?
G = C14.512+ (1+4) order 448 = 26·7
51st non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.512+ (1+4), C4⋊C4⋊11D14, (C2×Q8)⋊5D14, C22⋊Q8⋊11D7, D14.2(C2×Q8), (C22×D7)⋊3Q8, C22.7(Q8×D7), (Q8×C14)⋊8C22, C7⋊2(C23⋊2Q8), D14⋊3Q8⋊16C2, D14⋊2Q8⋊27C2, D14⋊Q8⋊21C2, (C2×C28).57C23, C4⋊Dic7⋊36C22, C22⋊C4.59D14, C28.48D4⋊46C2, C14.36(C22×Q8), (C2×C14).178C24, Dic7⋊C4⋊18C22, (C2×Dic14)⋊9C22, (C22×C4).240D14, C2.53(D4⋊6D14), C2.35(D4⋊8D14), D14⋊C4.147C22, C22⋊Dic14⋊24C2, (C2×Dic7).89C23, (C23×D7).53C22, C22.199(C23×D7), C23.191(C22×D7), C23.D7.34C22, (C22×C14).206C23, (C22×C28).315C22, (C22×D7).200C23, (C22×Dic7).119C22, C2.19(C2×Q8×D7), (C2×C14).7(C2×Q8), (C7×C4⋊C4)⋊20C22, (C7×C22⋊Q8)⋊14C2, (D7×C22⋊C4).2C2, (C2×C4×D7).98C22, (C2×D14⋊C4).20C2, (C2×C4).183(C22×D7), (C7×C22⋊C4).33C22, SmallGroup(448,1087)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1244 in 242 conjugacy classes, 103 normal (31 characteristic)
C1, C2 [×3], C2 [×6], C4 [×12], C22, C22 [×2], C22 [×14], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], Q8 [×4], C23, C23 [×8], D7 [×4], C14 [×3], C14 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4, C22×C4 [×5], C2×Q8, C2×Q8 [×3], C24, Dic7 [×6], C28 [×6], D14 [×4], D14 [×8], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×C22⋊C4 [×3], C22⋊Q8, C22⋊Q8 [×11], Dic14 [×3], C4×D7 [×4], C2×Dic7 [×6], C2×Dic7, C2×C28 [×2], C2×C28 [×4], C2×C28, C7×Q8, C22×D7 [×6], C22×D7 [×2], C22×C14, C23⋊2Q8, Dic7⋊C4 [×6], C4⋊Dic7, C4⋊Dic7 [×2], D14⋊C4 [×8], C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×Dic14, C2×Dic14 [×2], C2×C4×D7 [×4], C22×Dic7, C22×C28, Q8×C14, C23×D7, C22⋊Dic14 [×2], D7×C22⋊C4 [×2], D14⋊Q8 [×4], D14⋊2Q8 [×2], C28.48D4, C2×D14⋊C4, D14⋊3Q8 [×2], C7×C22⋊Q8, C14.512+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D7, C2×Q8 [×6], C24, D14 [×7], C22×Q8, 2+ (1+4) [×2], C22×D7 [×7], C23⋊2Q8, Q8×D7 [×2], C23×D7, D4⋊6D14, C2×Q8×D7, D4⋊8D14, C14.512+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, 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=a7b2d >
►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 67 20 74)(2 68 21 75)(3 69 22 76)(4 70 23 77)(5 57 24 78)(6 58 25 79)(7 59 26 80)(8 60 27 81)(9 61 28 82)(10 62 15 83)(11 63 16 84)(12 64 17 71)(13 65 18 72)(14 66 19 73)(29 97 55 101)(30 98 56 102)(31 85 43 103)(32 86 44 104)(33 87 45 105)(34 88 46 106)(35 89 47 107)(36 90 48 108)(37 91 49 109)(38 92 50 110)(39 93 51 111)(40 94 52 112)(41 95 53 99)(42 96 54 100)
(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)(85 110)(86 111)(87 112)(88 99)(89 100)(90 101)(91 102)(92 103)(93 104)(94 105)(95 106)(96 107)(97 108)(98 109)
(1 52 20 40)(2 51 21 39)(3 50 22 38)(4 49 23 37)(5 48 24 36)(6 47 25 35)(7 46 26 34)(8 45 27 33)(9 44 28 32)(10 43 15 31)(11 56 16 30)(12 55 17 29)(13 54 18 42)(14 53 19 41)(57 101 78 97)(58 100 79 96)(59 99 80 95)(60 112 81 94)(61 111 82 93)(62 110 83 92)(63 109 84 91)(64 108 71 90)(65 107 72 89)(66 106 73 88)(67 105 74 87)(68 104 75 86)(69 103 76 85)(70 102 77 98)
(1 40 8 33)(2 41 9 34)(3 42 10 35)(4 29 11 36)(5 30 12 37)(6 31 13 38)(7 32 14 39)(15 47 22 54)(16 48 23 55)(17 49 24 56)(18 50 25 43)(19 51 26 44)(20 52 27 45)(21 53 28 46)(57 91 64 98)(58 92 65 85)(59 93 66 86)(60 94 67 87)(61 95 68 88)(62 96 69 89)(63 97 70 90)(71 102 78 109)(72 103 79 110)(73 104 80 111)(74 105 81 112)(75 106 82 99)(76 107 83 100)(77 108 84 101)
