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## G = C2×C8⋊2D4order 128 = 27

### Direct product of C2 and C8⋊2D4

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C8⋊2D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C2×C8⋊2D4
 Lower central C1 — C2 — C2×C4 — C2×C8⋊2D4
 Upper central C1 — C23 — C23×C4 — C2×C8⋊2D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8⋊2D4

Generators and relations for C2×C82D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b3, dbd=b-1, dcd=c-1 >

Subgroups: 692 in 298 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, D4⋊C4, C4.Q8, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22×C8, C2×M4(2), C2×M4(2), C2×D8, C2×D8, C23×C4, C22×D4, C22×D4, C2×D4⋊C4, C2×C4.Q8, C82D4, C2×C4⋊D4, C22×M4(2), C22×D8, C2×C82D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C8⋊C22, C22×D4, C2×C4○D4, C82D4, C2×C4⋊D4, C2×C8⋊C22, C2×C82D4

Smallest permutation representation of C2×C82D4
On 64 points
Generators in S64
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)
(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 9 19 59)(2 12 20 62)(3 15 21 57)(4 10 22 60)(5 13 23 63)(6 16 24 58)(7 11 17 61)(8 14 18 64)(25 37 51 48)(26 40 52 43)(27 35 53 46)(28 38 54 41)(29 33 55 44)(30 36 56 47)(31 39 49 42)(32 34 50 45)
(2 8)(3 7)(4 6)(9 59)(10 58)(11 57)(12 64)(13 63)(14 62)(15 61)(16 60)(17 21)(18 20)(22 24)(26 32)(27 31)(28 30)(33 44)(34 43)(35 42)(36 41)(37 48)(38 47)(39 46)(40 45)(49 53)(50 52)(54 56)

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A ··· 8H order 1 2 ··· 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 4 4 8 8 8 8 2 2 2 2 4 4 8 8 8 8 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 D4 C4○D4 C8⋊C22 kernel C2×C8⋊2D4 C2×D4⋊C4 C2×C4.Q8 C8⋊2D4 C2×C4⋊D4 C22×M4(2) C22×D8 C2×C8 C22×C4 C24 C2×C4 C22 # reps 1 2 1 8 2 1 1 4 3 1 4 4

Matrix representation of C2×C82D4 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 4 9 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 16 15 0 0 0 0 0 0 1 1 0 0
,
 16 2 0 0 0 0 0 0 16 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 16 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 16 0 0 0 0 0 0 0 0 16 15 0 0 0 0 0 0 0 1

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,9,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16],[1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,15,1] >;

C2×C82D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_2D_4
% in TeX

G:=Group("C2xC8:2D4");
// GroupNames label

G:=SmallGroup(128,1784);
// by ID

G=gap.SmallGroup(128,1784);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,723,2804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^3,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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