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## G = D4⋊Dic7order 224 = 25·7

### 1st semidirect product of D4 and Dic7 acting via Dic7/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D4⋊Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C4⋊Dic7 — D4⋊Dic7
 Lower central C7 — C14 — C28 — D4⋊Dic7
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for D4⋊Dic7
G = < a,b,c,d | a4=b2=c14=1, d2=c7, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Smallest permutation representation of D4⋊Dic7
On 112 points
Generators in S112
(1 50 35 107)(2 51 36 108)(3 52 37 109)(4 53 38 110)(5 54 39 111)(6 55 40 112)(7 56 41 99)(8 43 42 100)(9 44 29 101)(10 45 30 102)(11 46 31 103)(12 47 32 104)(13 48 33 105)(14 49 34 106)(15 59 92 74)(16 60 93 75)(17 61 94 76)(18 62 95 77)(19 63 96 78)(20 64 97 79)(21 65 98 80)(22 66 85 81)(23 67 86 82)(24 68 87 83)(25 69 88 84)(26 70 89 71)(27 57 90 72)(28 58 91 73)
(1 107)(2 108)(3 109)(4 110)(5 111)(6 112)(7 99)(8 100)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 92)(16 93)(17 94)(18 95)(19 96)(20 97)(21 98)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 91)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 43)
(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 85 8 92)(2 98 9 91)(3 97 10 90)(4 96 11 89)(5 95 12 88)(6 94 13 87)(7 93 14 86)(15 35 22 42)(16 34 23 41)(17 33 24 40)(18 32 25 39)(19 31 26 38)(20 30 27 37)(21 29 28 36)(43 59 50 66)(44 58 51 65)(45 57 52 64)(46 70 53 63)(47 69 54 62)(48 68 55 61)(49 67 56 60)(71 110 78 103)(72 109 79 102)(73 108 80 101)(74 107 81 100)(75 106 82 99)(76 105 83 112)(77 104 84 111)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14U 28A ··· 28F order 1 2 2 2 2 2 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 4 4 2 2 28 28 2 2 2 14 14 14 14 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

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

Matrix representation of D4⋊Dic7 in GL6(𝔽113)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 43 42 0 0 0 0 112 70
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 5 1 0 0 0 0 0 0 43 42 0 0 0 0 69 70
,
 16 0 0 0 0 0 2 106 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 19 64 0 0 0 0 65 94 0 0 0 0 0 0 19 98 0 0 0 0 107 94 0 0 0 0 0 0 54 54 0 0 0 0 36 59

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,43,112,0,0,0,0,42,70],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,5,0,0,0,0,0,1,0,0,0,0,0,0,43,69,0,0,0,0,42,70],[16,2,0,0,0,0,0,106,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[19,65,0,0,0,0,64,94,0,0,0,0,0,0,19,107,0,0,0,0,98,94,0,0,0,0,0,0,54,36,0,0,0,0,54,59] >;

D4⋊Dic7 in GAP, Magma, Sage, TeX

D_4\rtimes {\rm Dic}_7
% in TeX

G:=Group("D4:Dic7");
// GroupNames label

G:=SmallGroup(224,38);
// by ID

G=gap.SmallGroup(224,38);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,579,297,69,6917]);
// Polycyclic

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

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