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## G = Dic7⋊D4order 224 = 25·7

### 2nd semidirect product of Dic7 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — Dic7⋊D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C7⋊D4 — Dic7⋊D4
 Lower central C7 — C2×C14 — Dic7⋊D4
 Upper central C1 — C22 — C2×D4

Generators and relations for Dic7⋊D4
G = < a,b,c,d | a14=c4=d2=1, b2=a7, bab-1=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd=c-1 >

Subgroups: 382 in 94 conjugacy classes, 35 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, Dic7, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C4⋊D4, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, Dic7⋊C4, D14⋊C4, C23.D7, C22×Dic7, C2×C7⋊D4, D4×C14, Dic7⋊D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C7⋊D4, C22×D7, D4×D7, D42D7, C2×C7⋊D4, Dic7⋊D4

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D7 C4○D4 D14 D14 C7⋊D4 D4×D7 D4⋊2D7 kernel Dic7⋊D4 Dic7⋊C4 D14⋊C4 C23.D7 C22×Dic7 C2×C7⋊D4 D4×C14 Dic7 C2×C14 C2×D4 C14 C2×C4 C23 C22 C2 C2 # reps 1 1 1 1 1 2 1 2 2 3 2 3 6 12 3 3

Matrix representation of Dic7⋊D4 in GL4(𝔽29) generated by

 0 28 0 0 1 22 0 0 0 0 1 0 0 0 0 1
,
 9 14 0 0 19 20 0 0 0 0 1 0 0 0 0 1
,
 9 14 0 0 15 20 0 0 0 0 11 5 0 0 22 18
,
 28 0 0 0 0 28 0 0 0 0 18 24 0 0 24 11
G:=sub<GL(4,GF(29))| [0,1,0,0,28,22,0,0,0,0,1,0,0,0,0,1],[9,19,0,0,14,20,0,0,0,0,1,0,0,0,0,1],[9,15,0,0,14,20,0,0,0,0,11,22,0,0,5,18],[28,0,0,0,0,28,0,0,0,0,18,24,0,0,24,11] >;

Dic7⋊D4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(224,134);
// by ID

G=gap.SmallGroup(224,134);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,218,188,6917]);
// Polycyclic

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

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