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## G = Dic7.D4order 224 = 25·7

### 1st non-split extension by Dic7 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — Dic7.D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — D14⋊C4 — Dic7.D4
 Lower central C7 — C2×C14 — Dic7.D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for Dic7.D4
G = < a,b,c,d | a14=c4=1, b2=d2=a7, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a7b, dcd-1=a7c-1 >

Subgroups: 326 in 76 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C2×D4, C2×Q8, Dic7, Dic7, C28, D14, C2×C14, C2×C14, C4.4D4, Dic14, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×C14, C4×Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C2×Dic14, C2×C7⋊D4, Dic7.D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, C22×D7, C4○D28, D4×D7, D42D7, 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 96 8 89)(2 95 9 88)(3 94 10 87)(4 93 11 86)(5 92 12 85)(6 91 13 98)(7 90 14 97)(15 75 22 82)(16 74 23 81)(17 73 24 80)(18 72 25 79)(19 71 26 78)(20 84 27 77)(21 83 28 76)(29 64 36 57)(30 63 37 70)(31 62 38 69)(32 61 39 68)(33 60 40 67)(34 59 41 66)(35 58 42 65)(43 99 50 106)(44 112 51 105)(45 111 52 104)(46 110 53 103)(47 109 54 102)(48 108 55 101)(49 107 56 100)
(1 73 44 38)(2 74 45 39)(3 75 46 40)(4 76 47 41)(5 77 48 42)(6 78 49 29)(7 79 50 30)(8 80 51 31)(9 81 52 32)(10 82 53 33)(11 83 54 34)(12 84 55 35)(13 71 56 36)(14 72 43 37)(15 103 60 87)(16 104 61 88)(17 105 62 89)(18 106 63 90)(19 107 64 91)(20 108 65 92)(21 109 66 93)(22 110 67 94)(23 111 68 95)(24 112 69 96)(25 99 70 97)(26 100 57 98)(27 101 58 85)(28 102 59 86)
(1 80 8 73)(2 79 9 72)(3 78 10 71)(4 77 11 84)(5 76 12 83)(6 75 13 82)(7 74 14 81)(15 98 22 91)(16 97 23 90)(17 96 24 89)(18 95 25 88)(19 94 26 87)(20 93 27 86)(21 92 28 85)(29 53 36 46)(30 52 37 45)(31 51 38 44)(32 50 39 43)(33 49 40 56)(34 48 41 55)(35 47 42 54)(57 103 64 110)(58 102 65 109)(59 101 66 108)(60 100 67 107)(61 99 68 106)(62 112 69 105)(63 111 70 104)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 14A ··· 14I 14J ··· 14O 28A ··· 28L order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 4 28 2 2 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 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D7 C4○D4 D14 D14 C4○D28 D4×D7 D4⋊2D7 kernel Dic7.D4 C4×Dic7 D14⋊C4 C23.D7 C7×C22⋊C4 C2×Dic14 C2×C7⋊D4 Dic7 C22⋊C4 C14 C2×C4 C23 C2 C2 C2 # reps 1 1 2 1 1 1 1 2 3 4 6 3 12 3 3

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

 26 28 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 5 2 0 0 16 24 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 17 27 0 0 0 12
,
 17 22 0 0 0 12 0 0 0 0 12 2 0 0 1 17
G:=sub<GL(4,GF(29))| [26,1,0,0,28,0,0,0,0,0,1,0,0,0,0,1],[5,16,0,0,2,24,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,17,0,0,0,27,12],[17,0,0,0,22,12,0,0,0,0,12,1,0,0,2,17] >;

Dic7.D4 in GAP, Magma, Sage, TeX

{\rm Dic}_7.D_4
% in TeX

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

G:=SmallGroup(224,80);
// by ID

G=gap.SmallGroup(224,80);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,55,506,188,6917]);
// Polycyclic

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

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