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## G = (C2×C4)⋊3D8order 128 = 27

### 2nd semidirect product of C2×C4 and D8 acting via D8/C4=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C4)⋊3D8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×D4⋊C4 — (C2×C4)⋊3D8
 Lower central C1 — C2 — C22×C4 — (C2×C4)⋊3D8
 Upper central C1 — C23 — C2×C42 — (C2×C4)⋊3D8
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C4)⋊3D8

Generators and relations for (C2×C4)⋊3D8
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=ab-1, dcd=c-1 >

Subgroups: 480 in 180 conjugacy classes, 52 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×5], C22 [×3], C22 [×4], C22 [×20], C8 [×3], C2×C4 [×2], C2×C4 [×6], C2×C4 [×11], D4 [×12], C23, C23 [×16], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×5], D8 [×8], C22×C4 [×3], C22×C4 [×2], C2×D4 [×4], C2×D4 [×10], C24 [×2], D4⋊C4 [×4], C4⋊C8 [×2], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4 [×2], C22×C8 [×2], C2×D8 [×6], C22×D4 [×2], C22.4Q16, C24.3C22 [×2], C2×D4⋊C4 [×2], C2×C4⋊C8, C22×D8, (C2×C4)⋊3D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, D8 [×2], C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C2×D8, C4○D8, C8⋊C22 [×2], C23.10D4, C4⋊D8 [×2], D4.2D4 [×2], C87D4, C82D4, (C2×C4)⋊3D8

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D8 C4○D4 C4○D8 C8⋊C22 kernel (C2×C4)⋊3D8 C22.4Q16 C24.3C22 C2×D4⋊C4 C2×C4⋊C8 C22×D8 C2×C8 C22×C4 C2×D4 C2×C4 C2×C4 C22 C22 # reps 1 1 2 2 1 1 2 2 4 4 6 4 2

Matrix representation of (C2×C4)⋊3D8 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 4 0 0 0 0 0 7 13
,
 14 3 0 0 0 0 14 14 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 7 9 0 0 0 0 2 10
,
 3 14 0 0 0 0 14 14 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 7 9 0 0 0 0 6 10

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,7,0,0,0,0,0,13],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,7,2,0,0,0,0,9,10],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,7,6,0,0,0,0,9,10] >;

(C2×C4)⋊3D8 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_3D_8
% in TeX

G:=Group("(C2xC4):3D8");
// GroupNames label

G:=SmallGroup(128,786);
// by ID

G=gap.SmallGroup(128,786);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2019,1018,248,2804,1411,172,4037,2028,124]);
// Polycyclic

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

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