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G = D28.8D4order 448 = 26·7

8th non-split extension by D28 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D28.8D4, D4.6D28, D145SD16, C4⋊C42D14, D14⋊C89C2, (C2×C8)⋊16D14, (C7×D4).1D4, C28.1(C2×D4), C4.2(C2×D28), C4.85(D4×D7), C14.D87C2, D4⋊C410D7, (C2×C56)⋊15C22, D142Q81C2, C72(C22⋊SD16), C2.11(D7×SD16), C14.20C22≀C2, (C2×D4).135D14, (C2×Dic7).20D4, C14.23(C2×SD16), (C22×D7).72D4, C22.174(D4×D7), C2.13(D8⋊D7), C14.31(C8⋊C22), (C2×C28).216C23, (D4×C14).37C22, (C2×D28).50C22, C2.23(C22⋊D28), (C2×Dic14)⋊13C22, (C2×D4×D7).5C2, (C2×C7⋊C8)⋊3C22, (C7×C4⋊C4)⋊4C22, (C2×D4.D7)⋊3C2, (C2×C56⋊C2)⋊14C2, (C2×C4×D7).9C22, (C7×D4⋊C4)⋊10C2, (C2×C14).229(C2×D4), (C2×C4).323(C22×D7), SmallGroup(448,310)

Series: Derived Chief Lower central Upper central

C1C2×C28 — D28.8D4
C1C7C14C28C2×C28C2×C4×D7C2×D4×D7 — D28.8D4
C7C14C2×C28 — D28.8D4
C1C22C2×C4D4⋊C4

Generators and relations for D28.8D4
 G = < a,b,c,d | a28=b2=c4=1, d2=a7, bab=a-1, cac-1=a15, ad=da, cbc-1=a7b, dbd-1=a21b, dcd-1=a7c-1 >

Subgroups: 1332 in 188 conjugacy classes, 45 normal (37 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×3], C22, C22 [×20], C7, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], D4 [×8], Q8 [×2], C23 [×11], D7 [×4], C14 [×3], C14 [×2], C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×D4 [×6], C2×Q8, C24, Dic7 [×2], C28 [×2], C28, D14 [×2], D14 [×14], C2×C14, C2×C14 [×4], C22⋊C8, D4⋊C4, D4⋊C4, C22⋊Q8, C2×SD16 [×2], C22×D4, C7⋊C8, C56, Dic14 [×2], C4×D7 [×2], D28 [×2], D28, C2×Dic7, C2×Dic7, C7⋊D4 [×4], C2×C28, C2×C28, C7×D4 [×2], C7×D4, C22×D7, C22×D7 [×9], C22×C14, C22⋊SD16, C56⋊C2 [×2], C2×C7⋊C8, C4⋊Dic7, D14⋊C4, D4.D7 [×2], C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, D4×D7 [×4], C2×C7⋊D4, D4×C14, C23×D7, C14.D8, D14⋊C8, C7×D4⋊C4, D142Q8, C2×C56⋊C2, C2×D4.D7, C2×D4×D7, D28.8D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D7, SD16 [×2], C2×D4 [×3], D14 [×3], C22≀C2, C2×SD16, C8⋊C22, D28 [×2], C22×D7, C22⋊SD16, C2×D28, D4×D7 [×2], C22⋊D28, D8⋊D7, D7×SD16, D28.8D4

Smallest permutation representation of D28.8D4
On 112 points
Generators in S112
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 28)(23 27)(24 26)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(49 56)(50 55)(51 54)(52 53)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(97 112)(98 111)(99 110)(100 109)(101 108)(102 107)(103 106)(104 105)
(1 105 82 53)(2 92 83 40)(3 107 84 55)(4 94 57 42)(5 109 58 29)(6 96 59 44)(7 111 60 31)(8 98 61 46)(9 85 62 33)(10 100 63 48)(11 87 64 35)(12 102 65 50)(13 89 66 37)(14 104 67 52)(15 91 68 39)(16 106 69 54)(17 93 70 41)(18 108 71 56)(19 95 72 43)(20 110 73 30)(21 97 74 45)(22 112 75 32)(23 99 76 47)(24 86 77 34)(25 101 78 49)(26 88 79 36)(27 103 80 51)(28 90 81 38)
(1 53 8 32 15 39 22 46)(2 54 9 33 16 40 23 47)(3 55 10 34 17 41 24 48)(4 56 11 35 18 42 25 49)(5 29 12 36 19 43 26 50)(6 30 13 37 20 44 27 51)(7 31 14 38 21 45 28 52)(57 108 64 87 71 94 78 101)(58 109 65 88 72 95 79 102)(59 110 66 89 73 96 80 103)(60 111 67 90 74 97 81 104)(61 112 68 91 75 98 82 105)(62 85 69 92 76 99 83 106)(63 86 70 93 77 100 84 107)

