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## G = C2×Q32⋊C2order 128 = 27

### Direct product of C2 and Q32⋊C2

direct product, p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C2×Q32⋊C2
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C22×Q16 — C2×Q32⋊C2
 Lower central C1 — C2 — C4 — C8 — C2×Q32⋊C2
 Upper central C1 — C22 — C22×C4 — C22×C8 — C2×Q32⋊C2
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C2×Q32⋊C2

Generators and relations for C2×Q32⋊C2
G = < a,b,c,d | a2=b16=d2=1, c2=b8, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, cd=dc >

Subgroups: 372 in 180 conjugacy classes, 92 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×7], Q8 [×13], C23, C23, C16 [×4], C2×C8 [×2], C2×C8 [×4], D8 [×2], D8, SD16 [×4], Q16 [×6], Q16 [×7], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8 [×10], C4○D4 [×6], C2×C16 [×2], M5(2) [×4], SD32 [×8], Q32 [×8], C22×C8, C2×D8, C2×SD16, C2×Q16, C2×Q16 [×6], C2×Q16 [×3], C4○D8 [×4], C4○D8 [×2], C22×Q8, C2×C4○D4, C2×M5(2), C2×SD32 [×2], C2×Q32 [×2], Q32⋊C2 [×8], C22×Q16, C2×C4○D8, C2×Q32⋊C2
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, Q32⋊C2 [×2], C22×D8, C2×Q32⋊C2

Smallest permutation representation of C2×Q32⋊C2
On 64 points
Generators in S64
(1 44)(2 45)(3 46)(4 47)(5 48)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)
(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 62 9 54)(2 61 10 53)(3 60 11 52)(4 59 12 51)(5 58 13 50)(6 57 14 49)(7 56 15 64)(8 55 16 63)(17 41 25 33)(18 40 26 48)(19 39 27 47)(20 38 28 46)(21 37 29 45)(22 36 30 44)(23 35 31 43)(24 34 32 42)
(1 53)(2 62)(3 55)(4 64)(5 57)(6 50)(7 59)(8 52)(9 61)(10 54)(11 63)(12 56)(13 49)(14 58)(15 51)(16 60)(17 48)(18 41)(19 34)(20 43)(21 36)(22 45)(23 38)(24 47)(25 40)(26 33)(27 42)(28 35)(29 44)(30 37)(31 46)(32 39)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4J 8A 8B 8C 8D 8E 8F 16A ··· 16H order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 8 8 8 8 8 8 16 ··· 16 size 1 1 1 1 2 2 8 8 2 2 2 2 8 ··· 8 2 2 2 2 4 4 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D8 D8 Q32⋊C2 kernel C2×Q32⋊C2 C2×M5(2) C2×SD32 C2×Q32 Q32⋊C2 C22×Q16 C2×C4○D8 C2×C8 C22×C4 C2×C4 C23 C2 # reps 1 1 2 2 8 1 1 3 1 6 2 4

Matrix representation of C2×Q32⋊C2 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 14 13 0 0 0 0 11 3 0 0 0 0 0 0 3 7 9 5 0 0 2 15 12 9 0 0 4 8 2 10 0 0 8 4 15 14
,
 3 4 0 0 0 0 15 14 0 0 0 0 0 0 10 3 1 12 0 0 2 2 12 16 0 0 9 4 15 3 0 0 13 8 2 7
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 2 0 0 0 1 0 0 2 0 0 0 0 0 16 0 0 0 0 16 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[14,11,0,0,0,0,13,3,0,0,0,0,0,0,3,2,4,8,0,0,7,15,8,4,0,0,9,12,2,15,0,0,5,9,10,14],[3,15,0,0,0,0,4,14,0,0,0,0,0,0,10,2,9,13,0,0,3,2,4,8,0,0,1,12,15,2,0,0,12,16,3,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,16,0,0,0,2,16,0] >;

C2×Q32⋊C2 in GAP, Magma, Sage, TeX

C_2\times Q_{32}\rtimes C_2
% in TeX

G:=Group("C2xQ32:C2");
// GroupNames label

G:=SmallGroup(128,2145);
// by ID

G=gap.SmallGroup(128,2145);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,456,1430,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^16=d^2=1,c^2=b^8,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^9,c*d=d*c>;
// generators/relations

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