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G = D45D8order 128 = 27

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

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

Aliases: D45D8, C42.483C23, C4.692- 1+4, (C8×D4)⋊9C2, (C4×D8)⋊14C2, C4.45(C2×D8), C814(C4○D4), D42(C2.D8), C87D415C2, C4⋊C4.266D4, C82Q817C2, D46D414C2, C22.5(C2×D8), D4⋊Q814C2, (C2×D4).354D4, (C4×C8).87C22, C2.54(Q8○D8), C2.20(C22×D8), C4⋊C4.239C23, C4⋊C8.299C22, (C2×C8).195C23, (C2×C4).526C24, C22⋊C4.110D4, C22.D89C2, C23.477(C2×D4), C4⋊Q8.161C22, C2.79(D46D4), C2.D8.61C22, (C4×D4).339C22, (C2×D4).248C23, (C2×D8).142C22, C4⋊D4.97C22, C22⋊C8.185C22, (C22×C8).164C22, C22.786(C22×D4), D4⋊C4.169C22, (C22×C4).1158C23, (C2×D4)(C2.D8), (C2×C2.D8)⋊28C2, C4.108(C2×C4○D4), (C2×C4).171(C2×D4), (C2×C4⋊C4).678C22, SmallGroup(128,2066)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D45D8
C1C2C4C2×C4C22×C4C2×C4⋊C4D46D4 — D45D8
C1C2C2×C4 — D45D8
C1C22C4×D4 — D45D8
C1C2C2C2×C4 — D45D8

Generators and relations for D45D8
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 424 in 210 conjugacy classes, 96 normal (24 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×4], C22 [×10], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×4], D4 [×12], Q8 [×4], C23 [×2], C23 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], D8 [×2], C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C4×C8, C22⋊C8 [×2], D4⋊C4 [×6], C4⋊C8, C2.D8, C2.D8 [×8], C2×C4⋊C4 [×4], C4×D4, C4×D4 [×2], C4⋊D4 [×4], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8 [×2], C22×C8 [×2], C2×D8, C2×C4○D4 [×2], C2×C2.D8 [×2], C8×D4, C4×D8, C87D4 [×2], D4⋊Q8 [×2], C22.D8 [×4], C82Q8, D46D4 [×2], D45D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C4○D4 [×2], C24, C2×D8 [×6], C22×D4, C2×C4○D4, 2- 1+4, D46D4, C22×D8, Q8○D8, D45D8

Smallest permutation representation of D45D8
On 64 points
Generators in S64
(1 10 59 38)(2 11 60 39)(3 12 61 40)(4 13 62 33)(5 14 63 34)(6 15 64 35)(7 16 57 36)(8 9 58 37)(17 55 43 31)(18 56 44 32)(19 49 45 25)(20 50 46 26)(21 51 47 27)(22 52 48 28)(23 53 41 29)(24 54 42 30)
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 49)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)
(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 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 42)(18 41)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 52)(26 51)(27 50)(28 49)(29 56)(30 55)(31 54)(32 53)(33 34)(35 40)(36 39)(37 38)(57 60)(58 59)(61 64)(62 63)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4K4L4M4N4O8A8B8C8D8E···8J
order122222222244444···4444488888···8
size111122228822224···4888822224···4

35 irreducible representations

dim1111111112222244
type+++++++++++++--
imageC1C2C2C2C2C2C2C2C2D4D4D4C4○D4D82- 1+4Q8○D8
kernelD45D8C2×C2.D8C8×D4C4×D8C87D4D4⋊Q8C22.D8C82Q8D46D4C22⋊C4C4⋊C4C2×D4C8D4C4C2
# reps1211224122114812

Matrix representation of D45D8 in GL4(𝔽17) generated by

1000
0100
00130
0044
,
16000
01600
0048
001313
,
14300
141400
0010
0001
,
14300
3300
0010
001616
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,13,4,0,0,0,4],[16,0,0,0,0,16,0,0,0,0,4,13,0,0,8,13],[14,14,0,0,3,14,0,0,0,0,1,0,0,0,0,1],[14,3,0,0,3,3,0,0,0,0,1,16,0,0,0,16] >;

D45D8 in GAP, Magma, Sage, TeX

D_4\rtimes_5D_8
% in TeX

G:=Group("D4:5D8");
// GroupNames label

G:=SmallGroup(128,2066);
// by ID

G=gap.SmallGroup(128,2066);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,100,346,4037,1027,124]);
// Polycyclic

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

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