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## G = C2×D8.C4order 128 = 27

### Direct product of C2 and D8.C4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×D8.C4
G = < a,b,c,d | a2=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=b5c >

Subgroups: 244 in 112 conjugacy classes, 52 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×2], C22 [×3], C22 [×6], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×5], D4 [×7], Q8 [×3], C23, C23, C16 [×2], C2×C8 [×6], C2×C8, M4(2) [×3], D8 [×2], D8, SD16 [×4], Q16 [×2], Q16, C22×C4, C22×C4, C2×D4 [×2], C2×Q8, C4○D4 [×6], C8.C4 [×2], C8.C4, C2×C16 [×2], C2×C16 [×2], C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4○D8 [×4], C4○D8 [×2], C2×C4○D4, D8.C4 [×4], C2×C8.C4, C22×C16, C2×C4○D8, C2×D8.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, D8.C4 [×2], C2×D4⋊C4, C2×D8.C4

Smallest permutation representation of C2×D8.C4
On 64 points
Generators in S64
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 56)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 33)(24 34)(41 61)(42 62)(43 63)(44 64)(45 57)(46 58)(47 59)(48 60)
(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 8)(2 7)(3 6)(4 5)(9 13)(10 12)(14 16)(17 20)(18 19)(21 24)(22 23)(25 30)(26 29)(27 28)(31 32)(33 40)(34 39)(35 38)(36 37)(42 48)(43 47)(44 46)(49 51)(52 56)(53 55)(58 64)(59 63)(60 62)
(1 12 23 58 5 16 19 62)(2 11 24 57 6 15 20 61)(3 10 17 64 7 14 21 60)(4 9 18 63 8 13 22 59)(25 54 38 41 29 50 34 45)(26 53 39 48 30 49 35 44)(27 52 40 47 31 56 36 43)(28 51 33 46 32 55 37 42)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D8 SD16 SD16 D8.C4 kernel C2×D8.C4 D8.C4 C2×C8.C4 C22×C16 C2×C4○D8 C2×D8 C2×Q16 C4○D8 C2×C8 C22×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 2 2 4 3 1 4 2 2 16

Matrix representation of C2×D8.C4 in GL3(𝔽17) generated by

 16 0 0 0 1 0 0 0 1
,
 1 0 0 0 0 6 0 14 6
,
 16 0 0 0 0 6 0 3 0
,
 13 0 0 0 16 13 0 14 1
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[1,0,0,0,0,14,0,6,6],[16,0,0,0,0,3,0,6,0],[13,0,0,0,16,14,0,13,1] >;

C2×D8.C4 in GAP, Magma, Sage, TeX

C_2\times D_8.C_4
% in TeX

G:=Group("C2xD8.C4");
// GroupNames label

G:=SmallGroup(128,874);
// by ID

G=gap.SmallGroup(128,874);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1123,570,360,172,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^5*c>;
// generators/relations

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