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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(D4×D7), C28⋊D4⋊1C2, C8⋊D14⋊5C2, C7⋊1(D4⋊4D4), C28.92(C2×D4), C4.D4⋊2D7, D4⋊6D14⋊2C2, D28⋊4C4⋊2C2, (C2×D4).14D14, (C2×C28).4C23, C14.16C22≀C2, (C2×D28)⋊11C22, (C4×Dic7)⋊2C22, C4○D28.2C22, (C22×C14).20D4, C22.11(C2×D28), (D4×C14).14C22, (C7×M4(2))⋊9C22, C2.19(C22⋊D28), (C7×C4.D4)⋊4C2, (C2×C14).21(C2×D4), (C2×C4).4(C22×D7), SmallGroup(448,281)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C28 — D28⋊1D4
Chief series
C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — D4⋊6D14 — D28⋊1D4
Lower central
C7 — C14 — C2×C28 — D28⋊1D4
Upper central
C1 — C2 — C2×C4 — 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, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C4.D4, C4≀C2, C4⋊1D4, C8⋊C22, 2+ 1+4, C56, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, D4⋊4D4, C56⋊C2, D56, C4×Dic7, C7×M4(2), C2×D28, C4○D28, D4×D7, D4⋊2D7, C2×C7⋊D4, D4×C14, D28⋊4C4, C7×C4.D4, C8⋊D14, C28⋊D4, D4⋊6D14, D28⋊1D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, D28, C22×D7, D4⋊4D4, C2×D28, D4×D7, 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⋊4D4D4×D7D28⋊1D4
kernelD28⋊1D4D28⋊4C4C7×C4.D4C8⋊D14C28⋊D4D4⋊6D14Dic14D28C22×C14C4.D4M4(2)C2×D4C23C7C4C1
# 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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