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), (1,67,20,74)(2,68,21,75)(3,69,22,76)(4,70,23,77)(5,57,24,78)(6,58,25,79)(7,59,26,80)(8,60,27,81)(9,61,28,82)(10,62,15,83)(11,63,16,84)(12,64,17,71)(13,65,18,72)(14,66,19,73)(29,97,55,101)(30,98,56,102)(31,85,43,103)(32,86,44,104)(33,87,45,105)(34,88,46,106)(35,89,47,107)(36,90,48,108)(37,91,49,109)(38,92,50,110)(39,93,51,111)(40,94,52,112)(41,95,53,99)(42,96,54,100), (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)(85,110)(86,111)(87,112)(88,99)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109), (1,52,20,40)(2,51,21,39)(3,50,22,38)(4,49,23,37)(5,48,24,36)(6,47,25,35)(7,46,26,34)(8,45,27,33)(9,44,28,32)(10,43,15,31)(11,56,16,30)(12,55,17,29)(13,54,18,42)(14,53,19,41)(57,101,78,97)(58,100,79,96)(59,99,80,95)(60,112,81,94)(61,111,82,93)(62,110,83,92)(63,109,84,91)(64,108,71,90)(65,107,72,89)(66,106,73,88)(67,105,74,87)(68,104,75,86)(69,103,76,85)(70,102,77,98), (1,40,8,33)(2,41,9,34)(3,42,10,35)(4,29,11,36)(5,30,12,37)(6,31,13,38)(7,32,14,39)(15,47,22,54)(16,48,23,55)(17,49,24,56)(18,50,25,43)(19,51,26,44)(20,52,27,45)(21,53,28,46)(57,91,64,98)(58,92,65,85)(59,93,66,86)(60,94,67,87)(61,95,68,88)(62,96,69,89)(63,97,70,90)(71,102,78,109)(72,103,79,110)(73,104,80,111)(74,105,81,112)(75,106,82,99)(76,107,83,100)(77,108,84,101)>;
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), (1,67,20,74)(2,68,21,75)(3,69,22,76)(4,70,23,77)(5,57,24,78)(6,58,25,79)(7,59,26,80)(8,60,27,81)(9,61,28,82)(10,62,15,83)(11,63,16,84)(12,64,17,71)(13,65,18,72)(14,66,19,73)(29,97,55,101)(30,98,56,102)(31,85,43,103)(32,86,44,104)(33,87,45,105)(34,88,46,106)(35,89,47,107)(36,90,48,108)(37,91,49,109)(38,92,50,110)(39,93,51,111)(40,94,52,112)(41,95,53,99)(42,96,54,100), (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)(85,110)(86,111)(87,112)(88,99)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109), (1,52,20,40)(2,51,21,39)(3,50,22,38)(4,49,23,37)(5,48,24,36)(6,47,25,35)(7,46,26,34)(8,45,27,33)(9,44,28,32)(10,43,15,31)(11,56,16,30)(12,55,17,29)(13,54,18,42)(14,53,19,41)(57,101,78,97)(58,100,79,96)(59,99,80,95)(60,112,81,94)(61,111,82,93)(62,110,83,92)(63,109,84,91)(64,108,71,90)(65,107,72,89)(66,106,73,88)(67,105,74,87)(68,104,75,86)(69,103,76,85)(70,102,77,98), (1,40,8,33)(2,41,9,34)(3,42,10,35)(4,29,11,36)(5,30,12,37)(6,31,13,38)(7,32,14,39)(15,47,22,54)(16,48,23,55)(17,49,24,56)(18,50,25,43)(19,51,26,44)(20,52,27,45)(21,53,28,46)(57,91,64,98)(58,92,65,85)(59,93,66,86)(60,94,67,87)(61,95,68,88)(62,96,69,89)(63,97,70,90)(71,102,78,109)(72,103,79,110)(73,104,80,111)(74,105,81,112)(75,106,82,99)(76,107,83,100)(77,108,84,101) );
