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G = D28:18D4  order 448 = 26·7

6th semidirect product of D28 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D28:18D4, Dic14:18D4, M4(2):5D14, (C7xD4):7D4, (C7xQ8):7D4, (C2xD4):4D14, C8:C22:1D7, C28:D4:7C2, D4:4(C7:D4), C7:4(D4:4D4), C4oD4.5D14, Q8:4(C7:D4), C4.104(D4xD7), D4:8D14:2C2, D28:4C4:9C2, C28.194(C2xD4), (D4xC14):4C22, (C22xD7).5D4, C22.35(D4xD7), C14.63C22wrC2, D4:2Dic7:6C2, D4.D14:5C2, C28.46D4:9C2, (C2xC28).13C23, (C4xDic7):5C22, C4.Dic7:8C22, C4oD28.23C22, C2.31(C23:D14), (C2xD28).128C22, (C7xM4(2)):15C22, (C7xC8:C22):5C2, C4.50(C2xC7:D4), (C2xC14).34(C2xD4), (C2xC4).13(C22xD7), (C7xC4oD4).11C22, SmallGroup(448,732)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC28 — D28:18D4
Chief series
C1 — C7 — C14 — C2xC14 — C2xC28 — C2xD28 — D4:8D14 — D28:18D4
Lower central
C7 — C14 — C2xC28 — D28:18D4
Upper central
C1 — C2 — C2xC4 — C8:C22

Generators and relations for D28:18D4
 G = < a,b,c,d | a28=b2=c4=d2=1, bab=dad=a-1, cac-1=a13, cbc-1=a5b, dbd=a26b, dcd=c-1 >

Subgroups: 1132 in 168 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, D7, C14, C14, C42, M4(2), M4(2), D8, SD16, C2xD4, C2xD4, C4oD4, C4oD4, Dic7, C28, C28, D14, C2xC14, C2xC14, C4.D4, C4wrC2, C4:1D4, C8:C22, C8:C22, 2+ 1+4, C7:C8, C56, Dic14, C4xD7, D28, D28, C2xDic7, C7:D4, C2xC28, C2xC28, C7xD4, C7xD4, C7xQ8, C22xD7, C22xD7, C22xC14, D4:4D4, C4.Dic7, C4xDic7, D4:D7, D4.D7, C7xM4(2), C7xD8, C7xSD16, C2xD28, C2xD28, C4oD28, C4oD28, D4xD7, Q8:2D7, C2xC7:D4, D4xC14, C7xC4oD4, C28.46D4, D28:4C4, D4:2Dic7, D4.D14, C28:D4, C7xC8:C22, D4:8D14, D28:18D4
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, D14, C22wrC2, C7:D4, C22xD7, D4:4D4, D4xD7, C2xC7:D4, C23:D14, D28:18D4

Smallest permutation representation of D28:18D4
►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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)

(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 36 43 50)(30 49 44 35)(31 34 45 48)(32 47 46 33)(37 56 51 42)(38 41 52 55)(39 54 53 40)

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

49 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F7A7B7C8A8B14A14B14C14D14E14F14G···14O28A···28F28G28H28I56A···56F
order122222224444447778814141414141414···1428···2828282856···56
size112482828282242828282228562224448···84···48888···8

49 irreducible representations

dim11111111222222222224448
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D4D7D14D14D14C7:D4C7:D4D4:4D4D4xD7D4xD7D28:18D4
kernelD28:18D4C28.46D4D28:4C4D4:2Dic7D4.D14C28:D4C7xC8:C22D4:8D14Dic14D28C7xD4C7xQ8C22xD7C8:C22M4(2)C2xD4C4oD4D4Q8C7C4C22C1
# reps11111111111123333662333

Matrix representation of D28:18D4 ►in GL8(F113)

124000000
89103000000
001240000
00891030000
0000111100
0000111200
0000001111
0000001112
,
0011200000
002410000
1120000000
241000000
0000001120
0000001121
0000112000
0000112100
,
10389000000
10310000000
0010240000
00101030000
0000112000
0000011200
0000001122
0000001121
,
1024000000
10103000000
0010240000
00101030000
0000111100
0000011200
0000001111
0000000112

G:=sub<GL(8,GF(113))| [1,89,0,0,0,0,0,0,24,103,0,0,0,0,0,0,0,0,1,89,0,0,0,0,0,0,24,103,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,111,112,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,111,112],[0,0,112,24,0,0,0,0,0,0,0,1,0,0,0,0,112,24,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,112,112,0,0,0,0,0,0,0,1,0,0,0,0,112,112,0,0,0,0,0,0,0,1,0,0],[103,103,0,0,0,0,0,0,89,10,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,0,24,103,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,112,0,0,0,0,0,0,2,1],[10,10,0,0,0,0,0,0,24,103,0,0,0,0,0,0,0,0,10,10,0,0,0,0,0,0,24,103,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,111,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,111,112] >;
 

D28:18D4 in GAP, Magma, Sage, TeX 

D_{28}\rtimes_{18}D_4
 
% in TeX
 
G:=Group("D28:18D4");
 
// GroupNames label
 
G:=SmallGroup(448,732);
 
// by ID
 
G=gap.SmallGroup(448,732);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,570,1684,851,438,102,18822]);
 
// 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^26*b,d*c*d=c^-1>;
 
// generators/relations
 

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