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G = (C2×Q8)⋊17D4order 128 = 27

13rd semidirect product of C2×Q8 and D4 acting via D4/C2=C22

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

Aliases: C4○D44D4, (C2×Q8)⋊17D4, Q8⋊D44C2, D4.48(C2×D4), Q8.48(C2×D4), C2.7(Q8○D8), D4⋊D418C2, C4.85C22≀C2, (C2×D4).296D4, C4⋊C4.15C23, C4.50(C22×D4), D4.7D418C2, (C2×C8).131C23, (C2×C4).232C24, C22⋊Q1614C2, C22.4C22≀C2, (C2×D4).32C23, C23.236(C2×D4), C2.11(D4○SD16), (C2×D8).115C22, C4⋊D4.18C22, C23.36D44C2, C22⋊C8.13C22, (C2×2- 1+4)⋊2C2, (C2×Q8).356C23, C22⋊Q8.18C22, D4⋊C4.19C22, (C22×C8).176C22, (C22×C4).280C23, Q8⋊C4.21C22, (C2×Q16).114C22, C22.492(C22×D4), C22.31C244C2, (C2×SD16).130C22, (C2×M4(2)).43C22, (C22×Q8).263C22, (C2×C4○D8)⋊4C2, (C2×C4).459(C2×D4), (C22×C8)⋊C28C2, (C2×Q8⋊C4)⋊28C2, C2.50(C2×C22≀C2), (C2×C8.C22)⋊10C2, (C2×C4⋊C4).582C22, (C2×C4○D4).103C22, SmallGroup(128,1745)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — (C2×Q8)⋊17D4
C1C2C22C2×C4C22×C4C2×C4○D4C2×2- 1+4 — (C2×Q8)⋊17D4
C1C2C2×C4 — (C2×Q8)⋊17D4
C1C22C2×C4○D4 — (C2×Q8)⋊17D4
C1C2C2C2×C4 — (C2×Q8)⋊17D4

Generators and relations for (C2×Q8)⋊17D4
 G = < a,b,c,d,e | a2=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, eae=ab2, cbc-1=b-1, dbd-1=ab-1c, ebe=abc, dcd-1=b2c, ce=ec, ede=d-1 >

Subgroups: 660 in 356 conjugacy classes, 108 normal (38 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×15], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×37], D4 [×2], D4 [×25], Q8 [×6], Q8 [×19], C23, C23 [×2], C23 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], D8 [×2], SD16 [×8], Q16 [×6], C22×C4, C22×C4 [×2], C22×C4 [×8], C2×D4 [×3], C2×D4 [×2], C2×D4 [×6], C2×Q8 [×3], C2×Q8 [×6], C2×Q8 [×21], C4○D4 [×4], C4○D4 [×42], C22⋊C8 [×4], D4⋊C4 [×2], Q8⋊C4 [×6], C2×C4⋊C4, C4⋊D4 [×2], C4⋊D4 [×3], C22⋊Q8 [×2], C22⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16 [×2], C2×SD16 [×2], C2×Q16, C2×Q16 [×2], C4○D8 [×4], C8.C22 [×4], C22×Q8, C22×Q8 [×2], C22×Q8, C2×C4○D4 [×3], C2×C4○D4 [×2], C2×C4○D4 [×3], 2- 1+4 [×8], (C22×C8)⋊C2, C2×Q8⋊C4, C23.36D4, Q8⋊D4 [×2], D4⋊D4 [×2], C22⋊Q16 [×2], D4.7D4 [×2], C22.31C24, C2×C4○D8, C2×C8.C22, C2×2- 1+4, (C2×Q8)⋊17D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, D4○SD16, Q8○D8, (C2×Q8)⋊17D4

Smallest permutation representation of (C2×Q8)⋊17D4
On 64 points
Generators in S64
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 45)(10 46)(11 47)(12 48)(13 52)(14 49)(15 50)(16 51)(17 37)(18 38)(19 39)(20 40)(21 44)(22 41)(23 42)(24 43)(25 62)(26 63)(27 64)(28 61)(29 33)(30 34)(31 35)(32 36)
(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 36 3 34)(2 35 4 33)(5 64 7 62)(6 63 8 61)(9 42 11 44)(10 41 12 43)(13 37 15 39)(14 40 16 38)(17 50 19 52)(18 49 20 51)(21 45 23 47)(22 48 24 46)(25 57 27 59)(26 60 28 58)(29 54 31 56)(30 53 32 55)
(1 9 64 38)(2 24 61 50)(3 11 62 40)(4 22 63 52)(5 14 34 42)(6 17 35 48)(7 16 36 44)(8 19 33 46)(10 60 39 29)(12 58 37 31)(13 56 41 26)(15 54 43 28)(18 53 45 27)(20 55 47 25)(21 59 51 32)(23 57 49 30)
(2 31)(4 29)(6 26)(8 28)(9 38)(10 52)(11 40)(12 50)(13 48)(14 42)(15 46)(16 44)(17 41)(18 47)(19 43)(20 45)(21 49)(22 39)(23 51)(24 37)(25 27)(30 32)(33 54)(35 56)(53 55)(57 59)(58 61)(60 63)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D4E···4L4M4N4O8A8B8C8D8E8F
order1222222222244444···4444888888
size1111224444822224···4888444488

32 irreducible representations

dim11111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D4○SD16Q8○D8
kernel(C2×Q8)⋊17D4(C22×C8)⋊C2C2×Q8⋊C4C23.36D4Q8⋊D4D4⋊D4C22⋊Q16D4.7D4C22.31C24C2×C4○D8C2×C8.C22C2×2- 1+4C2×D4C2×Q8C4○D4C2C2
# reps11112222111135422

Matrix representation of (C2×Q8)⋊17D4 in GL6(𝔽17)

100000
010000
0016161615
0000160
0001600
000111
,
0160000
1600000
0000013
004044
004404
0013000
,
1600000
0160000
0000160
001112
001000
001616016
,
0160000
100000
00120127
005050
00510510
00121200
,
100000
0160000
001000
0001600
000010
001601616

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16,0,16,1,0,0,16,16,0,1,0,0,15,0,0,1],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,4,4,13,0,0,0,0,4,0,0,0,0,4,0,0,0,0,13,4,4,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,1,16,0,0,0,1,0,16,0,0,16,1,0,0,0,0,0,2,0,16],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,12,5,5,12,0,0,0,0,10,12,0,0,12,5,5,0,0,0,7,0,10,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,16,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;

(C2×Q8)⋊17D4 in GAP, Magma, Sage, TeX

(C_2\times Q_8)\rtimes_{17}D_4
% in TeX

G:=Group("(C2xQ8):17D4");
// GroupNames label

G:=SmallGroup(128,1745);
// by ID

G=gap.SmallGroup(128,1745);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,521,248,2804,1411,172]);
// Polycyclic

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

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