metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊23D4, C42⋊14D14, C14.1032+ (1+4), C4⋊C4⋊47D14, (C4×D4)⋊12D7, (C4×D28)⋊28C2, (D4×C28)⋊14C2, 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, C14.41(C2×C4○D4), C2.45(C2×C4○D28), (C2×C7⋊D4)⋊3C22, (C7×C22⋊C4)⋊56C22, (C2×C4).158(C22×D7), SmallGroup(448,1003)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1908 in 334 conjugacy classes, 107 normal (51 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×27], C7, C2×C4 [×5], C2×C4 [×14], D4 [×18], Q8 [×2], C23 [×2], C23 [×14], D7 [×6], C14 [×3], C14 [×3], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic7 [×4], C28 [×2], C28 [×4], D14 [×4], D14 [×18], C2×C14, C2×C14 [×2], C2×C14 [×5], C2×C22⋊C4 [×2], C4×D4, C4×D4, C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic14 [×2], C4×D7 [×6], D28 [×4], D28 [×6], C2×Dic7 [×4], C7⋊D4 [×6], C2×C28 [×5], C2×C28 [×4], C7×D4 [×2], C22×D7 [×4], C22×D7 [×10], C22×C14 [×2], D4⋊5D4, Dic7⋊C4 [×2], C4⋊Dic7, D14⋊C4 [×8], C23.D7 [×2], C4×C28, C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14, C2×C4×D7 [×4], C2×D28 [×2], C2×D28 [×2], C2×D28 [×4], C4○D28 [×4], C2×C7⋊D4 [×4], C22×C28 [×2], D4×C14, C23×D7 [×2], C4×D28, C4.D28, D7×C22⋊C4 [×2], D14⋊D4 [×2], D14.5D4 [×2], C28.48D4, C28⋊7D4, C23⋊D14 [×2], D4×C28, C22×D28, C2×C4○D28, D28⋊23D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×2], C24, D14 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C22×D7 [×7], D4⋊5D4, C4○D28 [×2], D4×D7 [×2], C23×D7, C2×C4○D28, C2×D4×D7, D4⋊8D14, D28⋊23D4
Generators and relations
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 >
►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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(57 110)(58 109)(59 108)(60 107)(61 106)(62 105)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)(73 94)(74 93)(75 92)(76 91)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 112)(84 111)
(1 58 36 89)(2 59 37 90)(3 60 38 91)(4 61 39 92)(5 62 40 93)(6 63 41 94)(7 64 42 95)(8 65 43 96)(9 66 44 97)(10 67 45 98)(11 68 46 99)(12 69 47 100)(13 70 48 101)(14 71 49 102)(15 72 50 103)(16 73 51 104)(17 74 52 105)(18 75 53 106)(19 76 54 107)(20 77 55 108)(21 78 56 109)(22 79 29 110)(23 80 30 111)(24 81 31 112)(25 82 32 85)(26 83 33 86)(27 84 34 87)(28 57 35 88)
(57 88)(58 89)(59 90)(60 91)(61 92)(62 93)(63 94)(64 95)(65 96)(66 97)(67 98)(68 99)(69 100)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 112)(82 85)(83 86)(84 87)
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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,112)(84,111), (1,58,36,89)(2,59,37,90)(3,60,38,91)(4,61,39,92)(5,62,40,93)(6,63,41,94)(7,64,42,95)(8,65,43,96)(9,66,44,97)(10,67,45,98)(11,68,46,99)(12,69,47,100)(13,70,48,101)(14,71,49,102)(15,72,50,103)(16,73,51,104)(17,74,52,105)(18,75,53,106)(19,76,54,107)(20,77,55,108)(21,78,56,109)(22,79,29,110)(23,80,30,111)(24,81,31,112)(25,82,32,85)(26,83,33,86)(27,84,34,87)(28,57,35,88), (57,88)(58,89)(59,90)(60,91)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,85)(83,86)(84,87)>;
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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,112)(84,111), (1,58,36,89)(2,59,37,90)(3,60,38,91)(4,61,39,92)(5,62,40,93)(6,63,41,94)(7,64,42,95)(8,65,43,96)(9,66,44,97)(10,67,45,98)(11,68,46,99)(12,69,47,100)(13,70,48,101)(14,71,49,102)(15,72,50,103)(16,73,51,104)(17,74,52,105)(18,75,53,106)(19,76,54,107)(20,77,55,108)(21,78,56,109)(22,79,29,110)(23,80,30,111)(24,81,31,112)(25,82,32,85)(26,83,33,86)(27,84,34,87)(28,57,35,88), (57,88)(58,89)(59,90)(60,91)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,85)(83,86)(84,87) );
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)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,30),(14,29),(15,56),(16,55),(17,54),(18,53),(19,52),(20,51),(21,50),(22,49),(23,48),(24,47),(25,46),(26,45),(27,44),(28,43),(57,110),(58,109),(59,108),(60,107),(61,106),(62,105),(63,104),(64,103),(65,102),(66,101),(67,100),(68,99),(69,98),(70,97),(71,96),(72,95),(73,94),(74,93),(75,92),(76,91),(77,90),(78,89),(79,88),(80,87),(81,86),(82,85),(83,112),(84,111)], [(1,58,36,89),(2,59,37,90),(3,60,38,91),(4,61,39,92),(5,62,40,93),(6,63,41,94),(7,64,42,95),(8,65,43,96),(9,66,44,97),(10,67,45,98),(11,68,46,99),(12,69,47,100),(13,70,48,101),(14,71,49,102),(15,72,50,103),(16,73,51,104),(17,74,52,105),(18,75,53,106),(19,76,54,107),(20,77,55,108),(21,78,56,109),(22,79,29,110),(23,80,30,111),(24,81,31,112),(25,82,32,85),(26,83,33,86),(27,84,34,87),(28,57,35,88)], [(57,88),(58,89),(59,90),(60,91),(61,92),(62,93),(63,94),(64,95),(65,96),(66,97),(67,98),(68,99),(69,100),(70,101),(71,102),(72,103),(73,104),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111),(81,112),(82,85),(83,86),(84,87)])
Matrix representation ►G ⊆ 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] >;
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 |
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