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## G = D28.14D4order 448 = 26·7

### 14th non-split extension by D28 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D28.14D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — Q8.10D14 — D28.14D4
 Lower central C7 — C14 — C2×C28 — D28.14D4
 Upper central C1 — C2 — C2×C4 — C4.4D4

Generators and relations for D28.14D4
G = < a,b,c,d | a28=b2=d2=1, c4=a14, bab=cac-1=a-1, dad=a13, cbc-1=a19b, dbd=a26b, dcd=a14c3 >

Subgroups: 748 in 146 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C7⋊C8, Dic14, Dic14, C4×D7, D28, D28, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, D4.8D4, C4.Dic7, D4⋊D7, D4.D7, C4×C28, C7×C22⋊C4, C4○D28, C4○D28, Q8×D7, Q82D7, D4×C14, Q8×C14, Dic14⋊C4, C28.10D4, D4.D14, C7×C4.4D4, Q8.10D14, D28.14D4
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, D28.14D4

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

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

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

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

58 conjugacy classes

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

58 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 D14 C7⋊D4 D4.8D4 D4×D7 D28.14D4 kernel D28.14D4 Dic14⋊C4 C28.10D4 D4.D14 C7×C4.4D4 Q8.10D14 Dic14 D28 C2×C28 C4.4D4 C42 C2×D4 C2×Q8 C2×C4 C7 C4 C1 # reps 1 2 1 2 1 1 2 2 2 3 3 3 3 12 2 6 12

Matrix representation of D28.14D4 in GL4(𝔽113) generated by

 8 0 0 0 0 105 0 0 0 0 14 0 0 0 0 99
,
 0 0 0 99 0 0 99 0 0 8 0 0 8 0 0 0
,
 0 0 0 98 0 0 15 0 0 1 0 0 1 0 0 0
,
 0 0 98 0 0 0 0 15 15 0 0 0 0 98 0 0
G:=sub<GL(4,GF(113))| [8,0,0,0,0,105,0,0,0,0,14,0,0,0,0,99],[0,0,0,8,0,0,8,0,0,99,0,0,99,0,0,0],[0,0,0,1,0,0,1,0,0,15,0,0,98,0,0,0],[0,0,15,0,0,0,0,98,98,0,0,0,0,15,0,0] >;

D28.14D4 in GAP, Magma, Sage, TeX

D_{28}._{14}D_4
% in TeX

G:=Group("D28.14D4");
// GroupNames label

G:=SmallGroup(448,596);
// by ID

G=gap.SmallGroup(448,596);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,1123,570,297,136,1684,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^28=b^2=d^2=1,c^4=a^14,b*a*b=c*a*c^-1=a^-1,d*a*d=a^13,c*b*c^-1=a^19*b,d*b*d=a^26*b,d*c*d=a^14*c^3>;
// generators/relations

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