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G = D8:5Dic7order 448 = 26·7

The semidirect product of D8 and Dic7 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8:5Dic7, Q16:5Dic7, SD16:3Dic7, (C7xD8):3C4, C7:5(C8oD8), C7:C8.23D4, (C7xQ16):3C4, C4oD8.5D7, C56.20(C2xC4), (C8xDic7):2C2, (C7xSD16):4C4, C14.99(C4xD4), C4.217(D4xD7), C56.C4:8C2, C4oD4.22D14, (C2xC8).254D14, C28.376(C2xD4), Q8.Dic7:3C2, D4.3(C2xDic7), Q8.3(C2xDic7), C8.11(C2xDic7), C2.16(D4xDic7), D4:2Dic7:4C2, C28.77(C22xC4), (C2xC56).44C22, C4.7(C22xDic7), (C2xC28).467C23, C22.3(D4:2D7), C4.Dic7.22C22, (C4xDic7).246C22, (C7xC4oD8).2C2, (C7xD4).10(C2xC4), (C7xQ8).10(C2xC4), (C2xC7:C8).280C22, (C7xC4oD4).9C22, (C2xC14).11(C4oD4), (C2xC4).554(C22xD7), SmallGroup(448,730)

Series: Derived Chief Lower central Upper central

C1C28 — D8:5Dic7
C1C7C14C28C2xC28C2xC7:C8Q8.Dic7 — D8:5Dic7
C7C14C28 — D8:5Dic7
C1C4C2xC4C4oD8

Generators and relations for D8:5Dic7
 G = < a,b,c,d | a8=b2=c14=1, d2=c7, bab=a-1, ac=ca, ad=da, cbc-1=a4b, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 340 in 106 conjugacy classes, 53 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2xC4, C2xC4, D4, D4, Q8, C14, C14, C42, C2xC8, C2xC8, M4(2), D8, SD16, Q16, C4oD4, Dic7, C28, C28, C2xC14, C2xC14, C4xC8, C4wrC2, C8.C4, C8oD4, C4oD8, C7:C8, C7:C8, C56, C2xDic7, C2xC28, C2xC28, C7xD4, C7xD4, C7xQ8, C8oD8, C2xC7:C8, C2xC7:C8, C4.Dic7, C4.Dic7, C4xDic7, C2xC56, C7xD8, C7xSD16, C7xQ16, C7xC4oD4, C8xDic7, C56.C4, D4:2Dic7, Q8.Dic7, C7xC4oD8, D8:5Dic7
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D7, C22xC4, C2xD4, C4oD4, Dic7, D14, C4xD4, C2xDic7, C22xD7, C8oD8, D4xD7, D4:2D7, C22xDic7, D4xDic7, D8:5Dic7

Smallest permutation representation of D8:5Dic7
On 112 points
Generators in S112
(1 34 12 55 26 39 20 48)(2 35 13 56 27 40 21 49)(3 29 14 50 28 41 15 43)(4 30 8 51 22 42 16 44)(5 31 9 52 23 36 17 45)(6 32 10 53 24 37 18 46)(7 33 11 54 25 38 19 47)(57 86 72 104 64 93 79 111)(58 87 73 105 65 94 80 112)(59 88 74 106 66 95 81 99)(60 89 75 107 67 96 82 100)(61 90 76 108 68 97 83 101)(62 91 77 109 69 98 84 102)(63 92 78 110 70 85 71 103)
(1 106)(2 100)(3 108)(4 102)(5 110)(6 104)(7 112)(8 98)(9 92)(10 86)(11 94)(12 88)(13 96)(14 90)(15 97)(16 91)(17 85)(18 93)(19 87)(20 95)(21 89)(22 109)(23 103)(24 111)(25 105)(26 99)(27 107)(28 101)(29 76)(30 84)(31 78)(32 72)(33 80)(34 74)(35 82)(36 71)(37 79)(38 73)(39 81)(40 75)(41 83)(42 77)(43 68)(44 62)(45 70)(46 64)(47 58)(48 66)(49 60)(50 61)(51 69)(52 63)(53 57)(54 65)(55 59)(56 67)
(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)
(2 7)(3 6)(4 5)(8 9)(10 14)(11 13)(15 18)(16 17)(19 21)(22 23)(24 28)(25 27)(29 32)(30 31)(33 35)(36 42)(37 41)(38 40)(43 46)(44 45)(47 49)(50 53)(51 52)(54 56)(57 76 64 83)(58 75 65 82)(59 74 66 81)(60 73 67 80)(61 72 68 79)(62 71 69 78)(63 84 70 77)(85 109 92 102)(86 108 93 101)(87 107 94 100)(88 106 95 99)(89 105 96 112)(90 104 97 111)(91 103 98 110)

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

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

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

70 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I7A7B7C8A8B8C8D8E8F8G8H8I8J8K8L8M8N14A14B14C14D14E14F14G···14L28A···28F28G28H28I28J···28O56A···56L
order122224444444447778888888888888814141414141414···1428···2828282828···2856···56
size112441124414141414222222277771414282828282224448···82···24448···84···4

70 irreducible representations

dim111111111222222222444
type+++++++++---++-
imageC1C2C2C2C2C2C4C4C4D4D7C4oD4D14Dic7Dic7Dic7D14C8oD8D4xD7D4:2D7D8:5Dic7
kernelD8:5Dic7C8xDic7C56.C4D4:2Dic7Q8.Dic7C7xC4oD8C7xD8C7xSD16C7xQ16C7:C8C4oD8C2xC14C2xC8D8SD16Q16C4oD4C7C4C22C1
# reps1112212422323363683312

Matrix representation of D8:5Dic7 in GL4(F113) generated by

112000
011200
00440
003618
,
112000
011200
0018100
007795
,
34100
538800
001120
00321
,
010400
25000
00150
0021
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,44,36,0,0,0,18],[112,0,0,0,0,112,0,0,0,0,18,77,0,0,100,95],[34,53,0,0,1,88,0,0,0,0,112,32,0,0,0,1],[0,25,0,0,104,0,0,0,0,0,15,2,0,0,0,1] >;

D8:5Dic7 in GAP, Magma, Sage, TeX

D_8\rtimes_5{\rm Dic}_7
% in TeX

G:=Group("D8:5Dic7");
// GroupNames label

G:=SmallGroup(448,730);
// by ID

G=gap.SmallGroup(448,730);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,219,136,851,438,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=c^7,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^4*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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