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## G = C2×D8⋊C4order 128 = 27

### Direct product of C2 and D8⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×D8⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×D4 — C2×D8⋊C4
 Lower central C1 — C2 — C4 — C2×D8⋊C4
 Upper central C1 — C23 — C2×C42 — C2×D8⋊C4
 Jennings C1 — C2 — C2 — C2×C4 — C2×D8⋊C4

Generators and relations for C2×D8⋊C4
G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b4c >

Subgroups: 604 in 304 conjugacy classes, 148 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×32], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×28], D4 [×8], D4 [×12], C23, C23 [×20], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×8], C2×C8 [×2], D8 [×16], C22×C4, C22×C4 [×2], C22×C4 [×18], C2×D4 [×12], C2×D4 [×6], C24 [×2], C8⋊C4 [×4], D4⋊C4 [×8], C4.Q8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4×D4 [×8], C4×D4 [×4], C22×C8 [×2], C2×D8 [×12], C23×C4 [×2], C22×D4 [×2], C2×C8⋊C4, C2×D4⋊C4 [×2], C2×C4.Q8, D8⋊C4 [×8], C2×C4×D4 [×2], C22×D8, C2×D8⋊C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C8⋊C22 [×4], C23×C4, C22×D4, C2×C4○D4, D8⋊C4 [×4], C2×C4×D4, C2×C8⋊C22 [×2], 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 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 57)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(41 55)(42 56)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)
(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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 64)(10 63)(11 62)(12 61)(13 60)(14 59)(15 58)(16 57)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 40)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(48 56)
(1 38 49 12)(2 35 50 9)(3 40 51 14)(4 37 52 11)(5 34 53 16)(6 39 54 13)(7 36 55 10)(8 33 56 15)(17 46 60 31)(18 43 61 28)(19 48 62 25)(20 45 63 30)(21 42 64 27)(22 47 57 32)(23 44 58 29)(24 41 59 26)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4L 4M ··· 4T 8A ··· 8H order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 C8⋊C22 kernel C2×D8⋊C4 C2×C8⋊C4 C2×D4⋊C4 C2×C4.Q8 D8⋊C4 C2×C4×D4 C22×D8 C2×D8 C42 C22×C4 C2×C4 C22 # reps 1 1 2 1 8 2 1 16 2 2 4 4

Matrix representation of C2×D8⋊C4 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
,
 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 16 15 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 15 2 15 0 0 0 0 1 15 1 0 0 0 0 0 1 16 0 2 0 0 0 0 9 0 16 2
,
 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 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 1
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 1 15 2 0 0 0 0 0 1 16 0 2 0 0 0 0 1 0 16 2 0 0 0 0 0 1 16 1

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

C2×D8⋊C4 in GAP, Magma, Sage, TeX

C_2\times D_8\rtimes C_4
% in TeX

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

G:=SmallGroup(128,1674);
// by ID

G=gap.SmallGroup(128,1674);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,520,2804,1411,172]);
// Polycyclic

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

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