metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C28)⋊15D4, (C2×D4)⋊43D14, (C2×Q8)⋊32D14, D14⋊6(C4○D4), C28⋊2D4⋊46C2, C28.265(C2×D4), (C22×C4)⋊29D14, C23⋊D14⋊34C2, D14⋊3Q8⋊47C2, (D4×C14)⋊47C22, C4⋊Dic7⋊80C22, (Q8×C14)⋊39C22, (C2×C14).314C24, (C2×C28).651C23, Dic7⋊C4⋊76C22, (C22×C28)⋊42C22, C7⋊8(C22.19C24), (C4×Dic7)⋊60C22, C14.164(C22×D4), C23.D7⋊65C22, D14⋊C4.159C22, C23.137(C22×D7), C22.323(C23×D7), C23.18D14⋊34C2, C23.21D14⋊38C2, (C22×C14).240C23, (C2×Dic7).293C23, (C22×D7).243C23, (C23×D7).115C22, (C22×Dic7).236C22, (C2×C4○D4)⋊6D7, (D7×C22×C4)⋊7C2, (C14×C4○D4)⋊6C2, (C4×C7⋊D4)⋊60C2, (C2×C4)⋊14(C7⋊D4), C2.103(D7×C4○D4), (C2×C14).80(C2×D4), C4.100(C2×C7⋊D4), C14.215(C2×C4○D4), (C2×C4×D7).263C22, C2.37(C22×C7⋊D4), C22.23(C2×C7⋊D4), (C2×C4).639(C22×D7), (C2×C7⋊D4).140C22, SmallGroup(448,1281)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1428 in 330 conjugacy classes, 115 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×24], C7, C2×C4 [×2], C2×C4 [×6], C2×C4 [×20], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×8], D7 [×4], C14, C14 [×2], C14 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×9], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24, Dic7 [×6], C28 [×4], C28 [×2], D14 [×4], D14 [×12], C2×C14, C2×C14 [×2], C2×C14 [×8], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, C4×D7 [×8], C2×Dic7 [×6], C2×Dic7 [×2], C7⋊D4 [×8], C2×C28 [×2], C2×C28 [×6], C2×C28 [×4], C7×D4 [×6], C7×Q8 [×2], C22×D7 [×2], C22×D7 [×6], C22×C14, C22×C14 [×2], C22.19C24, C4×Dic7 [×2], Dic7⋊C4 [×4], C4⋊Dic7 [×2], D14⋊C4 [×4], C23.D7 [×6], C2×C4×D7 [×4], C2×C4×D7 [×4], C22×Dic7, C2×C7⋊D4 [×4], C22×C28, C22×C28 [×2], D4×C14, D4×C14 [×2], Q8×C14, C7×C4○D4 [×4], C23×D7, C23.21D14, C4×C7⋊D4 [×4], C23.18D14 [×2], C23⋊D14 [×2], C28⋊2D4 [×2], D14⋊3Q8 [×2], D7×C22×C4, C14×C4○D4, (C2×C28)⋊15D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×4], C24, D14 [×7], C22×D4, C2×C4○D4 [×2], C7⋊D4 [×4], C22×D7 [×7], C22.19C24, C2×C7⋊D4 [×6], C23×D7, D7×C4○D4 [×2], C22×C7⋊D4, (C2×C28)⋊15D4
Generators and relations
G = < a,b,c,d | a2=b28=c4=d2=1, ab=ba, cac-1=ab14, ad=da, cbc-1=dbd=b13, dcd=c-1 >
►On 112 points
(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 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(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 95 56 69)(2 108 29 82)(3 93 30 67)(4 106 31 80)(5 91 32 65)(6 104 33 78)(7 89 34 63)(8 102 35 76)(9 87 36 61)(10 100 37 74)(11 85 38 59)(12 98 39 72)(13 111 40 57)(14 96 41 70)(15 109 42 83)(16 94 43 68)(17 107 44 81)(18 92 45 66)(19 105 46 79)(20 90 47 64)(21 103 48 77)(22 88 49 62)(23 101 50 75)(24 86 51 60)(25 99 52 73)(26 112 53 58)(27 97 54 71)(28 110 55 84)
(1 15)(2 28)(3 13)(4 26)(5 11)(6 24)(7 9)(8 22)(10 20)(12 18)(14 16)(17 27)(19 25)(21 23)(29 55)(30 40)(31 53)(32 38)(33 51)(34 36)(35 49)(37 47)(39 45)(41 43)(42 56)(44 54)(46 52)(48 50)(57 93)(58 106)(59 91)(60 104)(61 89)(62 102)(63 87)(64 100)(65 85)(66 98)(67 111)(68 96)(69 109)(70 94)(71 107)(72 92)(73 105)(74 90)(75 103)(76 88)(77 101)(78 86)(79 99)(80 112)(81 97)(82 110)(83 95)(84 108)
G:=sub<Sym(112)| (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)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,95,56,69)(2,108,29,82)(3,93,30,67)(4,106,31,80)(5,91,32,65)(6,104,33,78)(7,89,34,63)(8,102,35,76)(9,87,36,61)(10,100,37,74)(11,85,38,59)(12,98,39,72)(13,111,40,57)(14,96,41,70)(15,109,42,83)(16,94,43,68)(17,107,44,81)(18,92,45,66)(19,105,46,79)(20,90,47,64)(21,103,48,77)(22,88,49,62)(23,101,50,75)(24,86,51,60)(25,99,52,73)(26,112,53,58)(27,97,54,71)(28,110,55,84), (1,15)(2,28)(3,13)(4,26)(5,11)(6,24)(7,9)(8,22)(10,20)(12,18)(14,16)(17,27)(19,25)(21,23)(29,55)(30,40)(31,53)(32,38)(33,51)(34,36)(35,49)(37,47)(39,45)(41,43)(42,56)(44,54)(46,52)(48,50)(57,93)(58,106)(59,91)(60,104)(61,89)(62,102)(63,87)(64,100)(65,85)(66,98)(67,111)(68,96)(69,109)(70,94)(71,107)(72,92)(73,105)(74,90)(75,103)(76,88)(77,101)(78,86)(79,99)(80,112)(81,97)(82,110)(83,95)(84,108)>;