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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(49,56)(50,55)(51,54)(52,53)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105), (1,105,82,53)(2,92,83,40)(3,107,84,55)(4,94,57,42)(5,109,58,29)(6,96,59,44)(7,111,60,31)(8,98,61,46)(9,85,62,33)(10,100,63,48)(11,87,64,35)(12,102,65,50)(13,89,66,37)(14,104,67,52)(15,91,68,39)(16,106,69,54)(17,93,70,41)(18,108,71,56)(19,95,72,43)(20,110,73,30)(21,97,74,45)(22,112,75,32)(23,99,76,47)(24,86,77,34)(25,101,78,49)(26,88,79,36)(27,103,80,51)(28,90,81,38), (1,53,8,32,15,39,22,46)(2,54,9,33,16,40,23,47)(3,55,10,34,17,41,24,48)(4,56,11,35,18,42,25,49)(5,29,12,36,19,43,26,50)(6,30,13,37,20,44,27,51)(7,31,14,38,21,45,28,52)(57,108,64,87,71,94,78,101)(58,109,65,88,72,95,79,102)(59,110,66,89,73,96,80,103)(60,111,67,90,74,97,81,104)(61,112,68,91,75,98,82,105)(62,85,69,92,76,99,83,106)(63,86,70,93,77,100,84,107)>;

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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(49,56)(50,55)(51,54)(52,53)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105), (1,105,82,53)(2,92,83,40)(3,107,84,55)(4,94,57,42)(5,109,58,29)(6,96,59,44)(7,111,60,31)(8,98,61,46)(9,85,62,33)(10,100,63,48)(11,87,64,35)(12,102,65,50)(13,89,66,37)(14,104,67,52)(15,91,68,39)(16,106,69,54)(17,93,70,41)(18,108,71,56)(19,95,72,43)(20,110,73,30)(21,97,74,45)(22,112,75,32)(23,99,76,47)(24,86,77,34)(25,101,78,49)(26,88,79,36)(27,103,80,51)(28,90,81,38), (1,53,8,32,15,39,22,46)(2,54,9,33,16,40,23,47)(3,55,10,34,17,41,24,48)(4,56,11,35,18,42,25,49)(5,29,12,36,19,43,26,50)(6,30,13,37,20,44,27,51)(7,31,14,38,21,45,28,52)(57,108,64,87,71,94,78,101)(58,109,65,88,72,95,79,102)(59,110,66,89,73,96,80,103)(60,111,67,90,74,97,81,104)(61,112,68,91,75,98,82,105)(62,85,69,92,76,99,83,106)(63,86,70,93,77,100,84,107) );

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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,28),(23,27),(24,26),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39),(49,56),(50,55),(51,54),(52,53),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(97,112),(98,111),(99,110),(100,109),(101,108),(102,107),(103,106),(104,105)], [(1,105,82,53),(2,92,83,40),(3,107,84,55),(4,94,57,42),(5,109,58,29),(6,96,59,44),(7,111,60,31),(8,98,61,46),(9,85,62,33),(10,100,63,48),(11,87,64,35),(12,102,65,50),(13,89,66,37),(14,104,67,52),(15,91,68,39),(16,106,69,54),(17,93,70,41),(18,108,71,56),(19,95,72,43),(20,110,73,30),(21,97,74,45),(22,112,75,32),(23,99,76,47),(24,86,77,34),(25,101,78,49),(26,88,79,36),(27,103,80,51),(28,90,81,38)], [(1,53,8,32,15,39,22,46),(2,54,9,33,16,40,23,47),(3,55,10,34,17,41,24,48),(4,56,11,35,18,42,25,49),(5,29,12,36,19,43,26,50),(6,30,13,37,20,44,27,51),(7,31,14,38,21,45,28,52),(57,108,64,87,71,94,78,101),(58,109,65,88,72,95,79,102),(59,110,66,89,73,96,80,103),(60,111,67,90,74,97,81,104),(61,112,68,91,75,98,82,105),(62,85,69,92,76,99,83,106),(63,86,70,93,77,100,84,107)])

61 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E7A7B7C8A8B8C8D14A···14I14J···14O28A···28F28G···28L56A···56L
order122222222244444777888814···1414···1428···2828···2856···56
size1111441414282822828562224428282···28···84···48···84···4

61 irreducible representations

dim11111111222222222244444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D7SD16D14D14D14D28C8⋊C22D4×D7D4×D7D8⋊D7D7×SD16
kernelD28.8D4C14.D8D14⋊C8C7×D4⋊C4D142Q8C2×C56⋊C2C2×D4.D7C2×D4×D7D28C2×Dic7C7×D4C22×D7D4⋊C4D14C4⋊C4C2×C8C2×D4D4C14C4C22C2C2
# reps111111112121343331213366

Matrix representation of D28.8D4 in GL4(𝔽113) generated by

1127200
91100
008979
00100
,
1000
2211200
00079
001030
,
268100
608700
001767
008096
,
08100
608700
009646
003317
G:=sub<GL(4,GF(113))| [112,91,0,0,72,1,0,0,0,0,89,10,0,0,79,0],[1,22,0,0,0,112,0,0,0,0,0,103,0,0,79,0],[26,60,0,0,81,87,0,0,0,0,17,80,0,0,67,96],[0,60,0,0,81,87,0,0,0,0,96,33,0,0,46,17] >;

D28.8D4 in GAP, Magma, Sage, TeX

D_{28}._8D_4
% in TeX

G:=Group("D28.8D4");
// GroupNames label

G:=SmallGroup(448,310);
// by ID

G=gap.SmallGroup(448,310);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,135,268,570,297,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^28=b^2=c^4=1,d^2=a^7,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^7*b,d*b*d^-1=a^21*b,d*c*d^-1=a^7*c^-1>;
// generators/relations

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