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G = D28:1D4  order 448 = 26·7

1st semidirect product of D28 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D28:1D4, Dic14:1D4, C23.7D28, M4(2):2D14, C4.79(D4xD7), C28:D4:1C2, C8:D14:5C2, C7:1(D4:4D4), C28.92(C2xD4), C4.D4:2D7, D4:6D14:2C2, D28:4C4:2C2, (C2xD4).14D14, (C2xC28).4C23, C14.16C22wrC2, (C2xD28):11C22, (C4xDic7):2C22, C4oD28.2C22, (C22xC14).20D4, C22.11(C2xD28), (D4xC14).14C22, (C7xM4(2)):9C22, C2.19(C22:D28), (C7xC4.D4):4C2, (C2xC14).21(C2xD4), (C2xC4).4(C22xD7), SmallGroup(448,281)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC28 — D28:1D4
Chief series
C1 — C7 — C14 — C28 — C2xC28 — C4oD28 — D4:6D14 — D28:1D4
Lower central
C7 — C14 — C2xC28 — D28:1D4
Upper central
C1 — C2 — C2xC4 — C4.D4

Generators and relations for D28:1D4
 G = < a,b,c,d | a28=b2=c4=d2=1, bab=dad=a-1, cac-1=a13, cbc-1=a5b, dbd=a19b, dcd=c-1 >

Subgroups: 1228 in 168 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2xC4, C2xC4, D4, Q8, C23, C23, D7, C14, C14, C42, M4(2), D8, SD16, C2xD4, C2xD4, C4oD4, Dic7, C28, D14, C2xC14, C2xC14, C4.D4, C4wrC2, C4:1D4, C8:C22, 2+ 1+4, C56, Dic14, C4xD7, D28, D28, C2xDic7, C7:D4, C2xC28, C7xD4, C22xD7, C22xC14, D4:4D4, C56:C2, D56, C4xDic7, C7xM4(2), C2xD28, C4oD28, D4xD7, D4:2D7, C2xC7:D4, D4xC14, D28:4C4, C7xC4.D4, C8:D14, C28:D4, D4:6D14, D28:1D4
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, D14, C22wrC2, D28, C22xD7, D4:4D4, C2xD28, D4xD7, C22:D28, D28:1D4

Smallest permutation representation of D28:1D4
►On 56 points
Generators in S56
(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)

(1 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 56)(13 55)(14 54)(15 53)(16 52)(17 51)(18 50)(19 49)(20 48)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 40)

(1 15)(2 28)(3 13)(4 26)(5 11)(6 24)(7 9)(8 22)(10 20)(12 18)(14 16)(17 27)(19 25)(21 23)(29 44 43 30)(31 42 45 56)(32 55 46 41)(33 40 47 54)(34 53 48 39)(35 38 49 52)(36 51 50 37)

(1 8)(2 7)(3 6)(4 5)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(52 56)(53 55)
 
G:=sub<Sym(56)| (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,40), (1,15)(2,28)(3,13)(4,26)(5,11)(6,24)(7,9)(8,22)(10,20)(12,18)(14,16)(17,27)(19,25)(21,23)(29,44,43,30)(31,42,45,56)(32,55,46,41)(33,40,47,54)(34,53,48,39)(35,38,49,52)(36,51,50,37), (1,8)(2,7)(3,6)(4,5)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(52,56)(53,55)>;
 
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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,40), (1,15)(2,28)(3,13)(4,26)(5,11)(6,24)(7,9)(8,22)(10,20)(12,18)(14,16)(17,27)(19,25)(21,23)(29,44,43,30)(31,42,45,56)(32,55,46,41)(33,40,47,54)(34,53,48,39)(35,38,49,52)(36,51,50,37), (1,8)(2,7)(3,6)(4,5)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(52,56)(53,55) );
 
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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,56),(13,55),(14,54),(15,53),(16,52),(17,51),(18,50),(19,49),(20,48),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(28,40)], [(1,15),(2,28),(3,13),(4,26),(5,11),(6,24),(7,9),(8,22),(10,20),(12,18),(14,16),(17,27),(19,25),(21,23),(29,44,43,30),(31,42,45,56),(32,55,46,41),(33,40,47,54),(34,53,48,39),(35,38,49,52),(36,51,50,37)], [(1,8),(2,7),(3,6),(4,5),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(52,56),(53,55)]])
 

49 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F7A7B7C8A8B14A14B14C14D14E14F14G···14L28A···28F56A···56L
order122222224444447778814141414141414···1428···2856···56
size112442828562228282828222882224448···84···48···8

49 irreducible representations

dim1111112222222448
type++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D7D14D14D28D4:4D4D4xD7D28:1D4
kernelD28:1D4D28:4C4C7xC4.D4C8:D14C28:D4D4:6D14Dic14D28C22xC14C4.D4M4(2)C2xD4C23C7C4C1
# reps12121122236312263

Matrix representation of D28:1D4 ►in GL8(F113)

10302400000
01030240000
890100000
089010000
00000100
0000112000
0000000112
00000010
,
2303600000
0230360000
2309000000
0230900000
00000001
0000001120
0000011200
00001000
,
0112000000
10000000
089010000
24011200000
0000112000
0000011200
00000001
0000001120
,
01000000
10000000
02401120000
24011200000
00000100
00001000
00000010
0000000112

G:=sub<GL(8,GF(113))| [103,0,89,0,0,0,0,0,0,103,0,89,0,0,0,0,24,0,1,0,0,0,0,0,0,24,0,1,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0],[23,0,23,0,0,0,0,0,0,23,0,23,0,0,0,0,36,0,90,0,0,0,0,0,0,36,0,90,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0],[0,1,0,24,0,0,0,0,112,0,89,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0],[0,1,0,24,0,0,0,0,1,0,24,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112] >;
 

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

D_{28}\rtimes_1D_4
 
% in TeX
 
G:=Group("D28:1D4");
 
// GroupNames label
 
G:=SmallGroup(448,281);
 
// by ID
 
G=gap.SmallGroup(448,281);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,226,1123,570,136,1684,438,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^5*b,d*b*d=a^19*b,d*c*d=c^-1>;
 
// generators/relations
 

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