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, (C23×D7).115C22, (C22×D7).243C23, (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, C22.23(C2×C7⋊D4), C2.37(C22×C7⋊D4), (C2×C4).639(C22×D7), (C2×C7⋊D4).140C22, SmallGroup(448,1281)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C28)⋊15D4
G = < a,b,c,d | a2=b28=c4=d2=1, ab=ba, cac-1=ab14, ad=da, cbc-1=dbd=b13, dcd=c-1 >
Subgroups: 1428 in 330 conjugacy classes, 115 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×D7, C22×C14, C22×C14, C22.19C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C2×C4×D7, C2×C4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C23×D7, C23.21D14, C4×C7⋊D4, C23.18D14, C23⋊D14, C28⋊2D4, D14⋊3Q8, D7×C22×C4, C14×C4○D4, (C2×C28)⋊15D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, C7⋊D4, C22×D7, C22.19C24, C2×C7⋊D4, C23×D7, D7×C4○D4, C22×C7⋊D4, (C2×C28)⋊15D4
►On 112 points
(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(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)
(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 79 110 39)(2 64 111 52)(3 77 112 37)(4 62 85 50)(5 75 86 35)(6 60 87 48)(7 73 88 33)(8 58 89 46)(9 71 90 31)(10 84 91 44)(11 69 92 29)(12 82 93 42)(13 67 94 55)(14 80 95 40)(15 65 96 53)(16 78 97 38)(17 63 98 51)(18 76 99 36)(19 61 100 49)(20 74 101 34)(21 59 102 47)(22 72 103 32)(23 57 104 45)(24 70 105 30)(25 83 106 43)(26 68 107 56)(27 81 108 41)(28 66 109 54)
(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 75)(30 60)(31 73)(32 58)(33 71)(34 84)(35 69)(36 82)(37 67)(38 80)(39 65)(40 78)(41 63)(42 76)(43 61)(44 74)(45 59)(46 72)(47 57)(48 70)(49 83)(50 68)(51 81)(52 66)(53 79)(54 64)(55 77)(56 62)(85 107)(86 92)(87 105)(88 90)(89 103)(91 101)(93 99)(94 112)(95 97)(96 110)(98 108)(100 106)(102 104)(109 111)
G:=sub<Sym(112)| (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(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), (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), (1,79,110,39)(2,64,111,52)(3,77,112,37)(4,62,85,50)(5,75,86,35)(6,60,87,48)(7,73,88,33)(8,58,89,46)(9,71,90,31)(10,84,91,44)(11,69,92,29)(12,82,93,42)(13,67,94,55)(14,80,95,40)(15,65,96,53)(16,78,97,38)(17,63,98,51)(18,76,99,36)(19,61,100,49)(20,74,101,34)(21,59,102,47)(22,72,103,32)(23,57,104,45)(24,70,105,30)(25,83,106,43)(26,68,107,56)(27,81,108,41)(28,66,109,54), (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)(29,75)(30,60)(31,73)(32,58)(33,71)(34,84)(35,69)(36,82)(37,67)(38,80)(39,65)(40,78)(41,63)(42,76)(43,61)(44,74)(45,59)(46,72)(47,57)(48,70)(49,83)(50,68)(51,81)(52,66)(53,79)(54,64)(55,77)(56,62)(85,107)(86,92)(87,105)(88,90)(89,103)(91,101)(93,99)(94,112)(95,97)(96,110)(98,108)(100,106)(102,104)(109,111)>;
G:=Group( (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(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), (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), (1,79,110,39)(2,64,111,52)(3,77,112,37)(4,62,85,50)(5,75,86,35)(6,60,87,48)(7,73,88,33)(8,58,89,46)(9,71,90,31)(10,84,91,44)(11,69,92,29)(12,82,93,42)(13,67,94,55)(14,80,95,40)(15,65,96,53)(16,78,97,38)(17,63,98,51)(18,76,99,36)(19,61,100,49)(20,74,101,34)(21,59,102,47)(22,72,103,32)(23,57,104,45)(24,70,105,30)(25,83,106,43)(26,68,107,56)(27,81,108,41)(28,66,109,54), (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)(29,75)(30,60)(31,73)(32,58)(33,71)(34,84)(35,69)(36,82)(37,67)(38,80)(39,65)(40,78)(41,63)(42,76)(43,61)(44,74)(45,59)(46,72)(47,57)(48,70)(49,83)(50,68)(51,81)(52,66)(53,79)(54,64)(55,77)(56,62)(85,107)(86,92)(87,105)(88,90)(89,103)(91,101)(93,99)(94,112)(95,97)(96,110)(98,108)(100,106)(102,104)(109,111) );
G=PermutationGroup([[(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(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)], [(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)], [(1,79,110,39),(2,64,111,52),(3,77,112,37),(4,62,85,50),(5,75,86,35),(6,60,87,48),(7,73,88,33),(8,58,89,46),(9,71,90,31),(10,84,91,44),(11,69,92,29),(12,82,93,42),(13,67,94,55),(14,80,95,40),(15,65,96,53),(16,78,97,38),(17,63,98,51),(18,76,99,36),(19,61,100,49),(20,74,101,34),(21,59,102,47),(22,72,103,32),(23,57,104,45),(24,70,105,30),(25,83,106,43),(26,68,107,56),(27,81,108,41),(28,66,109,54)], [(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),(29,75),(30,60),(31,73),(32,58),(33,71),(34,84),(35,69),(36,82),(37,67),(38,80),(39,65),(40,78),(41,63),(42,76),(43,61),(44,74),(45,59),(46,72),(47,57),(48,70),(49,83),(50,68),(51,81),(52,66),(53,79),(54,64),(55,77),(56,62),(85,107),(86,92),(87,105),(88,90),(89,103),(91,101),(93,99),(94,112),(95,97),(96,110),(98,108),(100,106),(102,104),(109,111)]])
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 |
Matrix representation of (C2×C28)⋊15D4 ►in 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] >;
(C2×C28)⋊15D4 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