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## G = C2×D4○C16order 128 = 27

### Direct product of C2 and D4○C16

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×D4○C16
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C2×C8○D4 — C2×D4○C16
 Lower central C1 — C2 — C2×D4○C16
 Upper central C1 — C2×C16 — C2×D4○C16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C2×D4○C16

Generators and relations for C2×D4○C16
G = < a,b,c,d | a2=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >

Subgroups: 196 in 184 conjugacy classes, 172 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C16, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C2×C16, C2×C16, M5(2), C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C22×C16, C2×M5(2), D4○C16, C2×C8○D4, C2×D4○C16
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C24, C22×C8, C23×C4, D4○C16, C23×C8, C2×D4○C16

Smallest permutation representation of C2×D4○C16
On 64 points
Generators in S64
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 49)(29 50)(30 51)(31 52)(32 53)
(1 63 9 55)(2 64 10 56)(3 49 11 57)(4 50 12 58)(5 51 13 59)(6 52 14 60)(7 53 15 61)(8 54 16 62)(17 39 25 47)(18 40 26 48)(19 41 27 33)(20 42 28 34)(21 43 29 35)(22 44 30 36)(23 45 31 37)(24 46 32 38)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)
(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)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 8A ··· 8H 8I ··· 8T 16A ··· 16P 16Q ··· 16AN order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 16 ··· 16 16 ··· 16 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 type + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 D4○C16 kernel C2×D4○C16 C22×C16 C2×M5(2) D4○C16 C2×C8○D4 C2×M4(2) C8○D4 C2×C4○D4 C2×D4 C2×Q8 C4○D4 C2 # reps 1 3 3 8 1 6 8 2 12 4 16 16

Matrix representation of C2×D4○C16 in GL3(𝔽17) generated by

 16 0 0 0 1 0 0 0 1
,
 1 0 0 0 0 16 0 1 0
,
 16 0 0 0 0 16 0 16 0
,
 16 0 0 0 14 0 0 0 14
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[1,0,0,0,0,1,0,16,0],[16,0,0,0,0,16,0,16,0],[16,0,0,0,14,0,0,0,14] >;

C2×D4○C16 in GAP, Magma, Sage, TeX

C_2\times D_4\circ C_{16}
% in TeX

G:=Group("C2xD4oC16");
// GroupNames label

G:=SmallGroup(128,2138);
// by ID

G=gap.SmallGroup(128,2138);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,112,723,102,124]);
// Polycyclic

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

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