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G = Q1612D4order 128 = 27

1st semidirect product of Q16 and D4 acting through Inn(Q16)

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

Aliases: Q1612D4, C42.452C23, C4.1392+ 1+4, C2.69D42, (C8×D4)⋊14C2, C8.81(C2×D4), C22⋊C42Q16, C87D410C2, C4⋊C4.256D4, (C4×Q16)⋊24C2, Q85D47C2, Q8.28(C2×D4), Q8⋊D433C2, D4.7D48C2, (C2×D4).231D4, C8.12D49C2, C224(C4○D8), Q8.D46C2, C2.42(Q8○D8), C4.99(C22×D4), C8.18D422C2, C4⋊C8.317C22, C4⋊C4.224C23, (C2×C8).345C23, (C4×C8).119C22, (C2×C4).483C24, C22⋊C4.194D4, (C22×Q16)⋊12C2, C23.467(C2×D4), (C2×D4).217C23, (C4×D4).326C22, (C2×D8).137C22, C4⋊D4.68C22, (C4×Q8).144C22, (C2×Q8).391C23, C2.D8.188C22, C22⋊Q8.67C22, D4⋊C4.11C22, C22⋊C8.222C22, (C22×C8).194C22, (C2×Q16).170C22, (C2×SD16).95C22, C4.4D4.56C22, C22.743(C22×D4), (C22×C4).1127C23, Q8⋊C4.179C22, (C22×Q8).335C22, C22⋊C4(C2×Q16), (C2×C4○D8)⋊13C2, C2.56(C2×C4○D8), (C2×C4).921(C2×D4), (C2×C4○D4).194C22, SmallGroup(128,2017)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q1612D4
C1C2C22C2×C4C2×Q8C22×Q8C22×Q16 — Q1612D4
C1C2C2×C4 — Q1612D4
C1C22C4×D4 — Q1612D4
C1C2C2C2×C4 — Q1612D4

Generators and relations for Q1612D4
 G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=cac-1=dad=a-1, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 472 in 241 conjugacy classes, 96 normal (44 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×11], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×18], D4 [×14], Q8 [×4], Q8 [×10], C23 [×2], C23 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×4], D8 [×2], SD16 [×6], Q16 [×4], Q16 [×6], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8 [×4], C2×Q8 [×8], C4○D4 [×6], C4×C8, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×4], C4⋊C8, C2.D8, C4×D4, C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C4.4D4 [×2], C22×C8 [×2], C2×D8, C2×SD16 [×4], C2×Q16 [×2], C2×Q16 [×2], C2×Q16 [×4], C4○D8 [×4], C22×Q8 [×2], C2×C4○D4 [×2], C8×D4, C4×Q16, Q8⋊D4 [×2], D4.7D4 [×2], Q8.D4 [×2], C87D4, C8.18D4, C8.12D4, Q85D4 [×2], C22×Q16, C2×C4○D8, Q1612D4
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, Q1612D4

Smallest permutation representation of Q1612D4
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 30 5 26)(2 29 6 25)(3 28 7 32)(4 27 8 31)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)
(1 34 19 48)(2 33 20 47)(3 40 21 46)(4 39 22 45)(5 38 23 44)(6 37 24 43)(7 36 17 42)(8 35 18 41)(9 56 58 31)(10 55 59 30)(11 54 60 29)(12 53 61 28)(13 52 62 27)(14 51 63 26)(15 50 64 25)(16 49 57 32)
(2 8)(3 7)(4 6)(9 64)(10 63)(11 62)(12 61)(13 60)(14 59)(15 58)(16 57)(17 21)(18 20)(22 24)(25 31)(26 30)(27 29)(33 41)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(50 56)(51 55)(52 54)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4F4G···4L4M4N4O4P8A8B8C8D8E···8J
order1222222224···44···4444488888···8
size1111224882···24···4888822224···4

35 irreducible representations

dim1111111111112222244
type+++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D4C4○D82+ 1+4Q8○D8
kernelQ1612D4C8×D4C4×Q16Q8⋊D4D4.7D4Q8.D4C87D4C8.18D4C8.12D4Q85D4C22×Q16C2×C4○D8C22⋊C4C4⋊C4Q16C2×D4C22C4C2
# reps1112221112112141812

Matrix representation of Q1612D4 in GL4(𝔽17) generated by

16000
01600
00011
00311
,
16000
01600
00138
0004
,
01600
1000
00160
00161
,
1000
01600
0010
00116
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,3,0,0,11,11],[16,0,0,0,0,16,0,0,0,0,13,0,0,0,8,4],[0,1,0,0,16,0,0,0,0,0,16,16,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;

Q1612D4 in GAP, Magma, Sage, TeX

Q_{16}\rtimes_{12}D_4
% in TeX

G:=Group("Q16:12D4");
// GroupNames label

G:=SmallGroup(128,2017);
// by ID

G=gap.SmallGroup(128,2017);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,2019,346,248,2804,1411,375,172]);
// Polycyclic

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

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