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G = Q84D8order 128 = 27

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

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

Aliases: Q84D8, C42.499C23, C4.202- 1+4, (C8×Q8)⋊7C2, (C4×D8)⋊16C2, (D4×Q8)⋊10C2, C4.46(C2×D8), D48(C4○D4), C4⋊D815C2, C4⋊C4.272D4, Q82(D4⋊C4), D4⋊Q816C2, Q86D410C2, C4.4D817C2, (C4×C8).91C22, (C2×Q8).266D4, C2.21(C22×D8), C4⋊C4.426C23, C4⋊C8.302C22, (C2×C4).550C24, (C2×C8).204C23, C4⋊Q8.179C22, C2.58(Q85D4), C2.96(D4○SD16), (C4×D4).190C22, (C2×D8).145C22, (C2×D4).265C23, C41D4.94C22, (C4×Q8).304C22, C2.D8.198C22, D4⋊C4.17C22, C22.810(C22×D4), C4.251(C2×C4○D4), (C2×C4).1098(C2×D4), SmallGroup(128,2090)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q84D8
C1C2C4C2×C4C42C4×D4D4×Q8 — Q84D8
C1C2C2×C4 — Q84D8
C1C22C4×Q8 — Q84D8
C1C2C2C2×C4 — Q84D8

Generators and relations for Q84D8
 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 [×3], C2 [×5], C4 [×2], C4 [×6], C4 [×7], C22, C22 [×13], C8 [×4], C2×C4, C2×C4 [×6], C2×C4 [×16], D4 [×2], D4 [×16], Q8 [×4], Q8 [×6], C23 [×4], C42 [×3], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×6], C4⋊C4 [×3], C2×C8, C2×C8 [×3], D8 [×6], C22×C4 [×6], C2×D4, C2×D4 [×3], C2×D4 [×6], C2×Q8, C2×Q8 [×7], C4○D4 [×4], C4×C8 [×3], D4⋊C4, D4⋊C4 [×9], C4⋊C8 [×3], C2.D8 [×3], C4×D4 [×6], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C41D4 [×3], C4⋊Q8 [×3], C2×D8 [×3], C22×Q8, C2×C4○D4, C4×D8 [×3], C8×Q8, C4⋊D8 [×3], D4⋊Q8 [×3], C4.4D8 [×3], D4×Q8, Q86D4, Q84D8
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, Q85D4, C22×D8, D4○SD16, Q84D8

Smallest permutation representation of Q84D8
On 64 points
Generators in S64
(1 49 62 31)(2 50 63 32)(3 51 64 25)(4 52 57 26)(5 53 58 27)(6 54 59 28)(7 55 60 29)(8 56 61 30)(9 19 38 47)(10 20 39 48)(11 21 40 41)(12 22 33 42)(13 23 34 43)(14 24 35 44)(15 17 36 45)(16 18 37 46)
(1 46 62 18)(2 19 63 47)(3 48 64 20)(4 21 57 41)(5 42 58 22)(6 23 59 43)(7 44 60 24)(8 17 61 45)(9 32 38 50)(10 51 39 25)(11 26 40 52)(12 53 33 27)(13 28 34 54)(14 55 35 29)(15 30 36 56)(16 49 37 31)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 40)(7 39)(8 38)(9 61)(10 60)(11 59)(12 58)(13 57)(14 64)(15 63)(16 62)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 56)(48 55)

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

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

G=PermutationGroup([(1,49,62,31),(2,50,63,32),(3,51,64,25),(4,52,57,26),(5,53,58,27),(6,54,59,28),(7,55,60,29),(8,56,61,30),(9,19,38,47),(10,20,39,48),(11,21,40,41),(12,22,33,42),(13,23,34,43),(14,24,35,44),(15,17,36,45),(16,18,37,46)], [(1,46,62,18),(2,19,63,47),(3,48,64,20),(4,21,57,41),(5,42,58,22),(6,23,59,43),(7,44,60,24),(8,17,61,45),(9,32,38,50),(10,51,39,25),(11,26,40,52),(12,53,33,27),(13,28,34,54),(14,55,35,29),(15,30,36,56),(16,49,37,31)], [(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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,40),(7,39),(8,38),(9,61),(10,60),(11,59),(12,58),(13,57),(14,64),(15,63),(16,62),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25),(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+++++++++++-
imageC1C2C2C2C2C2C2C2D4D4C4○D4D82- 1+4D4○SD16
kernelQ84D8C4×D8C8×Q8C4⋊D8D4⋊Q8C4.4D8D4×Q8Q86D4C4⋊C4C2×Q8D4Q8C4C2
# reps13133311314812

Matrix representation of Q84D8 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] >;

Q84D8 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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