G = C14.532+ 1+4 order 448 = 26·7
53rd non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.532+ 1+4, C4⋊C4⋊12D14, (C2×Q8)⋊6D14, C22⋊Q8⋊14D7, D28⋊C4⋊28C2, (Q8×C14)⋊9C22, D14⋊C4⋊68C22, D14⋊3Q8⋊19C2, C22⋊D28.2C2, (C2×C28).60C23, C4⋊Dic7⋊37C22, C22⋊C4.62D14, D14.19(C4○D4), D14.5D4⋊19C2, C28.23D4⋊14C2, (C2×C14).181C24, Dic7⋊C4⋊19C22, (C4×Dic7)⋊29C22, (C22×C4).243D14, C2.55(D4⋊6D14), C7⋊5(C22.45C24), (C2×D28).150C22, C22.D28⋊16C2, C23.11D14⋊8C2, C22.9(Q8⋊2D7), (C2×Dic7).92C23, (C23×D7).54C22, (C22×D7).74C23, C22.202(C23×D7), C23.194(C22×D7), (C22×C14).209C23, (C22×C28).381C22, C23.D7.121C22, (C22×Dic7).122C22, (C4×C7⋊D4)⋊57C2, (D7×C22⋊C4)⋊9C2, C2.52(D7×C4○D4), (C2×C4×D7)⋊51C22, (C2×D14⋊C4)⋊37C2, C4⋊C4⋊7D7⋊26C2, C4⋊C4⋊D7⋊17C2, (C7×C4⋊C4)⋊21C22, (C7×C22⋊Q8)⋊17C2, C14.164(C2×C4○D4), C2.18(C2×Q8⋊2D7), (C2×C4).51(C22×D7), (C2×C14).26(C4○D4), (C2×C7⋊D4).128C22, (C7×C22⋊C4).36C22, SmallGroup(448,1090)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.532+ 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=a7b2, e2=a7, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a7b-1, dbd-1=ebe-1=a7b, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 1276 in 248 conjugacy classes, 97 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, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×Q8, C22×D7, C22×D7, C22×C14, C22.45C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, Q8×C14, C23×D7, C23.11D14, D7×C22⋊C4, C22⋊D28, C22.D28, C4⋊C4⋊7D7, D28⋊C4, D14.5D4, C4⋊C4⋊D7, C2×D14⋊C4, C4×C7⋊D4, D14⋊3Q8, C28.23D4, C7×C22⋊Q8, C14.532+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.45C24, Q8⋊2D7, C23×D7, D4⋊6D14, C2×Q8⋊2D7, D7×C4○D4, C14.532+ 1+4
►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 57 25 81)(2 58 26 82)(3 59 27 83)(4 60 28 84)(5 61 15 71)(6 62 16 72)(7 63 17 73)(8 64 18 74)(9 65 19 75)(10 66 20 76)(11 67 21 77)(12 68 22 78)(13 69 23 79)(14 70 24 80)(29 90 56 106)(30 91 43 107)(31 92 44 108)(32 93 45 109)(33 94 46 110)(34 95 47 111)(35 96 48 112)(36 97 49 99)(37 98 50 100)(38 85 51 101)(39 86 52 102)(40 87 53 103)(41 88 54 104)(42 89 55 105)
(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 81)(65 82)(66 83)(67 84)(68 71)(69 72)(70 73)(85 108)(86 109)(87 110)(88 111)(89 112)(90 99)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)
(1 54 18 34)(2 53 19 33)(3 52 20 32)(4 51 21 31)(5 50 22 30)(6 49 23 29)(7 48 24 42)(8 47 25 41)(9 46 26 40)(10 45 27 39)(11 44 28 38)(12 43 15 37)(13 56 16 36)(14 55 17 35)(57 111 74 88)(58 110 75 87)(59 109 76 86)(60 108 77 85)(61 107 78 98)(62 106 79 97)(63 105 80 96)(64 104 81 95)(65 103 82 94)(66 102 83 93)(67 101 84 92)(68 100 71 91)(69 99 72 90)(70 112 73 89)
(1 34 8 41)(2 35 9 42)(3 36 10 29)(4 37 11 30)(5 38 12 31)(6 39 13 32)(7 40 14 33)(15 51 22 44)(16 52 23 45)(17 53 24 46)(18 54 25 47)(19 55 26 48)(20 56 27 49)(21 43 28 50)(57 88 64 95)(58 89 65 96)(59 90 66 97)(60 91 67 98)(61 92 68 85)(62 93 69 86)(63 94 70 87)(71 108 78 101)(72 109 79 102)(73 110 80 103)(74 111 81 104)(75 112 82 105)(76 99 83 106)(77 100 84 107)
