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## G = D8⋊2Dic7order 448 = 26·7

### 2nd semidirect product of D8 and Dic7 acting via Dic7/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — D8⋊2Dic7
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C56 — C8⋊Dic7 — D8⋊2Dic7
 Lower central C7 — C14 — C28 — C56 — D8⋊2Dic7
 Upper central C1 — C2 — C2×C4 — C2×C8 — C4○D8

Generators and relations for D82Dic7
G = < a,b,c,d | a8=b2=c14=1, d2=c7, bab=a-1, ac=ca, dad-1=a3, cbc-1=a4b, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 244 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C14, C14, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C4.Q8, M5(2), C4○D8, C56, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×Q8, D82C4, C7⋊C16, C4⋊Dic7, C2×C56, C7×D8, C7×SD16, C7×Q16, C7×C4○D4, C28.C8, C8⋊Dic7, C7×C4○D8, D82Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, Dic7, D14, D4⋊C4, C2×Dic7, C7⋊D4, D82C4, D4⋊D7, D4.D7, C23.D7, D4⋊Dic7, D82Dic7

Smallest permutation representation of D82Dic7
On 112 points
Generators in S112
(1 50 27 40 13 43 21 34)(2 51 28 41 14 44 15 35)(3 52 22 42 8 45 16 29)(4 53 23 36 9 46 17 30)(5 54 24 37 10 47 18 31)(6 55 25 38 11 48 19 32)(7 56 26 39 12 49 20 33)(57 98 84 103 64 91 77 110)(58 85 71 104 65 92 78 111)(59 86 72 105 66 93 79 112)(60 87 73 106 67 94 80 99)(61 88 74 107 68 95 81 100)(62 89 75 108 69 96 82 101)(63 90 76 109 70 97 83 102)
(1 100)(2 108)(3 102)(4 110)(5 104)(6 112)(7 106)(8 109)(9 103)(10 111)(11 105)(12 99)(13 107)(14 101)(15 96)(16 90)(17 98)(18 92)(19 86)(20 94)(21 88)(22 97)(23 91)(24 85)(25 93)(26 87)(27 95)(28 89)(29 63)(30 57)(31 65)(32 59)(33 67)(34 61)(35 69)(36 64)(37 58)(38 66)(39 60)(40 68)(41 62)(42 70)(43 74)(44 82)(45 76)(46 84)(47 78)(48 72)(49 80)(50 81)(51 75)(52 83)(53 77)(54 71)(55 79)(56 73)
(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 11)(9 10)(12 14)(15 26)(16 25)(17 24)(18 23)(19 22)(20 28)(21 27)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 49)(36 54)(37 53)(38 52)(39 51)(40 50)(41 56)(42 55)(57 92 64 85)(58 91 65 98)(59 90 66 97)(60 89 67 96)(61 88 68 95)(62 87 69 94)(63 86 70 93)(71 103 78 110)(72 102 79 109)(73 101 80 108)(74 100 81 107)(75 99 82 106)(76 112 83 105)(77 111 84 104)

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

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

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

58 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 14A 14B 14C 14D 14E 14F 14G ··· 14L 16A 16B 16C 16D 28A ··· 28F 28G 28H 28I 28J ··· 28O 56A ··· 56L order 1 2 2 2 4 4 4 4 4 7 7 7 8 8 8 14 14 14 14 14 14 14 ··· 14 16 16 16 16 28 ··· 28 28 28 28 28 ··· 28 56 ··· 56 size 1 1 2 8 2 2 8 56 56 2 2 2 2 2 4 2 2 2 4 4 4 8 ··· 8 28 28 28 28 2 ··· 2 4 4 4 8 ··· 8 4 ··· 4

58 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - - - + image C1 C2 C2 C2 C4 C4 D4 D4 D7 SD16 D8 D14 Dic7 Dic7 C7⋊D4 C7⋊D4 D8⋊2C4 D4.D7 D4⋊D7 D8⋊2Dic7 kernel D8⋊2Dic7 C28.C8 C8⋊Dic7 C7×C4○D8 C7×D8 C7×Q16 C56 C2×C28 C4○D8 C28 C2×C14 C2×C8 D8 Q16 C8 C2×C4 C7 C4 C22 C1 # reps 1 1 1 1 2 2 1 1 3 2 2 3 3 3 6 6 2 3 3 12

Matrix representation of D82Dic7 in GL4(𝔽113) generated by

 104 71 0 0 34 96 0 0 23 19 89 71 94 0 42 50
,
 96 42 111 3 79 104 65 36 23 19 89 71 94 0 42 50
,
 34 1 0 0 53 88 0 0 71 0 0 112 41 71 1 104
,
 0 104 0 0 25 0 0 0 17 101 50 42 98 27 40 63
G:=sub<GL(4,GF(113))| [104,34,23,94,71,96,19,0,0,0,89,42,0,0,71,50],[96,79,23,94,42,104,19,0,111,65,89,42,3,36,71,50],[34,53,71,41,1,88,0,71,0,0,0,1,0,0,112,104],[0,25,17,98,104,0,101,27,0,0,50,40,0,0,42,63] >;

D82Dic7 in GAP, Magma, Sage, TeX

D_8\rtimes_2{\rm Dic}_7
% in TeX

G:=Group("D8:2Dic7");
// GroupNames label

G:=SmallGroup(448,123);
// by ID

G=gap.SmallGroup(448,123);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,387,675,794,80,1684,851,102,18822]);
// 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,d*a*d^-1=a^3,c*b*c^-1=a^4*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

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