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## G = C2×C8.D4order 128 = 27

### Direct product of C2 and C8.D4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C8.D4
G = < a,b,c,d | a2=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b3, dbd-1=b-1, dcd-1=b4c-1 >

Subgroups: 436 in 246 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, Q8⋊C4, C4.Q8, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C22⋊Q8, C22×C8, C2×M4(2), C2×M4(2), C2×Q16, C2×Q16, C23×C4, C22×Q8, C2×Q8⋊C4, C2×C4.Q8, C8.D4, C2×C22⋊Q8, C22×M4(2), C22×Q16, C2×C8.D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C8.C22, C22×D4, C2×C4○D4, C8.D4, C2×C4⋊D4, C2×C8.C22, C2×C8.D4

Smallest permutation representation of C2×C8.D4
On 64 points
Generators in S64
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)
(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 30 43 54)(2 25 44 49)(3 28 45 52)(4 31 46 55)(5 26 47 50)(6 29 48 53)(7 32 41 56)(8 27 42 51)(9 64 20 35)(10 59 21 38)(11 62 22 33)(12 57 23 36)(13 60 24 39)(14 63 17 34)(15 58 18 37)(16 61 19 40)
(1 40 5 36)(2 39 6 35)(3 38 7 34)(4 37 8 33)(9 53 13 49)(10 52 14 56)(11 51 15 55)(12 50 16 54)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(41 63 45 59)(42 62 46 58)(43 61 47 57)(44 60 48 64)

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G ··· 4N 8A ··· 8H order 1 2 ··· 2 2 2 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 4 2 2 2 2 4 4 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.D4 C2×Q8⋊C4 C2×C4.Q8 C8.D4 C2×C22⋊Q8 C22×M4(2) C22×Q16 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×C8.D4 in GL8(𝔽17)

 16 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 16 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 16 0 0 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 16 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 0 1 0 0 0 0 0 0 16 0 0 0
,
 7 2 0 0 0 0 0 0 9 10 0 0 0 0 0 0 0 0 10 8 0 0 0 0 0 0 11 7 0 0 0 0 0 0 0 0 0 0 7 16 0 0 0 0 0 0 16 10 0 0 0 0 10 1 0 0 0 0 0 0 1 7 0 0
,
 7 2 0 0 0 0 0 0 10 10 0 0 0 0 0 0 0 0 10 8 0 0 0 0 0 0 11 7 0 0 0 0 0 0 0 0 0 0 10 1 0 0 0 0 0 0 1 7 0 0 0 0 10 1 0 0 0 0 0 0 1 7 0 0

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[7,9,0,0,0,0,0,0,2,10,0,0,0,0,0,0,0,0,10,11,0,0,0,0,0,0,8,7,0,0,0,0,0,0,0,0,0,0,10,1,0,0,0,0,0,0,1,7,0,0,0,0,7,16,0,0,0,0,0,0,16,10,0,0],[7,10,0,0,0,0,0,0,2,10,0,0,0,0,0,0,0,0,10,11,0,0,0,0,0,0,8,7,0,0,0,0,0,0,0,0,0,0,10,1,0,0,0,0,0,0,1,7,0,0,0,0,10,1,0,0,0,0,0,0,1,7,0,0] >;

C2×C8.D4 in GAP, Magma, Sage, TeX

C_2\times C_8.D_4
% in TeX

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

G:=SmallGroup(128,1785);
// by ID

G=gap.SmallGroup(128,1785);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,723,2804,172]);
// Polycyclic

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

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