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G = D28.2D4order 448 = 26·7

2nd non-split extension by D28 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D28.2D4, Dic14.2D4, M4(2).1D14, C7⋊C8.24D4, D28.C45C2, C4.148(D4×D7), C4.D43D7, C28.93(C2×D4), (C2×D4).15D14, C8.D146C2, C71(D4.3D4), (C2×C28).5C23, C28.53D41C2, C4.12D286C2, C14.9(C4⋊D4), C4○D28.3C22, D4.D14.1C2, (D4×C14).15C22, C2.12(D14⋊D4), C4.Dic7.2C22, C22.13(C4○D28), (C2×Dic14).45C22, (C7×M4(2)).10C22, (C2×D4.D7)⋊1C2, (C2×C7⋊C8).1C22, (C7×C4.D4)⋊1C2, (C2×C4).5(C22×D7), (C2×C14).30(C4○D4), SmallGroup(448,282)

Series: Derived Chief Lower central Upper central

C1C2×C28 — D28.2D4
C1C7C14C28C2×C28C4○D28D28.C4 — D28.2D4
C7C14C2×C28 — D28.2D4
C1C2C2×C4C4.D4

Generators and relations for D28.2D4
 G = < a,b,c,d | a28=b2=1, c4=a14, d2=a21, bab=a-1, cac-1=a15, ad=da, cbc-1=a14b, dbd-1=a7b, dcd-1=a7c3 >

Subgroups: 524 in 104 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, Q16, C2×D4, C2×Q8, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C7⋊C8, C7⋊C8, C56, Dic14, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×C14, D4.3D4, C8×D7, C8⋊D7, C56⋊C2, Dic28, C2×C7⋊C8, C4.Dic7, D4⋊D7, D4.D7, C7×M4(2), C2×Dic14, C4○D28, D4×C14, C28.53D4, C4.12D28, C7×C4.D4, D28.C4, C8.D14, D4.D14, C2×D4.D7, D28.2D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C22×D7, D4.3D4, C4○D28, D4×D7, D14⋊D4, D28.2D4

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

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

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

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

49 conjugacy classes

class 1 2A2B2C2D4A4B4C4D7A7B7C8A8B8C8D8E8F8G14A14B14C14D14E14F14G···14L28A···28F56A···56L
order122224444777888888814141414141414···1428···2856···56
size112828222856222448141428562224448···84···48···8

49 irreducible representations

dim1111111122222222448
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D4D7C4○D4D14D14C4○D28D4.3D4D4×D7D28.2D4
kernelD28.2D4C28.53D4C4.12D28C7×C4.D4D28.C4C8.D14D4.D14C2×D4.D7C7⋊C8Dic14D28C4.D4C2×C14M4(2)C2×D4C22C7C4C1
# reps11111111211326312263

Matrix representation of D28.2D4 in GL8(𝔽113)

80010400000
08001040000
9011200000
0901120000
00000100
0000112000
0000000112
00000010
,
850500000
085050000
9202800000
0920280000
00000013100
0000001313
000010010000
00001310000
,
0112000000
10000000
0001120000
00100000
00000010
00000001
00000100
0000112000
,
0112000000
1120000000
0001120000
0011200000
00000010
0000000112
00000100
00001000

G:=sub<GL(8,GF(113))| [80,0,9,0,0,0,0,0,0,80,0,9,0,0,0,0,104,0,112,0,0,0,0,0,0,104,0,112,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0],[85,0,92,0,0,0,0,0,0,85,0,92,0,0,0,0,5,0,28,0,0,0,0,0,0,5,0,28,0,0,0,0,0,0,0,0,0,0,100,13,0,0,0,0,0,0,100,100,0,0,0,0,13,13,0,0,0,0,0,0,100,13,0,0],[0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0] >;

D28.2D4 in GAP, Magma, Sage, TeX

D_{28}._2D_4
% in TeX

G:=Group("D28.2D4");
// GroupNames label

G:=SmallGroup(448,282);
// by ID

G=gap.SmallGroup(448,282);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,64,590,219,297,136,1684,851,18822]);
// Polycyclic

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

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