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G = C28.48D4order 224 = 25·7

5th non-split extension by C28 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C28.48D4, C222Dic14, C23.20D14, (C2×C14)⋊3Q8, C4⋊Dic78C2, C74(C22⋊Q8), C14.8(C2×Q8), Dic7⋊C42C2, C14.39(C2×D4), (C2×C4).83D14, (C22×C4).5D7, (C2×Dic14)⋊6C2, C4.23(C7⋊D4), (C22×C28).6C2, C2.9(C2×Dic14), C23.D7.4C2, C2.17(C4○D28), C14.15(C4○D4), (C2×C14).42C23, (C2×C28).91C22, C22.54(C22×D7), (C22×C14).34C22, (C2×Dic7).14C22, C2.5(C2×C7⋊D4), SmallGroup(224,119)

Series: Derived Chief Lower central Upper central

C1C2×C14 — C28.48D4
C1C7C14C2×C14C2×Dic7C2×Dic14 — C28.48D4
C7C2×C14 — C28.48D4
C1C22C22×C4

Generators and relations for C28.48D4
 G = < a,b,c | a28=b4=1, c2=a14, bab-1=cac-1=a-1, cbc-1=a14b-1 >

Subgroups: 238 in 74 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊Q8, Dic14, C2×Dic7, C2×C28, C2×C28, C22×C14, Dic7⋊C4, C4⋊Dic7, C23.D7, C2×Dic14, C22×C28, C28.48D4
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, Dic14, C7⋊D4, C22×D7, C2×Dic14, C4○D28, C2×C7⋊D4, C28.48D4

Smallest permutation representation of C28.48D4
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)
(1 104 33 63)(2 103 34 62)(3 102 35 61)(4 101 36 60)(5 100 37 59)(6 99 38 58)(7 98 39 57)(8 97 40 84)(9 96 41 83)(10 95 42 82)(11 94 43 81)(12 93 44 80)(13 92 45 79)(14 91 46 78)(15 90 47 77)(16 89 48 76)(17 88 49 75)(18 87 50 74)(19 86 51 73)(20 85 52 72)(21 112 53 71)(22 111 54 70)(23 110 55 69)(24 109 56 68)(25 108 29 67)(26 107 30 66)(27 106 31 65)(28 105 32 64)
(1 77 15 63)(2 76 16 62)(3 75 17 61)(4 74 18 60)(5 73 19 59)(6 72 20 58)(7 71 21 57)(8 70 22 84)(9 69 23 83)(10 68 24 82)(11 67 25 81)(12 66 26 80)(13 65 27 79)(14 64 28 78)(29 94 43 108)(30 93 44 107)(31 92 45 106)(32 91 46 105)(33 90 47 104)(34 89 48 103)(35 88 49 102)(36 87 50 101)(37 86 51 100)(38 85 52 99)(39 112 53 98)(40 111 54 97)(41 110 55 96)(42 109 56 95)

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

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

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

C28.48D4 is a maximal subgroup of
C4⋊Dic7⋊C4  C23.30D28  C23.34D28  C23.35D28  C23.10D28  D28.31D4  D28.32D4  C22⋊Dic28  C4⋊C4.230D14  C4⋊C4.231D14  C4⋊C4.233D14  C7⋊C823D4  C7⋊C85D4  C7⋊C8.29D4  C7⋊C8.6D4  C5630D4  C56.82D4  C562D4  C56.4D4  (C7×D4).31D4  (C2×C14)⋊8Q16  (C7×D4).32D4  C42.274D14  C42.277D14  C232Dic14  C24.30D14  C24.31D14  C14.72+ 1+4  C14.102+ 1+4  C14.52- 1+4  C14.62- 1+4  C42.89D14  C42.94D14  C42.98D14  C42.99D14  D4×Dic14  D45Dic14  C42.105D14  C42.106D14  D46Dic14  D2823D4  D2824D4  Dic1423D4  C4216D14  C42.115D14  C42.118D14  C4⋊C4.178D14  C14.352+ 1+4  C14.712- 1+4  C4⋊C421D14  C14.722- 1+4  C14.462+ 1+4  C14.492+ 1+4  (Q8×Dic7)⋊C2  C14.752- 1+4  C14.152- 1+4  D7×C22⋊Q8  C14.162- 1+4  C14.512+ 1+4  C14.582+ 1+4  C14.602+ 1+4  C14.832- 1+4  C14.842- 1+4  C14.862- 1+4  C24.72D14  C24.42D14  Q8×C7⋊D4  C14.1042- 1+4  C14.1052- 1+4  C14.1072- 1+4  C14.1082- 1+4
C28.48D4 is a maximal quotient of
C284(C4⋊C4)  (C2×C28)⋊10Q8  (C2×C42).D7  C23⋊Dic14  C24.6D14  C24.7D14  (C2×C4)⋊Dic14  (C2×C28).54D4  (C2×C28).55D4  C28.50D8  C28.38SD16  D4.3Dic14  C28.48SD16  C28.23Q16  Q8.3Dic14  C24.62D14  C23.27D28

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H7A7B7C14A···14U28A···28X
order1222224444444477714···1428···28
size1111222222282828282222···22···2

62 irreducible representations

dim111111222222222
type+++++++-+++-
imageC1C2C2C2C2C2D4Q8D7C4○D4D14D14C7⋊D4Dic14C4○D28
kernelC28.48D4Dic7⋊C4C4⋊Dic7C23.D7C2×Dic14C22×C28C28C2×C14C22×C4C14C2×C4C23C4C22C2
# reps121211223263121212

Matrix representation of C28.48D4 in GL4(𝔽29) generated by

21000
231800
00160
00020
,
221100
22700
0001
00280
,
221100
6700
0001
0010
G:=sub<GL(4,GF(29))| [21,23,0,0,0,18,0,0,0,0,16,0,0,0,0,20],[22,22,0,0,11,7,0,0,0,0,0,28,0,0,1,0],[22,6,0,0,11,7,0,0,0,0,0,1,0,0,1,0] >;

C28.48D4 in GAP, Magma, Sage, TeX

C_{28}._{48}D_4
% in TeX

G:=Group("C28.48D4");
// GroupNames label

G:=SmallGroup(224,119);
// by ID

G=gap.SmallGroup(224,119);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,217,103,218,6917]);
// Polycyclic

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

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