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G = D28.3D4order 448 = 26·7

3rd non-split extension by D28 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D28.3D4, Dic14.3D4, M4(2).2D14, C7⋊C8.25D4, C8⋊D146C2, D28.C46C2, C28.94(C2×D4), C4.D44D7, C4.149(D4×D7), (C2×D4).16D14, C71(D4.4D4), (C2×C28).6C23, D4.D141C2, C28.53D42C2, C28.46D46C2, C4○D28.4C22, C14.10(C4⋊D4), (C2×D28).39C22, (D4×C14).16C22, C2.13(D14⋊D4), C4.Dic7.3C22, C22.14(C4○D28), (C7×M4(2)).11C22, (C2×D4⋊D7)⋊1C2, (C2×C7⋊C8).2C22, (C7×C4.D4)⋊2C2, (C2×C4).6(C22×D7), (C2×C14).31(C4○D4), SmallGroup(448,283)

Series: Derived Chief Lower central Upper central

C1C2×C28 — D28.3D4
C1C7C14C28C2×C28C4○D28D28.C4 — D28.3D4
C7C14C2×C28 — D28.3D4
C1C2C2×C4C4.D4

Generators and relations for D28.3D4
 G = < a,b,c,d | a28=b2=1, c4=a14, d2=a7, bab=a-1, cac-1=a15, ad=da, cbc-1=a14b, dbd-1=a7b, dcd-1=a21c3 >

Subgroups: 652 in 108 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C4.D4, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C7⋊C8, C7⋊C8, C56, Dic14, C4×D7, D28, D28, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, D4.4D4, C8×D7, C8⋊D7, C56⋊C2, D56, C2×C7⋊C8, C4.Dic7, D4⋊D7, D4.D7, C7×M4(2), C2×D28, C4○D28, D4×C14, C28.53D4, C28.46D4, C7×C4.D4, D28.C4, C8⋊D14, C2×D4⋊D7, D4.D14, D28.3D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C22×D7, D4.4D4, C4○D28, D4×D7, D14⋊D4, D28.3D4

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

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

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

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

49 conjugacy classes

class 1 2A2B2C2D2E4A4B4C7A7B7C8A8B8C8D8E8F8G14A14B14C14D14E14F14G···14L28A···28F56A···56L
order122222444777888888814141414141414···1428···2856···56
size112828562228222448141428562224448···84···48···8

49 irreducible representations

dim1111111122222222448
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D7C4○D4D14D14C4○D28D4.4D4D4×D7D28.3D4
kernelD28.3D4C28.53D4C28.46D4C7×C4.D4D28.C4C8⋊D14C2×D4⋊D7D4.D14C7⋊C8Dic14D28C4.D4C2×C14M4(2)C2×D4C22C7C4C1
# reps11111111211326312263

Matrix representation of D28.3D4 in GL8(𝔽113)

3303400000
0330340000
71010400000
07101040000
000017200
00009111200
00006201122
00000281121
,
800100000
080010000
4203300000
0420330000
000011206251
0000001090
000002800
00000281121
,
01000000
1120000000
00010000
0011200000
00000010
00001090091
000017200
0000136820
,
01000000
10000000
00010000
00100000
00001120051
00000040
0000628500
0000628501

G:=sub<GL(8,GF(113))| [33,0,71,0,0,0,0,0,0,33,0,71,0,0,0,0,34,0,104,0,0,0,0,0,0,34,0,104,0,0,0,0,0,0,0,0,1,91,62,0,0,0,0,0,72,112,0,28,0,0,0,0,0,0,112,112,0,0,0,0,0,0,2,1],[80,0,42,0,0,0,0,0,0,80,0,42,0,0,0,0,1,0,33,0,0,0,0,0,0,1,0,33,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,28,28,0,0,0,0,62,109,0,112,0,0,0,0,51,0,0,1],[0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,109,1,1,0,0,0,0,0,0,72,36,0,0,0,0,1,0,0,82,0,0,0,0,0,91,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,112,0,62,62,0,0,0,0,0,0,85,85,0,0,0,0,0,4,0,0,0,0,0,0,51,0,0,1] >;

D28.3D4 in GAP, Magma, Sage, TeX

D_{28}._3D_4
% in TeX

G:=Group("D28.3D4");
// GroupNames label

G:=SmallGroup(448,283);
// by ID

G=gap.SmallGroup(448,283);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,64,590,555,297,136,1684,851,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^28=b^2=1,c^4=a^14,d^2=a^7,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^14*b,d*b*d^-1=a^7*b,d*c*d^-1=a^21*c^3>;
// generators/relations

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