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## G = C22×SD32order 128 = 27

### Direct product of C22 and SD32

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C22×SD32
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C22×D8 — C22×SD32
 Lower central C1 — C2 — C4 — C8 — C22×SD32
 Upper central C1 — C23 — C22×C4 — C22×C8 — C22×SD32
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C22×SD32

Generators and relations for C22×SD32
G = < a,b,c,d | a2=b2=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >

Subgroups: 500 in 200 conjugacy classes, 100 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C2×C8, D8, D8, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2×C16, SD32, C22×C8, C2×D8, C2×D8, C2×Q16, C2×Q16, C22×D4, C22×Q8, C22×C16, C2×SD32, C22×D8, C22×Q16, C22×SD32
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, SD32, C2×D8, C22×D4, C2×SD32, C22×D8, C22×SD32

Smallest permutation representation of C22×SD32
On 64 points
Generators in S64
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(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 41)(2 48)(3 39)(4 46)(5 37)(6 44)(7 35)(8 42)(9 33)(10 40)(11 47)(12 38)(13 45)(14 36)(15 43)(16 34)(17 57)(18 64)(19 55)(20 62)(21 53)(22 60)(23 51)(24 58)(25 49)(26 56)(27 63)(28 54)(29 61)(30 52)(31 59)(32 50)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D8 D8 SD32 kernel C22×SD32 C22×C16 C2×SD32 C22×D8 C22×Q16 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 1 12 1 1 3 1 6 2 16

Matrix representation of C22×SD32 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 16 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 1 0 0 0 0 7 1 0 0 16 7
,
 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 1
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,7,16,0,0,1,7],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1] >;

C22×SD32 in GAP, Magma, Sage, TeX

C_2^2\times {\rm SD}_{32}
% in TeX

G:=Group("C2^2xSD32");
// GroupNames label

G:=SmallGroup(128,2141);
// by ID

G=gap.SmallGroup(128,2141);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,448,253,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations

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