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), (1,57,25,81)(2,58,26,82)(3,59,27,83)(4,60,28,84)(5,61,15,71)(6,62,16,72)(7,63,17,73)(8,64,18,74)(9,65,19,75)(10,66,20,76)(11,67,21,77)(12,68,22,78)(13,69,23,79)(14,70,24,80)(29,90,56,106)(30,91,43,107)(31,92,44,108)(32,93,45,109)(33,94,46,110)(34,95,47,111)(35,96,48,112)(36,97,49,99)(37,98,50,100)(38,85,51,101)(39,86,52,102)(40,87,53,103)(41,88,54,104)(42,89,55,105), (57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,81)(65,82)(66,83)(67,84)(68,71)(69,72)(70,73)(85,108)(86,109)(87,110)(88,111)(89,112)(90,99)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107), (1,54,18,34)(2,53,19,33)(3,52,20,32)(4,51,21,31)(5,50,22,30)(6,49,23,29)(7,48,24,42)(8,47,25,41)(9,46,26,40)(10,45,27,39)(11,44,28,38)(12,43,15,37)(13,56,16,36)(14,55,17,35)(57,111,74,88)(58,110,75,87)(59,109,76,86)(60,108,77,85)(61,107,78,98)(62,106,79,97)(63,105,80,96)(64,104,81,95)(65,103,82,94)(66,102,83,93)(67,101,84,92)(68,100,71,91)(69,99,72,90)(70,112,73,89), (1,34,8,41)(2,35,9,42)(3,36,10,29)(4,37,11,30)(5,38,12,31)(6,39,13,32)(7,40,14,33)(15,51,22,44)(16,52,23,45)(17,53,24,46)(18,54,25,47)(19,55,26,48)(20,56,27,49)(21,43,28,50)(57,88,64,95)(58,89,65,96)(59,90,66,97)(60,91,67,98)(61,92,68,85)(62,93,69,86)(63,94,70,87)(71,108,78,101)(72,109,79,102)(73,110,80,103)(74,111,81,104)(75,112,82,105)(76,99,83,106)(77,100,84,107)>;
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), (1,57,25,81)(2,58,26,82)(3,59,27,83)(4,60,28,84)(5,61,15,71)(6,62,16,72)(7,63,17,73)(8,64,18,74)(9,65,19,75)(10,66,20,76)(11,67,21,77)(12,68,22,78)(13,69,23,79)(14,70,24,80)(29,90,56,106)(30,91,43,107)(31,92,44,108)(32,93,45,109)(33,94,46,110)(34,95,47,111)(35,96,48,112)(36,97,49,99)(37,98,50,100)(38,85,51,101)(39,86,52,102)(40,87,53,103)(41,88,54,104)(42,89,55,105), (57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,81)(65,82)(66,83)(67,84)(68,71)(69,72)(70,73)(85,108)(86,109)(87,110)(88,111)(89,112)(90,99)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107), (1,54,18,34)(2,53,19,33)(3,52,20,32)(4,51,21,31)(5,50,22,30)(6,49,23,29)(7,48,24,42)(8,47,25,41)(9,46,26,40)(10,45,27,39)(11,44,28,38)(12,43,15,37)(13,56,16,36)(14,55,17,35)(57,111,74,88)(58,110,75,87)(59,109,76,86)(60,108,77,85)(61,107,78,98)(62,106,79,97)(63,105,80,96)(64,104,81,95)(65,103,82,94)(66,102,83,93)(67,101,84,92)(68,100,71,91)(69,99,72,90)(70,112,73,89), (1,34,8,41)(2,35,9,42)(3,36,10,29)(4,37,11,30)(5,38,12,31)(6,39,13,32)(7,40,14,33)(15,51,22,44)(16,52,23,45)(17,53,24,46)(18,54,25,47)(19,55,26,48)(20,56,27,49)(21,43,28,50)(57,88,64,95)(58,89,65,96)(59,90,66,97)(60,91,67,98)(61,92,68,85)(62,93,69,86)(63,94,70,87)(71,108,78,101)(72,109,79,102)(73,110,80,103)(74,111,81,104)(75,112,82,105)(76,99,83,106)(77,100,84,107) );
