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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, (C7×D4)⋊7D4, (C7×Q8)⋊7D4, (C2×D4)⋊4D14, C8⋊C22⋊1D7, C28⋊D4⋊7C2, D4⋊4(C7⋊D4), C7⋊4(D4⋊4D4), C4○D4.5D14, Q8⋊4(C7⋊D4), C4.104(D4×D7), D4⋊8D14⋊2C2, D28⋊4C4⋊9C2, C28.194(C2×D4), (D4×C14)⋊4C22, (C22×D7).5D4, C22.35(D4×D7), C14.63C22≀C2, D4⋊2Dic7⋊6C2, D4.D14⋊5C2, C28.46D4⋊9C2, (C2×C28).13C23, (C4×Dic7)⋊5C22, C4.Dic7⋊8C22, C4○D28.23C22, C2.31(C23⋊D14), (C2×D28).128C22, (C7×M4(2))⋊15C22, (C7×C8⋊C22)⋊5C2, C4.50(C2×C7⋊D4), (C2×C14).34(C2×D4), (C2×C4).13(C22×D7), (C7×C4○D4).11C22, SmallGroup(448,732)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C28 — D28⋊18D4
Chief series
C1 — C7 — C14 — C2×C14 — C2×C28 — C2×D28 — D4⋊8D14 — D28⋊18D4
Lower central
C7 — C14 — C2×C28 — D28⋊18D4
Upper central
C1 — C2 — C2×C4 — 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, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C42, M4(2), M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.D4, C4≀C2, C4⋊1D4, C8⋊C22, C8⋊C22, 2+ 1+4, C7⋊C8, C56, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, C22×C14, D4⋊4D4, C4.Dic7, C4×Dic7, D4⋊D7, D4.D7, C7×M4(2), C7×D8, C7×SD16, C2×D28, C2×D28, C4○D28, C4○D28, D4×D7, Q8⋊2D7, C2×C7⋊D4, D4×C14, C7×C4○D4, C28.46D4, D28⋊4C4, D4⋊2Dic7, D4.D14, C28⋊D4, C7×C8⋊C22, D4⋊8D14, D28⋊18D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, D4⋊4D4, D4×D7, C2×C7⋊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⋊4D4D4×D7D4×D7D28⋊18D4
kernelD28⋊18D4C28.46D4D28⋊4C4D4⋊2Dic7D4.D14C28⋊D4C7×C8⋊C22D4⋊8D14Dic14D28C7×D4C7×Q8C22×D7C8⋊C22M4(2)C2×D4C4○D4D4Q8C7C4C22C1
# 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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