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G = Q8:4D8order 128 = 27

1st semidirect product of Q8 and D8 acting through Inn(Q8)

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

Aliases: Q8:4D8, C42.499C23, C4.202- 1+4, (C8xQ8):7C2, (C4xD8):16C2, (D4xQ8):10C2, C4.46(C2xD8), D4:8(C4oD4), C4:D8:15C2, C4:C4.272D4, Q8o2(D4:C4), D4:Q8:16C2, Q8:6D4:10C2, C4.4D8:17C2, (C4xC8).91C22, (C2xQ8).266D4, C2.21(C22xD8), C4:C4.426C23, C4:C8.302C22, (C2xC4).550C24, (C2xC8).204C23, C4:Q8.179C22, C2.58(Q8:5D4), C2.96(D4oSD16), (C4xD4).190C22, (C2xD8).145C22, (C2xD4).265C23, C4:1D4.94C22, (C4xQ8).304C22, C2.D8.198C22, D4:C4.17C22, C22.810(C22xD4), C4.251(C2xC4oD4), (C2xC4).1098(C2xD4), SmallGroup(128,2090)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — Q8:4D8
C1C2C4C2xC4C42C4xD4D4xQ8 — Q8:4D8
C1C2C2xC4 — Q8:4D8
C1C22C4xQ8 — Q8:4D8
C1C2C2C2xC4 — Q8:4D8

Generators and relations for Q8:4D8
 G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 464 in 213 conjugacy classes, 96 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C42, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, D8, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C2xQ8, C4oD4, C4xC8, D4:C4, D4:C4, C4:C8, C2.D8, C4xD4, C4xQ8, C4:D4, C22:Q8, C4:1D4, C4:Q8, C2xD8, C22xQ8, C2xC4oD4, C4xD8, C8xQ8, C4:D8, D4:Q8, C4.4D8, D4xQ8, Q8:6D4, Q8:4D8
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C4oD4, C24, C2xD8, C22xD4, C2xC4oD4, 2- 1+4, Q8:5D4, C22xD8, D4oSD16, Q8:4D8

Smallest permutation representation of Q8:4D8
On 64 points
Generators in S64
(1 49 62 29)(2 50 63 30)(3 51 64 31)(4 52 57 32)(5 53 58 25)(6 54 59 26)(7 55 60 27)(8 56 61 28)(9 23 40 47)(10 24 33 48)(11 17 34 41)(12 18 35 42)(13 19 36 43)(14 20 37 44)(15 21 38 45)(16 22 39 46)
(1 46 62 22)(2 23 63 47)(3 48 64 24)(4 17 57 41)(5 42 58 18)(6 19 59 43)(7 44 60 20)(8 21 61 45)(9 30 40 50)(10 51 33 31)(11 32 34 52)(12 53 35 25)(13 26 36 54)(14 55 37 27)(15 28 38 56)(16 49 39 29)
(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 40)(9 61)(10 60)(11 59)(12 58)(13 57)(14 64)(15 63)(16 62)(17 26)(18 25)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 56)(48 55)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4H4I···4M4N4O4P8A8B8C8D8E···8J
order1222222224···44···444488888···8
size1111448882···24···488822224···4

35 irreducible representations

dim11111111222244
type+++++++++++-
imageC1C2C2C2C2C2C2C2D4D4C4oD4D82- 1+4D4oSD16
kernelQ8:4D8C4xD8C8xQ8C4:D8D4:Q8C4.4D8D4xQ8Q8:6D4C4:C4C2xQ8D4Q8C4C2
# reps13133311314812

Matrix representation of Q8:4D8 in GL4(F17) generated by

16000
01600
0001
00160
,
1000
0100
00130
0004
,
0600
14600
0004
00130
,
16000
16100
0004
00130
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,4],[0,14,0,0,6,6,0,0,0,0,0,13,0,0,4,0],[16,16,0,0,0,1,0,0,0,0,0,13,0,0,4,0] >;

Q8:4D8 in GAP, Magma, Sage, TeX

Q_8\rtimes_4D_8
% in TeX

G:=Group("Q8:4D8");
// GroupNames label

G:=SmallGroup(128,2090);
// by ID

G=gap.SmallGroup(128,2090);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,346,80,4037,1027,124]);
// Polycyclic

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

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