G:=Group( (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)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,95,56,69)(2,108,29,82)(3,93,30,67)(4,106,31,80)(5,91,32,65)(6,104,33,78)(7,89,34,63)(8,102,35,76)(9,87,36,61)(10,100,37,74)(11,85,38,59)(12,98,39,72)(13,111,40,57)(14,96,41,70)(15,109,42,83)(16,94,43,68)(17,107,44,81)(18,92,45,66)(19,105,46,79)(20,90,47,64)(21,103,48,77)(22,88,49,62)(23,101,50,75)(24,86,51,60)(25,99,52,73)(26,112,53,58)(27,97,54,71)(28,110,55,84), (1,15)(2,28)(3,13)(4,26)(5,11)(6,24)(7,9)(8,22)(10,20)(12,18)(14,16)(17,27)(19,25)(21,23)(29,55)(30,40)(31,53)(32,38)(33,51)(34,36)(35,49)(37,47)(39,45)(41,43)(42,56)(44,54)(46,52)(48,50)(57,93)(58,106)(59,91)(60,104)(61,89)(62,102)(63,87)(64,100)(65,85)(66,98)(67,111)(68,96)(69,109)(70,94)(71,107)(72,92)(73,105)(74,90)(75,103)(76,88)(77,101)(78,86)(79,99)(80,112)(81,97)(82,110)(83,95)(84,108) );
G=PermutationGroup([(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),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)], [(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)], [(1,95,56,69),(2,108,29,82),(3,93,30,67),(4,106,31,80),(5,91,32,65),(6,104,33,78),(7,89,34,63),(8,102,35,76),(9,87,36,61),(10,100,37,74),(11,85,38,59),(12,98,39,72),(13,111,40,57),(14,96,41,70),(15,109,42,83),(16,94,43,68),(17,107,44,81),(18,92,45,66),(19,105,46,79),(20,90,47,64),(21,103,48,77),(22,88,49,62),(23,101,50,75),(24,86,51,60),(25,99,52,73),(26,112,53,58),(27,97,54,71),(28,110,55,84)], [(1,15),(2,28),(3,13),(4,26),(5,11),(6,24),(7,9),(8,22),(10,20),(12,18),(14,16),(17,27),(19,25),(21,23),(29,55),(30,40),(31,53),(32,38),(33,51),(34,36),(35,49),(37,47),(39,45),(41,43),(42,56),(44,54),(46,52),(48,50),(57,93),(58,106),(59,91),(60,104),(61,89),(62,102),(63,87),(64,100),(65,85),(66,98),(67,111),(68,96),(69,109),(70,94),(71,107),(72,92),(73,105),(74,90),(75,103),(76,88),(77,101),(78,86),(79,99),(80,112),(81,97),(82,110),(83,95),(84,108)])
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 | 0 | 18 | 0 | 0 |
| 0 | 0 | 8 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 22 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 25 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 13 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,18,3,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[8,22,0,0,0,0,1,21,0,0,0,0,0,0,28,0,0,0,0,0,25,1,0,0,0,0,0,0,0,28,0,0,0,0,28,0],[1,13,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,4,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28] >;
88 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 | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 4 | 4 | 14 | 14 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | D7×C4○D4 |
| kernel | (C2×C28)⋊15D4 | C23.21D14 | C4×C7⋊D4 | C23.18D14 | C23⋊D14 | C28⋊2D4 | D14⋊3Q8 | D7×C22×C4 | C14×C4○D4 | C2×C28 | C2×C4○D4 | D14 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 3 | 8 | 9 | 9 | 3 | 24 | 12 |
In GAP, Magma, Sage, TeX
(C_2\times C_{28})\rtimes_{15}D_4 % in TeX
G:=Group("(C2xC28):15D4"); // GroupNames label
G:=SmallGroup(448,1281);
// by ID
G=gap.SmallGroup(448,1281);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,675,570,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^14,a*d=d*a,c*b*c^-1=d*b*d=b^13,d*c*d=c^-1>;
// generators/relations