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## G = D8.Dic7order 448 = 26·7

### 2nd non-split extension by D8 of Dic7 acting via Dic7/C14=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for D8.Dic7
G = < a,b,c,d | a8=b2=1, c14=a4, d2=a4c7, bab=a-1, ac=ca, dad-1=a3, cbc-1=a4b, dbd-1=a5b, dcd-1=c13 >

Subgroups: 228 in 62 conjugacy classes, 27 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C8, C2×C4, D4, C23, C14, C14, C16, C2×C8, M4(2), D8, D8, C2×D4, C28, C2×C14, C2×C14, C8.C4, M5(2), C2×D8, C7⋊C8, C56, C2×C28, C7×D4, C22×C14, M5(2)⋊C2, C7⋊C16, C4.Dic7, C2×C56, C7×D8, C7×D8, D4×C14, C28.C8, C56.C4, C14×D8, D8.Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, Dic7, D14, D4⋊C4, C2×Dic7, C7⋊D4, M5(2)⋊C2, D4⋊D7, D4.D7, C23.D7, D4⋊Dic7, D8.Dic7

Smallest permutation representation of D8.Dic7
On 112 points
Generators in S112
(1 35 8 42 15 49 22 56)(2 36 9 43 16 50 23 29)(3 37 10 44 17 51 24 30)(4 38 11 45 18 52 25 31)(5 39 12 46 19 53 26 32)(6 40 13 47 20 54 27 33)(7 41 14 48 21 55 28 34)(57 106 78 99 71 92 64 85)(58 107 79 100 72 93 65 86)(59 108 80 101 73 94 66 87)(60 109 81 102 74 95 67 88)(61 110 82 103 75 96 68 89)(62 111 83 104 76 97 69 90)(63 112 84 105 77 98 70 91)
(1 56)(2 43)(3 30)(4 45)(5 32)(6 47)(7 34)(8 49)(9 36)(10 51)(11 38)(12 53)(13 40)(14 55)(15 42)(16 29)(17 44)(18 31)(19 46)(20 33)(21 48)(22 35)(23 50)(24 37)(25 52)(26 39)(27 54)(28 41)(57 78)(58 65)(59 80)(60 67)(61 82)(62 69)(63 84)(64 71)(66 73)(68 75)(70 77)(72 79)(74 81)(76 83)(85 99)(87 101)(89 103)(91 105)(93 107)(95 109)(97 111)
(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 78 22 71 15 64 8 57)(2 63 23 84 16 77 9 70)(3 76 24 69 17 62 10 83)(4 61 25 82 18 75 11 68)(5 74 26 67 19 60 12 81)(6 59 27 80 20 73 13 66)(7 72 28 65 21 58 14 79)(29 98 50 91 43 112 36 105)(30 111 51 104 44 97 37 90)(31 96 52 89 45 110 38 103)(32 109 53 102 46 95 39 88)(33 94 54 87 47 108 40 101)(34 107 55 100 48 93 41 86)(35 92 56 85 49 106 42 99)

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

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

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

58 conjugacy classes

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

58 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + - image C1 C2 C2 C2 C4 D4 D4 D7 D8 SD16 D14 Dic7 C7⋊D4 C7⋊D4 M5(2)⋊C2 D4⋊D7 D4.D7 D8.Dic7 kernel D8.Dic7 C28.C8 C56.C4 C14×D8 C7×D8 C56 C2×C28 C2×D8 C28 C2×C14 C2×C8 D8 C8 C2×C4 C7 C4 C22 C1 # reps 1 1 1 1 4 1 1 3 2 2 3 6 6 6 2 3 3 12

Matrix representation of D8.Dic7 in GL4(𝔽113) generated by

 62 51 0 0 31 0 0 0 16 97 0 107 68 0 19 51
,
 62 51 0 0 31 51 0 0 85 97 1 80 0 0 0 112
,
 97 32 0 0 97 16 0 0 5 52 7 108 19 0 110 106
,
 35 0 48 0 0 0 0 1 111 33 78 0 112 1 0 0
G:=sub<GL(4,GF(113))| [62,31,16,68,51,0,97,0,0,0,0,19,0,0,107,51],[62,31,85,0,51,51,97,0,0,0,1,0,0,0,80,112],[97,97,5,19,32,16,52,0,0,0,7,110,0,0,108,106],[35,0,111,112,0,0,33,1,48,0,78,0,0,1,0,0] >;

D8.Dic7 in GAP, Magma, Sage, TeX

D_8.{\rm Dic}_7
% in TeX

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

G:=SmallGroup(448,120);
// by ID

G=gap.SmallGroup(448,120);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,387,184,675,794,80,1684,851,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^14=a^4,d^2=a^4*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^13>;
// generators/relations

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