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## G = (C2×C8)⋊20D4order 128 = 27

### 16th semidirect product of C2×C8 and D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C8)⋊20D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C2×C4⋊1D4 — (C2×C8)⋊20D4
 Lower central C1 — C2 — C22×C4 — (C2×C8)⋊20D4
 Upper central C1 — C23 — C2×C42 — (C2×C8)⋊20D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8)⋊20D4

Generators and relations for (C2×C8)⋊20D4
G = < a,b,c,d | a2=b8=c4=d2=1, cbc-1=ab=ba, ac=ca, ad=da, dbd=b3, dcd=c-1 >

Subgroups: 616 in 237 conjugacy classes, 58 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×7], C22 [×3], C22 [×4], C22 [×20], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×11], D4 [×28], Q8 [×6], C23, C23 [×16], C42 [×2], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×4], SD16 [×16], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×4], C2×D4 [×26], C2×Q8 [×2], C2×Q8 [×5], C24 [×2], C2.C42 [×2], D4⋊C4 [×4], C2×C42, C2×C4⋊C4, C41D4 [×4], C22×C8 [×2], C2×SD16 [×12], C22×D4 [×2], C22×D4 [×2], C22×Q8, C22.7C42, C23.67C23, C2×D4⋊C4 [×2], C2×C41D4, C22×SD16 [×2], (C2×C8)⋊20D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, SD16 [×4], C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C2×SD16 [×2], C8⋊C22 [×2], C232D4, C22⋊SD16 [×2], C4⋊SD16 [×2], C85D4, C83D4, (C2×C8)⋊20D4

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

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

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

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

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 D4 SD16 C4○D4 C8⋊C22 kernel (C2×C8)⋊20D4 C22.7C42 C23.67C23 C2×D4⋊C4 C2×C4⋊1D4 C22×SD16 C2×C8 C22×C4 C2×D4 C2×Q8 C2×C4 C2×C4 C22 # reps 1 1 1 2 1 2 4 2 4 2 8 2 2

Matrix representation of (C2×C8)⋊20D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 5 14 0 0 0 0 3 12 0 0 0 0 0 0 0 7 0 0 0 0 5 7 0 0 0 0 0 0 12 12 0 0 0 0 5 12
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 16 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,3,0,0,0,0,14,12,0,0,0,0,0,0,0,5,0,0,0,0,7,7,0,0,0,0,0,0,12,5,0,0,0,0,12,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

(C2×C8)⋊20D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes_{20}D_4
% in TeX

G:=Group("(C2xC8):20D4");
// GroupNames label

G:=SmallGroup(128,746);
// by ID

G=gap.SmallGroup(128,746);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,2019,1018,521,248,2804,1411,718,172]);
// Polycyclic

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

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