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)], [(1,57,25,81),(2,58,26,82),(3,59,27,83),(4,60,28,84),(5,61,15,71),(6,62,16,72),(7,63,17,73),(8,64,18,74),(9,65,19,75),(10,66,20,76),(11,67,21,77),(12,68,22,78),(13,69,23,79),(14,70,24,80),(29,90,56,106),(30,91,43,107),(31,92,44,108),(32,93,45,109),(33,94,46,110),(34,95,47,111),(35,96,48,112),(36,97,49,99),(37,98,50,100),(38,85,51,101),(39,86,52,102),(40,87,53,103),(41,88,54,104),(42,89,55,105)], [(57,74),(58,75),(59,76),(60,77),(61,78),(62,79),(63,80),(64,81),(65,82),(66,83),(67,84),(68,71),(69,72),(70,73),(85,108),(86,109),(87,110),(88,111),(89,112),(90,99),(91,100),(92,101),(93,102),(94,103),(95,104),(96,105),(97,106),(98,107)], [(1,54,18,34),(2,53,19,33),(3,52,20,32),(4,51,21,31),(5,50,22,30),(6,49,23,29),(7,48,24,42),(8,47,25,41),(9,46,26,40),(10,45,27,39),(11,44,28,38),(12,43,15,37),(13,56,16,36),(14,55,17,35),(57,111,74,88),(58,110,75,87),(59,109,76,86),(60,108,77,85),(61,107,78,98),(62,106,79,97),(63,105,80,96),(64,104,81,95),(65,103,82,94),(66,102,83,93),(67,101,84,92),(68,100,71,91),(69,99,72,90),(70,112,73,89)], [(1,34,8,41),(2,35,9,42),(3,36,10,29),(4,37,11,30),(5,38,12,31),(6,39,13,32),(7,40,14,33),(15,51,22,44),(16,52,23,45),(17,53,24,46),(18,54,25,47),(19,55,26,48),(20,56,27,49),(21,43,28,50),(57,88,64,95),(58,89,65,96),(59,90,66,97),(60,91,67,98),(61,92,68,85),(62,93,69,86),(63,94,70,87),(71,108,78,101),(72,109,79,102),(73,110,80,103),(74,111,81,104),(75,112,82,105),(76,99,83,106),(77,100,84,107)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | ··· | 4G | 4H | ··· | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | ··· | 4 | 14 | ··· | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | Q8⋊2D7 | D4⋊6D14 | D7×C4○D4 |
| kernel | C14.532+ 1+4 | C23.11D14 | D7×C22⋊C4 | C22⋊D28 | C22.D28 | C4⋊C4⋊7D7 | D28⋊C4 | D14.5D4 | C4⋊C4⋊D7 | C2×D14⋊C4 | C4×C7⋊D4 | D14⋊3Q8 | C28.23D4 | C7×C22⋊Q8 | C22⋊Q8 | D14 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 6 | 9 | 3 | 3 | 1 | 6 | 6 | 6 |
Matrix representation of C14.532+ 1+4 ►in GL6(𝔽29)
| 28 | 21 | 0 | 0 | 0 | 0 |
| 17 | 19 | 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 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 27 |
| 0 | 0 | 0 | 0 | 28 | 12 |
| 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 | 17 | 28 |
| 1 | 8 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 28 | 12 |
G:=sub<GL(6,GF(29))| [28,17,0,0,0,0,21,19,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,0,17,0,0,0,0,0,0,17,28,0,0,0,0,27,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,28],[1,0,0,0,0,0,8,28,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,12,1,0,0,0,0,0,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,17,28,0,0,0,0,0,12] >;
C14.532+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._{53}2_+^{1+4} % in TeX
G:=Group("C14.53ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1090);
// by ID
G=gap.SmallGroup(448,1090);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,1571,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=a^7*b^2,e^2=a^7,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=a^7*b^-1,d*b*d^-1=e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations