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## G = D28.38D4order 448 = 26·7

### 8th non-split extension by D28 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D28.38D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — D4.10D14 — D28.38D4
 Lower central C7 — C14 — C2×C28 — D28.38D4
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for D28.38D4
G = < a,b,c,d | a28=b2=1, c4=d2=a14, bab=a-1, cac-1=a15, ad=da, cbc-1=a7b, bd=db, dcd-1=c3 >

Subgroups: 748 in 146 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C42, C22⋊C4, M4(2), M4(2), D8, SD16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, C8⋊C22, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×C14, D4.8D4, C4.Dic7, C4×Dic7, D4⋊D7, D4.D7, C23.D7, C7×M4(2), C7×D8, C7×SD16, C2×Dic14, C2×Dic14, C4○D28, C4○D28, D42D7, Q8×D7, D4×C14, C7×C4○D4, C4.12D28, D284C4, D42Dic7, D4.D14, C28.17D4, C7×C8⋊C22, D4.10D14, D28.38D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, D4.8D4, D4×D7, C2×C7⋊D4, C23⋊D14, D28.38D4

Smallest permutation representation of D28.38D4
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 95)(2 94)(3 93)(4 92)(5 91)(6 90)(7 89)(8 88)(9 87)(10 86)(11 85)(12 112)(13 111)(14 110)(15 109)(16 108)(17 107)(18 106)(19 105)(20 104)(21 103)(22 102)(23 101)(24 100)(25 99)(26 98)(27 97)(28 96)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)(41 62)(42 61)(43 60)(44 59)(45 58)(46 57)(47 84)(48 83)(49 82)(50 81)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)
(1 89 22 96 15 103 8 110)(2 104 23 111 16 90 9 97)(3 91 24 98 17 105 10 112)(4 106 25 85 18 92 11 99)(5 93 26 100 19 107 12 86)(6 108 27 87 20 94 13 101)(7 95 28 102 21 109 14 88)(29 66 50 73 43 80 36 59)(30 81 51 60 44 67 37 74)(31 68 52 75 45 82 38 61)(32 83 53 62 46 69 39 76)(33 70 54 77 47 84 40 63)(34 57 55 64 48 71 41 78)(35 72 56 79 49 58 42 65)
(1 74 15 60)(2 75 16 61)(3 76 17 62)(4 77 18 63)(5 78 19 64)(6 79 20 65)(7 80 21 66)(8 81 22 67)(9 82 23 68)(10 83 24 69)(11 84 25 70)(12 57 26 71)(13 58 27 72)(14 59 28 73)(29 109 43 95)(30 110 44 96)(31 111 45 97)(32 112 46 98)(33 85 47 99)(34 86 48 100)(35 87 49 101)(36 88 50 102)(37 89 51 103)(38 90 52 104)(39 91 53 105)(40 92 54 106)(41 93 55 107)(42 94 56 108)

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

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

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

49 conjugacy classes

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

49 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4 D7 D14 D14 D14 C7⋊D4 C7⋊D4 D4.8D4 D4×D7 D4×D7 D28.38D4 kernel D28.38D4 C4.12D28 D28⋊4C4 D4⋊2Dic7 D4.D14 C28.17D4 C7×C8⋊C22 D4.10D14 Dic14 D28 C2×Dic7 C7×D4 C7×Q8 C8⋊C22 M4(2) C2×D4 C4○D4 D4 Q8 C7 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 2 1 1 3 3 3 3 6 6 2 3 3 3

Matrix representation of D28.38D4 in GL8(𝔽113)

 33 9 0 0 0 0 0 0 104 1 0 0 0 0 0 0 0 0 34 9 0 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 98 0 0 0 0 0 0 0 56 15 0 0 0 0 0 36 0 0 98
,
 41 15 43 0 0 0 0 0 45 72 106 70 0 0 0 0 74 0 14 15 0 0 0 0 30 39 31 99 0 0 0 0 0 0 0 0 13 31 0 8 0 0 0 0 111 30 8 0 0 0 0 0 39 79 83 2 0 0 0 0 15 17 82 100
,
 63 13 0 65 0 0 0 0 100 67 55 63 0 0 0 0 104 56 36 100 0 0 0 0 58 87 102 60 0 0 0 0 0 0 0 0 31 100 0 106 0 0 0 0 30 2 106 0 0 0 0 0 93 60 111 83 0 0 0 0 31 84 13 82
,
 101 13 0 103 0 0 0 0 100 105 35 112 0 0 0 0 90 83 111 100 0 0 0 0 20 87 102 22 0 0 0 0 0 0 0 0 30 2 106 0 0 0 0 0 31 100 0 106 0 0 0 0 73 21 83 111 0 0 0 0 43 17 82 13

G:=sub<GL(8,GF(113))| [33,104,0,0,0,0,0,0,9,1,0,0,0,0,0,0,0,0,34,25,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,15,0,0,36,0,0,0,0,0,98,56,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,98],[41,45,74,30,0,0,0,0,15,72,0,39,0,0,0,0,43,106,14,31,0,0,0,0,0,70,15,99,0,0,0,0,0,0,0,0,13,111,39,15,0,0,0,0,31,30,79,17,0,0,0,0,0,8,83,82,0,0,0,0,8,0,2,100],[63,100,104,58,0,0,0,0,13,67,56,87,0,0,0,0,0,55,36,102,0,0,0,0,65,63,100,60,0,0,0,0,0,0,0,0,31,30,93,31,0,0,0,0,100,2,60,84,0,0,0,0,0,106,111,13,0,0,0,0,106,0,83,82],[101,100,90,20,0,0,0,0,13,105,83,87,0,0,0,0,0,35,111,102,0,0,0,0,103,112,100,22,0,0,0,0,0,0,0,0,30,31,73,43,0,0,0,0,2,100,21,17,0,0,0,0,106,0,83,82,0,0,0,0,0,106,111,13] >;

D28.38D4 in GAP, Magma, Sage, TeX

D_{28}._{38}D_4
% in TeX

G:=Group("D28.38D4");
// GroupNames label

G:=SmallGroup(448,735);
// by ID

G=gap.SmallGroup(448,735);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,1123,297,136,851,438,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^28=b^2=1,c^4=d^2=a^14,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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