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## G = C2×C4○D16order 128 = 27

### Direct product of C2 and C4○D16

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C2×C4○D16
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C2×C4○D8 — C2×C4○D16
 Lower central C1 — C2 — C4 — C8 — C2×C4○D16
 Upper central C1 — C2×C4 — C22×C4 — C22×C8 — C2×C4○D16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C2×C4○D16

Generators and relations for C2×C4○D16
G = < a,b,c,d | a2=b4=d2=1, c8=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c7 >

Subgroups: 404 in 184 conjugacy classes, 92 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C2×C8, C2×C8, D8, D8, SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C16, C2×C16, D16, SD32, Q32, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, C22×C16, C2×D16, C2×SD32, C2×Q32, C4○D16, C2×C4○D8, C2×C4○D16
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C22×D4, C4○D16, C22×D8, C2×C4○D16

Smallest permutation representation of C2×C4○D16
On 64 points
Generators in S64
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(1 18 9 26)(2 19 10 27)(3 20 11 28)(4 21 12 29)(5 22 13 30)(6 23 14 31)(7 24 15 32)(8 25 16 17)(33 58 41 50)(34 59 42 51)(35 60 43 52)(36 61 44 53)(37 62 45 54)(38 63 46 55)(39 64 47 56)(40 49 48 57)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 19)(20 32)(21 31)(22 30)(23 29)(24 28)(25 27)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(49 51)(52 64)(53 63)(54 62)(55 61)(56 60)(57 59)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D8 D8 C4○D16 kernel C2×C4○D16 C22×C16 C2×D16 C2×SD32 C2×Q32 C4○D16 C2×C4○D8 C2×C8 C22×C4 C2×C4 C23 C2 # reps 1 1 1 2 1 8 2 3 1 6 2 16

Matrix representation of C2×C4○D16 in GL3(𝔽17) generated by

 16 0 0 0 16 0 0 0 16
,
 16 0 0 0 4 0 0 0 4
,
 16 0 0 0 7 12 0 11 2
,
 16 0 0 0 1 0 0 1 16
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[16,0,0,0,4,0,0,0,4],[16,0,0,0,7,11,0,12,2],[16,0,0,0,1,1,0,0,16] >;

C2×C4○D16 in GAP, Magma, Sage, TeX

C_2\times C_4\circ D_{16}
% in TeX

G:=Group("C2xC4oD16");
// GroupNames label

G:=SmallGroup(128,2143);
// by ID

G=gap.SmallGroup(128,2143);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,352,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^7>;
// generators/relations

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