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G = D813D4order 128 = 27

2nd semidirect product of D8 and D4 acting through Inn(D8)

p-group, metabelian, nilpotent (class 3), monomial

Aliases: D813D4, C42.450C23, C4.1372+ 1+4, C4⋊C42D8, C2.67D42, (C8×D4)⋊12C2, (C4×D8)⋊10C2, C42(C4○D8), C8.80(C2×D4), D46D45C2, C4⋊C4.405D4, D4.29(C2×D4), C4⋊Q1611C2, D4.7D46C2, (C2×D4).229D4, (C4×C8).80C22, C2.41(Q8○D8), C22⋊C4.94D4, C4.97(C22×D4), D4.D442C2, C8.18D421C2, C4⋊C8.341C22, C4⋊C4.222C23, (C2×C8).569C23, (C2×C4).481C24, C23.103(C2×D4), C4⋊Q8.137C22, (C4×D4).151C22, (C2×D4).419C23, (C2×D8).175C22, (C2×Q16).35C22, (C2×Q8).202C23, C2.D8.187C22, C22⋊Q8.65C22, C22⋊C8.198C22, (C22×C8).192C22, C22.741(C22×D4), D4⋊C4.166C22, (C22×C4).1125C23, Q8⋊C4.111C22, (C2×SD16).152C22, C4⋊C4(C2×D8), (C2×C4○D8)⋊11C2, C2.54(C2×C4○D8), (C2×C4).919(C2×D4), (C2×C4○D4).192C22, SmallGroup(128,2015)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D813D4
C1C2C22C2×C4C2×D4C2×C4○D4C2×C4○D8 — D813D4
C1C2C2×C4 — D813D4
C1C22C4×D4 — D813D4
C1C2C2C2×C4 — D813D4

Generators and relations for D813D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 472 in 242 conjugacy classes, 96 normal (24 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×14], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×4], D4 [×12], Q8 [×8], C23 [×2], C23 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], D8 [×4], SD16 [×8], Q16 [×6], C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8 [×4], C4○D4 [×12], C4×C8, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×4], C4⋊C8, C2.D8, C2×C4⋊C4 [×2], C4×D4, C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×4], C22.D4 [×4], C4⋊Q8 [×2], C22×C8 [×2], C2×D8, C2×SD16 [×4], C2×Q16 [×4], C4○D8 [×8], C2×C4○D4 [×4], C8×D4, C4×D8, D4.7D4 [×4], D4.D4 [×2], C8.18D4 [×2], C4⋊Q16, D46D4 [×2], C2×C4○D8 [×2], D813D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C4○D8 [×2], C22×D4 [×2], 2+ 1+4, D42, C2×C4○D8, Q8○D8, D813D4

Smallest permutation representation of D813D4
On 64 points
Generators in S64
(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)
(1 9)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 42)(18 41)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 35)(26 34)(27 33)(28 40)(29 39)(30 38)(31 37)(32 36)(49 57)(50 64)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)
(1 19 10 41)(2 20 11 42)(3 21 12 43)(4 22 13 44)(5 23 14 45)(6 24 15 46)(7 17 16 47)(8 18 9 48)(25 56 36 59)(26 49 37 60)(27 50 38 61)(28 51 39 62)(29 52 40 63)(30 53 33 64)(31 54 34 57)(32 55 35 58)
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 64)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)

G:=sub<Sym(64)| (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), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,42)(18,41)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,35)(26,34)(27,33)(28,40)(29,39)(30,38)(31,37)(32,36)(49,57)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58), (1,19,10,41)(2,20,11,42)(3,21,12,43)(4,22,13,44)(5,23,14,45)(6,24,15,46)(7,17,16,47)(8,18,9,48)(25,56,36,59)(26,49,37,60)(27,50,38,61)(28,51,39,62)(29,52,40,63)(30,53,33,64)(31,54,34,57)(32,55,35,58), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)>;

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)(57,58,59,60,61,62,63,64), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,42)(18,41)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,35)(26,34)(27,33)(28,40)(29,39)(30,38)(31,37)(32,36)(49,57)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58), (1,19,10,41)(2,20,11,42)(3,21,12,43)(4,22,13,44)(5,23,14,45)(6,24,15,46)(7,17,16,47)(8,18,9,48)(25,56,36,59)(26,49,37,60)(27,50,38,61)(28,51,39,62)(29,52,40,63)(30,53,33,64)(31,54,34,57)(32,55,35,58), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47) );

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),(57,58,59,60,61,62,63,64)], [(1,9),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,42),(18,41),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(25,35),(26,34),(27,33),(28,40),(29,39),(30,38),(31,37),(32,36),(49,57),(50,64),(51,63),(52,62),(53,61),(54,60),(55,59),(56,58)], [(1,19,10,41),(2,20,11,42),(3,21,12,43),(4,22,13,44),(5,23,14,45),(6,24,15,46),(7,17,16,47),(8,18,9,48),(25,56,36,59),(26,49,37,60),(27,50,38,61),(28,51,39,62),(29,52,40,63),(30,53,33,64),(31,54,34,57),(32,55,35,58)], [(1,54),(2,55),(3,56),(4,49),(5,50),(6,51),(7,52),(8,53),(9,64),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28),(33,48),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47)])

35 conjugacy classes

class 1 2A2B2C2D···2I4A···4H4I4J···4O8A8B8C8D8E···8J
order12222···24···444···488888···8
size11114···42···248···822224···4

35 irreducible representations

dim1111111112222244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D4D4D4C4○D82+ 1+4Q8○D8
kernelD813D4C8×D4C4×D8D4.7D4D4.D4C8.18D4C4⋊Q16D46D4C2×C4○D8C22⋊C4C4⋊C4D8C2×D4C4C4C2
# reps1114221222141812

Matrix representation of D813D4 in GL4(𝔽17) generated by

16000
01600
0006
00146
,
1000
0100
0006
0030
,
5300
141200
00160
00016
,
141200
5300
00138
00134
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,14,0,0,6,6],[1,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0],[5,14,0,0,3,12,0,0,0,0,16,0,0,0,0,16],[14,5,0,0,12,3,0,0,0,0,13,13,0,0,8,4] >;

D813D4 in GAP, Magma, Sage, TeX

D_8\rtimes_{13}D_4
% in TeX

G:=Group("D8:13D4");
// GroupNames label

G:=SmallGroup(128,2015);
// by ID

G=gap.SmallGroup(128,2015);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,352,2019,346,2804,1411,375,172]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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