metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊7D4, C22⋊1D28, C23.24D14, (C2×C14)⋊5D4, D14⋊C4⋊3C2, (C2×D28)⋊6C2, C7⋊3(C4⋊D4), C4⋊3(C7⋊D4), C4⋊Dic7⋊9C2, (C22×C4)⋊4D7, (C22×C28)⋊6C2, C2.17(C2×D28), (C2×C4).85D14, C14.43(C2×D4), C2.19(C4○D28), C14.19(C4○D4), (C2×C14).48C23, (C2×C28).94C22, C22.56(C22×D7), (C22×C14).40C22, (C2×Dic7).16C22, (C22×D7).10C22, (C2×C7⋊D4)⋊3C2, C2.7(C2×C7⋊D4), SmallGroup(224,125)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊7D4
G = < a,b,c | a28=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 430 in 94 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C4⋊D4, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C4⋊Dic7, D14⋊C4, C2×D28, C2×C7⋊D4, C22×C28, C28⋊7D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, D28, C7⋊D4, C22×D7, C2×D28, C4○D28, C2×C7⋊D4, C28⋊7D4
►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 81 91 40)(2 80 92 39)(3 79 93 38)(4 78 94 37)(5 77 95 36)(6 76 96 35)(7 75 97 34)(8 74 98 33)(9 73 99 32)(10 72 100 31)(11 71 101 30)(12 70 102 29)(13 69 103 56)(14 68 104 55)(15 67 105 54)(16 66 106 53)(17 65 107 52)(18 64 108 51)(19 63 109 50)(20 62 110 49)(21 61 111 48)(22 60 112 47)(23 59 85 46)(24 58 86 45)(25 57 87 44)(26 84 88 43)(27 83 89 42)(28 82 90 41)
(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(29 64)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 57)(37 84)(38 83)(39 82)(40 81)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(98 112)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)
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), (1,81,91,40)(2,80,92,39)(3,79,93,38)(4,78,94,37)(5,77,95,36)(6,76,96,35)(7,75,97,34)(8,74,98,33)(9,73,99,32)(10,72,100,31)(11,71,101,30)(12,70,102,29)(13,69,103,56)(14,68,104,55)(15,67,105,54)(16,66,106,53)(17,65,107,52)(18,64,108,51)(19,63,109,50)(20,62,110,49)(21,61,111,48)(22,60,112,47)(23,59,85,46)(24,58,86,45)(25,57,87,44)(26,84,88,43)(27,83,89,42)(28,82,90,41), (2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)>;
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), (1,81,91,40)(2,80,92,39)(3,79,93,38)(4,78,94,37)(5,77,95,36)(6,76,96,35)(7,75,97,34)(8,74,98,33)(9,73,99,32)(10,72,100,31)(11,71,101,30)(12,70,102,29)(13,69,103,56)(14,68,104,55)(15,67,105,54)(16,66,106,53)(17,65,107,52)(18,64,108,51)(19,63,109,50)(20,62,110,49)(21,61,111,48)(22,60,112,47)(23,59,85,46)(24,58,86,45)(25,57,87,44)(26,84,88,43)(27,83,89,42)(28,82,90,41), (2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106) );
