 27 | 0 | 0 |
| 0 | 0 | 2 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 27 |
| 0 | 0 | 0 | 0 | 2 | 11 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,25,4,0,0,0,0,25,11,0,0,0,0,0,0,25,4,0,0,0,0,25,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,27,28,0,0,0,0,0,0,18,3,0,0,0,0,27,11,0,0,0,0,0,0,18,3,0,0,0,0,27,11],[1,0,0,0,0,0,27,28,0,0,0,0,0,0,18,2,0,0,0,0,27,11,0,0,0,0,0,0,18,2,0,0,0,0,27,11] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4×D7 | D4⋊6D14 | D4⋊8D14 |
| kernel | C14.382+ (1+4) | C22⋊D28 | Dic7.D4 | C4⋊C4⋊7D7 | C28⋊1D4 | C28⋊7D4 | C23⋊D14 | C28⋊2D4 | C28⋊D4 | C7×C4⋊D4 | C2×C4○D28 | C2×D4×D7 | C4×D7 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 3 | 6 | 3 | 3 | 9 | 2 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{38}2_+^{(1+4)} % in TeX
G:=Group("C14.38ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1060);
// by ID
G=gap.SmallGroup(448,1060);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,675,570,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=a^7*c,e*d*e=b^2*d>;
// generators/relations