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)], [(1,67,20,74),(2,68,21,75),(3,69,22,76),(4,70,23,77),(5,57,24,78),(6,58,25,79),(7,59,26,80),(8,60,27,81),(9,61,28,82),(10,62,15,83),(11,63,16,84),(12,64,17,71),(13,65,18,72),(14,66,19,73),(29,97,55,101),(30,98,56,102),(31,85,43,103),(32,86,44,104),(33,87,45,105),(34,88,46,106),(35,89,47,107),(36,90,48,108),(37,91,49,109),(38,92,50,110),(39,93,51,111),(40,94,52,112),(41,95,53,99),(42,96,54,100)], [(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),(85,110),(86,111),(87,112),(88,99),(89,100),(90,101),(91,102),(92,103),(93,104),(94,105),(95,106),(96,107),(97,108),(98,109)], [(1,52,20,40),(2,51,21,39),(3,50,22,38),(4,49,23,37),(5,48,24,36),(6,47,25,35),(7,46,26,34),(8,45,27,33),(9,44,28,32),(10,43,15,31),(11,56,16,30),(12,55,17,29),(13,54,18,42),(14,53,19,41),(57,101,78,97),(58,100,79,96),(59,99,80,95),(60,112,81,94),(61,111,82,93),(62,110,83,92),(63,109,84,91),(64,108,71,90),(65,107,72,89),(66,106,73,88),(67,105,74,87),(68,104,75,86),(69,103,76,85),(70,102,77,98)], [(1,40,8,33),(2,41,9,34),(3,42,10,35),(4,29,11,36),(5,30,12,37),(6,31,13,38),(7,32,14,39),(15,47,22,54),(16,48,23,55),(17,49,24,56),(18,50,25,43),(19,51,26,44),(20,52,27,45),(21,53,28,46),(57,91,64,98),(58,92,65,85),(59,93,66,86),(60,94,67,87),(61,95,68,88),(62,96,69,89),(63,97,70,90),(71,102,78,109),(72,103,79,110),(73,104,80,111),(74,105,81,112),(75,106,82,99),(76,107,83,100),(77,108,84,101)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 18 | 0 | 26 |
| 0 | 0 | 8 | 0 | 11 | 18 |
| 0 | 0 | 0 | 0 | 26 | 21 |
| 0 | 0 | 0 | 0 | 8 | 21 |
| 2 | 13 | 0 | 0 | 0 | 0 |
| 13 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 25 | 21 | 20 |
| 0 | 0 | 0 | 19 | 15 | 15 |
| 0 | 0 | 28 | 27 | 5 | 5 |
| 0 | 0 | 1 | 3 | 5 | 5 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 8 | 8 |
| 0 | 0 | 0 | 1 | 20 | 20 |
| 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 | 22 | 19 | 20 | 0 |
| 0 | 0 | 5 | 7 | 25 | 4 |
| 0 | 0 | 0 | 0 | 5 | 2 |
| 0 | 0 | 0 | 0 | 16 | 24 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 22 | 0 | 6 |
| 0 | 0 | 13 | 18 | 7 | 22 |
| 0 | 0 | 0 | 0 | 24 | 16 |
| 0 | 0 | 0 | 0 | 13 | 5 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,18,8,0,0,0,0,18,0,0,0,0,0,0,11,26,8,0,0,26,18,21,21],[2,13,0,0,0,0,13,27,0,0,0,0,0,0,0,0,28,1,0,0,25,19,27,3,0,0,21,15,5,5,0,0,20,15,5,5],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,8,20,28,0,0,0,8,20,0,28],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,22,5,0,0,0,0,19,7,0,0,0,0,20,25,5,16,0,0,0,4,2,24],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,11,13,0,0,0,0,22,18,0,0,0,0,0,7,24,13,0,0,6,22,16,5] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | ··· | 4L | 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 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 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 | Q8 | D7 | D14 | D14 | D14 | D14 | 2+ (1+4) | Q8×D7 | D4⋊6D14 | D4⋊8D14 |
| kernel | C14.512+ (1+4) | C22⋊Dic14 | D7×C22⋊C4 | D14⋊Q8 | D14⋊2Q8 | C28.48D4 | C2×D14⋊C4 | D14⋊3Q8 | C7×C22⋊Q8 | C22×D7 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 2 | 1 | 4 | 3 | 6 | 9 | 3 | 3 | 2 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{51}2_+^{(1+4)} % in TeX
G:=Group("C14.51ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1087);
// by ID
G=gap.SmallGroup(448,1087);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,675,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=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=a^7*b^2*d>;
// generators/relations