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)], [(1,81,91,40),(2,80,92,39),(3,79,93,38),(4,78,94,37),(5,77,95,36),(6,76,96,35),(7,75,97,34),(8,74,98,33),(9,73,99,32),(10,72,100,31),(11,71,101,30),(12,70,102,29),(13,69,103,56),(14,68,104,55),(15,67,105,54),(16,66,106,53),(17,65,107,52),(18,64,108,51),(19,63,109,50),(20,62,110,49),(21,61,111,48),(22,60,112,47),(23,59,85,46),(24,58,86,45),(25,57,87,44),(26,84,88,43),(27,83,89,42),(28,82,90,41)], [(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(29,64),(30,63),(31,62),(32,61),(33,60),(34,59),(35,58),(36,57),(37,84),(38,83),(39,82),(40,81),(41,80),(42,79),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92),(98,112),(99,111),(100,110),(101,109),(102,108),(103,107),(104,106)]])
C28⋊7D4 is a maximal subgroup of
C14.C4≀C2 C22.2D56 D28⋊13D4 D28⋊14D4 C23.38D28 C22.D56 C23.13D28 Dic14⋊14D4 (C2×C14).40D8 C4⋊C4.228D14 C4⋊C4.236D14 C7⋊C8⋊22D4 C4⋊D4⋊D7 C7⋊C8⋊24D4 C7⋊C8⋊6D4 C56⋊30D4 C56⋊29D4 C56⋊2D4 C56⋊3D4 (C2×C14)⋊8D8 (C7×Q8)⋊13D4 (C7×D4)⋊14D4 C42.276D14 C42.277D14 C24.27D14 C23⋊3D28 C24.30D14 C14.2- 1+4 C14.2+ 1+4 C14.112+ 1+4 C14.62- 1+4 C42.95D14 C42.97D14 C42.99D14 C42.100D14 C42.104D14 D4×D28 D28⋊23D4 Dic14⋊23D4 Dic14⋊24D4 D4⋊5D28 D4⋊6D28 C42⋊17D14 C42.116D14 C42.117D14 C42.119D14 C28⋊(C4○D4) C14.682- 1+4 D7×C4⋊D4 C14.372+ 1+4 C14.382+ 1+4 C14.472+ 1+4 C14.482+ 1+4 C22⋊Q8⋊25D7 C4⋊C4⋊26D14 C14.172- 1+4 C14.242- 1+4 C14.562+ 1+4 C14.572+ 1+4 C14.262- 1+4 C14.612+ 1+4 C14.662+ 1+4 C14.682+ 1+4 C14.862- 1+4 C24.72D14 D4×C7⋊D4 C14.452- 1+4 C14.1452+ 1+4 C14.1462+ 1+4 C14.1082- 1+4 C14.1482+ 1+4
C28⋊7D4 is a maximal quotient of
C28⋊4(C4⋊C4) (C2×C4)⋊6D28 (C2×C42)⋊D7 C24.10D14 C23⋊2D28 C23.16D28 (C2×C4).44D28 (C2×C4)⋊3D28 (C2×C4).45D28 C28⋊7D8 D4.1D28 D4.2D28 Q8⋊D28 Q8.1D28 C28⋊7Q16 C56⋊30D4 C56⋊29D4 C56.82D4 C56⋊2D4 C56⋊3D4 C56.4D4 D4.3D28 D4.4D28 D4.5D28 C23.27D28 C23.28D28
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | D28 | C4○D28 |
| kernel | C28⋊7D4 | C4⋊Dic7 | D14⋊C4 | C2×D28 | C2×C7⋊D4 | C22×C28 | C28 | C2×C14 | C22×C4 | C14 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 3 | 2 | 6 | 3 | 12 | 12 | 12 |
Matrix representation of C28⋊7D4 ►in GL4(𝔽29) generated by
| 25 | 5 | 0 | 0 |
| 22 | 23 | 0 | 0 |
| 0 | 0 | 27 | 24 |
| 0 | 0 | 5 | 12 |
| 0 | 26 | 0 | 0 |
| 19 | 0 | 0 | 0 |
| 0 | 0 | 5 | 16 |
| 0 | 0 | 2 | 24 |
| 0 | 26 | 0 | 0 |
| 19 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 3 | 28 |
G:=sub<GL(4,GF(29))| [25,22,0,0,5,23,0,0,0,0,27,5,0,0,24,12],[0,19,0,0,26,0,0,0,0,0,5,2,0,0,16,24],[0,19,0,0,26,0,0,0,0,0,1,3,0,0,0,28] >;
C28⋊7D4 in GAP, Magma, Sage, TeX
C_{28}\rtimes_7D_4 % in TeX
G:=Group("C28:7D4"); // GroupNames label
G:=SmallGroup(224,125);
// by ID
G=gap.SmallGroup(224,125);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,103,218,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations