metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊23D4, C42⋊14D14, C14.1032+ 1+4, C4⋊C4⋊47D14, (C4×D4)⋊12D7, (D4×C28)⋊14C2, (C4×D28)⋊28C2, C7⋊2(D4⋊5D4), C4.139(D4×D7), D14⋊D4⋊7C2, C28⋊7D4⋊18C2, (C4×C28)⋊19C22, C22⋊C4⋊46D14, D14.14(C2×D4), C28.345(C2×D4), (C22×D28)⋊9C2, (C22×C4)⋊12D14, C23⋊D14⋊20C2, D14⋊C4⋊30C22, D14.5D4⋊7C2, (C2×D4).213D14, C4.D28⋊16C2, C22⋊3(C4○D28), (C2×C14).94C24, Dic7⋊C4⋊3C22, C4⋊Dic7⋊59C22, C14.49(C22×D4), C28.48D4⋊10C2, (C2×C28).782C23, (C22×C28)⋊16C22, C2.15(D4⋊8D14), C23.94(C22×D7), (C2×Dic14)⋊53C22, (D4×C14).305C22, (C2×D28).210C22, (C2×Dic7).40C23, (C23×D7).39C22, C22.119(C23×D7), C23.D7.11C22, (C22×C14).164C23, (C22×D7).172C23, C2.22(C2×D4×D7), (C2×C4○D28)⋊7C2, (C2×C4×D7)⋊48C22, (C2×C14)⋊2(C4○D4), (C7×C4⋊C4)⋊59C22, (D7×C22⋊C4)⋊28C2, C2.45(C2×C4○D28), C14.41(C2×C4○D4), (C2×C7⋊D4)⋊3C22, (C7×C22⋊C4)⋊56C22, (C2×C4).158(C22×D7), SmallGroup(448,1003)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28⋊23D4
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a14b, bd=db, dcd=c-1 >
Subgroups: 1908 in 334 conjugacy classes, 107 normal (51 characteristic)
C1, C2, C2, C4, 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×D4, C2×Q8, C4○D4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, D4⋊5D4, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C2×D28, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C4×D28, C4.D28, D7×C22⋊C4, D14⋊D4, D14.5D4, C28.48D4, C28⋊7D4, C23⋊D14, D4×C28, C22×D28, C2×C4○D28, D28⋊23D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2+ 1+4, C22×D7, D4⋊5D4, C4○D28, D4×D7, C23×D7, C2×C4○D28, C2×D4×D7, D4⋊8D14, D28⋊23D4
►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 91)(2 90)(3 89)(4 88)(5 87)(6 86)(7 85)(8 112)(9 111)(10 110)(11 109)(12 108)(13 107)(14 106)(15 105)(16 104)(17 103)(18 102)(19 101)(20 100)(21 99)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(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)
(1 71 85 30)(2 72 86 31)(3 73 87 32)(4 74 88 33)(5 75 89 34)(6 76 90 35)(7 77 91 36)(8 78 92 37)(9 79 93 38)(10 80 94 39)(11 81 95 40)(12 82 96 41)(13 83 97 42)(14 84 98 43)(15 57 99 44)(16 58 100 45)(17 59 101 46)(18 60 102 47)(19 61 103 48)(20 62 104 49)(21 63 105 50)(22 64 106 51)(23 65 107 52)(24 66 108 53)(25 67 109 54)(26 68 110 55)(27 69 111 56)(28 70 112 29)
(29 70)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 84)(44 57)(45 58)(46 59)(47 60)(48 61)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)
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), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,112)(9,111)(10,110)(11,109)(12,108)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(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), (1,71,85,30)(2,72,86,31)(3,73,87,32)(4,74,88,33)(5,75,89,34)(6,76,90,35)(7,77,91,36)(8,78,92,37)(9,79,93,38)(10,80,94,39)(11,81,95,40)(12,82,96,41)(13,83,97,42)(14,84,98,43)(15,57,99,44)(16,58,100,45)(17,59,101,46)(18,60,102,47)(19,61,103,48)(20,62,104,49)(21,63,105,50)(22,64,106,51)(23,65,107,52)(24,66,108,53)(25,67,109,54)(26,68,110,55)(27,69,111,56)(28,70,112,29), (29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,57)(45,58)(46,59)(47,60)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)>;
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), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,112)(9,111)(10,110)(11,109)(12,108)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(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), (1,71,85,30)(2,72,86,31)(3,73,87,32)(4,74,88,33)(5,75,89,34)(6,76,90,35)(7,77,91,36)(8,78,92,37)(9,79,93,38)(10,80,94,39)(11,81,95,40)(12,82,96,41)(13,83,97,42)(14,84,98,43)(15,57,99,44)(16,58,100,45)(17,59,101,46)(18,60,102,47)(19,61,103,48)(20,62,104,49)(21,63,105,50)(22,64,106,51)(23,65,107,52)(24,66,108,53)(25,67,109,54)(26,68,110,55)(27,69,111,56)(28,70,112,29), (29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,57)(45,58)(46,59)(47,60)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69) );
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)], [(1,91),(2,90),(3,89),(4,88),(5,87),(6,86),(7,85),(8,112),(9,111),(10,110),(11,109),(12,108),(13,107),(14,106),(15,105),(16,104),(17,103),(18,102),(19,101),(20,100),(21,99),(22,98),(23,97),(24,96),(25,95),(26,94),(27,93),(28,92),(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)], [(1,71,85,30),(2,72,86,31),(3,73,87,32),(4,74,88,33),(5,75,89,34),(6,76,90,35),(7,77,91,36),(8,78,92,37),(9,79,93,38),(10,80,94,39),(11,81,95,40),(12,82,96,41),(13,83,97,42),(14,84,98,43),(15,57,99,44),(16,58,100,45),(17,59,101,46),(18,60,102,47),(19,61,103,48),(20,62,104,49),(21,63,105,50),(22,64,106,51),(23,65,107,52),(24,66,108,53),(25,67,109,54),(26,68,110,55),(27,69,111,56),(28,70,112,29)], [(29,70),(30,71),(31,72),(32,73),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,81),(41,82),(42,83),(43,84),(44,57),(45,58),(46,59),(47,60),(48,61),(49,62),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 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 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | D14 | C4○D28 | 2+ 1+4 | D4×D7 | D4⋊8D14 |
| kernel | D28⋊23D4 | C4×D28 | C4.D28 | D7×C22⋊C4 | D14⋊D4 | D14.5D4 | C28.48D4 | C28⋊7D4 | C23⋊D14 | D4×C28 | C22×D28 | C2×C4○D28 | D28 | C4×D4 | C2×C14 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 3 | 4 | 3 | 6 | 3 | 6 | 3 | 24 | 1 | 6 | 6 |
Matrix representation of D28⋊23D4 ►in GL4(𝔽29) generated by
| 9 | 4 | 0 | 0 |
| 25 | 8 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 10 | 7 | 0 | 0 |
| 19 | 19 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 9 | 14 | 0 | 0 |
| 15 | 20 | 0 | 0 |
| 0 | 0 | 1 | 27 |
| 0 | 0 | 1 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 28 |
G:=sub<GL(4,GF(29))| [9,25,0,0,4,8,0,0,0,0,28,0,0,0,0,28],[10,19,0,0,7,19,0,0,0,0,28,0,0,0,0,28],[9,15,0,0,14,20,0,0,0,0,1,1,0,0,27,28],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,28] >;
D28⋊23D4 in GAP, Magma, Sage, TeX
D_{28}\rtimes_{23}D_4 % in TeX
G:=Group("D28:23D4"); // GroupNames label
G:=SmallGroup(448,1003);
// by ID
G=gap.SmallGroup(448,1003);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,100,675,570,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^14*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations