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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×Q8)⋊17D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2- 1+4 — (C2×Q8)⋊17D4
 Lower central C1 — C2 — C2×C4 — (C2×Q8)⋊17D4
 Upper central C1 — C22 — C2×C4○D4 — (C2×Q8)⋊17D4
 Jennings C1 — C2 — C2 — C2×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 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E ··· 4L 4M 4N 4O 8A 8B 8C 8D 8E 8F order 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 8 8 8 8 8 8 size 1 1 1 1 2 2 4 4 4 4 8 2 2 2 2 4 ··· 4 8 8 8 4 4 4 4 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4○SD16 Q8○D8 kernel (C2×Q8)⋊17D4 (C22×C8)⋊C2 C2×Q8⋊C4 C23.36D4 Q8⋊D4 D4⋊D4 C22⋊Q16 D4.7D4 C22.31C24 C2×C4○D8 C2×C8.C22 C2×2- 1+4 C2×D4 C2×Q8 C4○D4 C2 C2 # reps 1 1 1 1 2 2 2 2 1 1 1 1 3 5 4 2 2

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 16 16 15 0 0 0 0 16 0 0 0 0 16 0 0 0 0 0 1 1 1
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 13 0 0 4 0 4 4 0 0 4 4 0 4 0 0 13 0 0 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 1 1 1 2 0 0 1 0 0 0 0 0 16 16 0 16
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 12 0 12 7 0 0 5 0 5 0 0 0 5 10 5 10 0 0 12 12 0 0
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 16 0 16 16

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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