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## G = D8⋊5Dic7order 448 = 26·7

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D8⋊5Dic7
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C7⋊C8 — Q8.Dic7 — D8⋊5Dic7
 Lower central C7 — C14 — C28 — D8⋊5Dic7
 Upper central C1 — C4 — C2×C4 — C4○D8

Generators and relations for D85Dic7
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, C2×C4, C2×C4, D4, D4, Q8, C14, C14, C42, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C4×C8, C4≀C2, C8.C4, C8○D4, C4○D8, C7⋊C8, C7⋊C8, C56, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C8○D8, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, C4×Dic7, C2×C56, C7×D8, C7×SD16, C7×Q16, C7×C4○D4, C8×Dic7, C56.C4, D42Dic7, Q8.Dic7, C7×C4○D8, D85Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, Dic7, D14, C4×D4, C2×Dic7, C22×D7, C8○D8, D4×D7, D42D7, C22×Dic7, D4×Dic7, D85Dic7

Smallest permutation representation of D85Dic7
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 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 8M 8N 14A 14B 14C 14D 14E 14F 14G ··· 14L 28A ··· 28F 28G 28H 28I 28J ··· 28O 56A ··· 56L order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 14 14 14 14 14 14 14 ··· 14 28 ··· 28 28 28 28 28 ··· 28 56 ··· 56 size 1 1 2 4 4 1 1 2 4 4 14 14 14 14 2 2 2 2 2 2 2 7 7 7 7 14 14 28 28 28 28 2 2 2 4 4 4 8 ··· 8 2 ··· 2 4 4 4 8 ··· 8 4 ··· 4

70 irreducible representations

 dim 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 C4 C4 C4 D4 D7 C4○D4 D14 Dic7 Dic7 Dic7 D14 C8○D8 D4×D7 D4⋊2D7 D8⋊5Dic7 kernel D8⋊5Dic7 C8×Dic7 C56.C4 D4⋊2Dic7 Q8.Dic7 C7×C4○D8 C7×D8 C7×SD16 C7×Q16 C7⋊C8 C4○D8 C2×C14 C2×C8 D8 SD16 Q16 C4○D4 C7 C4 C22 C1 # reps 1 1 1 2 2 1 2 4 2 2 3 2 3 3 6 3 6 8 3 3 12

Matrix representation of D85Dic7 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 44 0 0 0 36 18
,
 112 0 0 0 0 112 0 0 0 0 18 100 0 0 77 95
,
 34 1 0 0 53 88 0 0 0 0 112 0 0 0 32 1
,
 0 104 0 0 25 0 0 0 0 0 15 0 0 0 2 1
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] >;

D85Dic7 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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