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G = D28.1D4order 448 = 26·7

1st non-split extension by D28 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D28.1D4, C23.6D28, M4(2)⋊1D14, Dic14.1D4, C4.78(D4×D7), C4.D41D7, C28.91(C2×D4), D284C41C2, (C2×D4).13D14, C8.D145C2, C71(D4.9D4), (C2×C28).3C23, C14.15C22≀C2, D46D14.2C2, C28.17D41C2, (C4×Dic7)⋊1C22, C4○D28.1C22, (C22×C14).19D4, C22.10(C2×D28), (D4×C14).13C22, (C7×M4(2))⋊8C22, C2.18(C22⋊D28), (C2×Dic14)⋊12C22, (C7×C4.D4)⋊3C2, (C2×C14).20(C2×D4), (C2×C4).3(C22×D7), SmallGroup(448,280)

Series: Derived Chief Lower central Upper central

C1C2×C28 — D28.1D4
C1C7C14C28C2×C28C4○D28D46D14 — D28.1D4
C7C14C2×C28 — D28.1D4
C1C2C2×C4C4.D4

Generators and relations for D28.1D4
 G = < a,b,c,d | a28=b2=1, c4=a14, d2=a21, bab=a-1, cac-1=a15, ad=da, cbc-1=a21b, dbd-1=a7b, dcd-1=a7c3 >

Subgroups: 972 in 152 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, M4(2), SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C4.D4, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C56, Dic14, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, D4.9D4, C56⋊C2, Dic28, C4×Dic7, C23.D7, C7×M4(2), C2×Dic14, C4○D28, D4×D7, D42D7, C2×C7⋊D4, D4×C14, D284C4, C7×C4.D4, C8.D14, C28.17D4, D46D14, D28.1D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, D28, C22×D7, D4.9D4, C2×D28, D4×D7, C22⋊D28, D28.1D4

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

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

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

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

49 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G7A7B7C8A8B14A14B14C14D14E14F14G···14L28A···28F56A···56L
order122222244444447778814141414141414···1428···2856···56
size112442828222828282856222882224448···84···48···8

49 irreducible representations

dim1111112222222448
type++++++++++++++-
imageC1C2C2C2C2C2D4D4D4D7D14D14D28D4.9D4D4×D7D28.1D4
kernelD28.1D4D284C4C7×C4.D4C8.D14C28.17D4D46D14Dic14D28C22×C14C4.D4M4(2)C2×D4C23C7C4C1
# reps12121122236312263

Matrix representation of D28.1D4 in GL6(𝔽113)

34890000
59880000
0015000
00989800
00980980
0067557915
,
01030000
7900000
00988300
00151500
0021741963
00511017594
,
11200000
01120000
0011211100
008100
0024205472
007676859
,
11200000
01120000
006675582
0016193972
0024205472
00474887

G:=sub<GL(6,GF(113))| [34,59,0,0,0,0,89,88,0,0,0,0,0,0,15,98,98,67,0,0,0,98,0,55,0,0,0,0,98,79,0,0,0,0,0,15],[0,79,0,0,0,0,103,0,0,0,0,0,0,0,98,15,21,51,0,0,83,15,74,101,0,0,0,0,19,75,0,0,0,0,63,94],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,8,24,7,0,0,111,1,20,67,0,0,0,0,54,68,0,0,0,0,72,59],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,66,16,24,4,0,0,75,19,20,74,0,0,5,39,54,8,0,0,82,72,72,87] >;

D28.1D4 in GAP, Magma, Sage, TeX

D_{28}._1D_4
% in TeX

G:=Group("D28.1D4");
// GroupNames label

G:=SmallGroup(448,280);
// by ID

G=gap.SmallGroup(448,280);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,58,1123,570,136,1684,438,18822]);
// 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^21*b,d*b*d^-1=a^7*b,d*c*d^-1=a^7*c^3>;
// generators/relations

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