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G = C28:D4order 224 = 25·7

3rd semidirect product of C28 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C28:3D4, Dic7:1D4, C23.10D14, (C2xD4):6D7, (D4xC14):4C2, (C2xD28):9C2, C4:1(C7:D4), C7:2(C4:1D4), C2.28(D4xD7), (C4xDic7):6C2, C14.52(C2xD4), (C2xC4).52D14, (C2xC14).55C23, (C2xC28).35C22, C22.62(C22xD7), (C22xC14).22C22, (C2xDic7).39C22, (C22xD7).12C22, (C2xC7:D4):7C2, C2.16(C2xC7:D4), SmallGroup(224,135)

Series: Derived Chief Lower central Upper central

C1C2xC14 — C28:D4
C1C7C14C2xC14C22xD7C2xD28 — C28:D4
C7C2xC14 — C28:D4
C1C22C2xD4

Generators and relations for C28:D4
 G = < a,b,c | a28=b4=c2=1, bab-1=a13, cac=a-1, cbc=b-1 >

Subgroups: 510 in 108 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2xC4, C2xC4, D4, C23, C23, D7, C14, C14, C14, C42, C2xD4, C2xD4, Dic7, C28, D14, C2xC14, C2xC14, C4:1D4, D28, C2xDic7, C7:D4, C2xC28, C7xD4, C22xD7, C22xC14, C4xDic7, C2xD28, C2xC7:D4, D4xC14, C28:D4
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, D14, C4:1D4, C7:D4, C22xD7, D4xD7, C2xC7:D4, C28:D4

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

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

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

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

C28:D4 is a maximal subgroup of
C23.2D28  D28:1D4  Dic7.SD16  Dic14:2D4  C4:C4.D14  D28:3D4  Dic7:D8  C56:5D4  C56:11D4  Dic7:5SD16  C56:15D4  C56:9D4  D28:18D4  2+ 1+4:D7  C42.228D14  Dic14:24D4  C42.114D14  C42.116D14  C24:3D14  C24.34D14  C24.36D14  C28:(C4oD4)  Dic14:20D4  C14.382+ 1+4  D28:19D4  C14.442+ 1+4  C14.472+ 1+4  C14.482+ 1+4  C14.662+ 1+4  C14.672+ 1+4  C14.682+ 1+4  C42.233D14  C42:18D14  C42.143D14  D7xC4:1D4  C42:26D14  Dic14:11D4  D4xC7:D4  C24.41D14  C14.1462+ 1+4  (C2xC28):17D4  C14.1482+ 1+4
C28:D4 is a maximal quotient of
C24.7D14  C24.13D14  C23:2D28  (C4xDic7):8C4  (C2xD28):10C4  (C2xC28).289D4  C42.64D14  C42.214D14  C42.65D14  C28:D8  C42.74D14  C28:4SD16  C28:6SD16  C42.80D14  C28:3Q16  C56:5D4  C56:11D4  C56.22D4  C56.31D4  C56.43D4  C56:15D4  C56:9D4  C56.26D4  C56.37D4  C56.28D4  C24.19D14  C24.21D14

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F7A7B7C14A···14I14J···14U28A···28F
order1222222244444477714···1414···1428···28
size111144282822141414142222···24···44···4

44 irreducible representations

dim111112222224
type+++++++++++
imageC1C2C2C2C2D4D4D7D14D14C7:D4D4xD7
kernelC28:D4C4xDic7C2xD28C2xC7:D4D4xC14Dic7C28C2xD4C2xC4C23C4C2
# reps1114142336126

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

12600
32100
002817
0051
,
51600
22400
00280
00028
,
1000
32800
0010
002428
G:=sub<GL(4,GF(29))| [1,3,0,0,26,21,0,0,0,0,28,5,0,0,17,1],[5,2,0,0,16,24,0,0,0,0,28,0,0,0,0,28],[1,3,0,0,0,28,0,0,0,0,1,24,0,0,0,28] >;

C28:D4 in GAP, Magma, Sage, TeX

C_{28}\rtimes D_4
% in TeX

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

G:=SmallGroup(224,135);
// by ID

G=gap.SmallGroup(224,135);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,103,218,188,6917]);
// Polycyclic

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

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