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G = (C2×C4)⋊2D8order 128 = 27

1st 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)⋊2D8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C2×C4⋊1D4 — (C2×C4)⋊2D8
 Lower central C1 — C2 — C22×C4 — (C2×C4)⋊2D8
 Upper central C1 — C23 — C2×C42 — (C2×C4)⋊2D8
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C4)⋊2D8

Generators and relations for (C2×C4)⋊2D8
G = < a,b,c,d | a2=b4=c8=d2=1, dbd=ab=ba, ac=ca, ad=da, cbc-1=ab-1, dcd=c-1 >

Subgroups: 712 in 255 conjugacy classes, 58 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×4], C4 [×5], C22 [×3], C22 [×4], C22 [×30], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×7], D4 [×34], C23, C23 [×24], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×4], D8 [×16], C22×C4, C22×C4 [×2], C22×C4, C2×D4 [×6], C2×D4 [×31], C24 [×3], D4⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C41D4 [×4], C22×C8 [×2], C2×D8 [×12], C22×D4, C22×D4 [×2], C22×D4 [×2], C22.7C42, C24.3C22, C2×D4⋊C4 [×2], C2×C41D4, C22×D8 [×2], (C2×C4)⋊2D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, D8 [×4], C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C2×D8 [×2], C8⋊C22 [×2], C232D4, C22⋊D8 [×2], C4⋊D8 [×2], C84D4, C83D4, (C2×C4)⋊2D8

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D8 C4○D4 C8⋊C22 kernel (C2×C4)⋊2D8 C22.7C42 C24.3C22 C2×D4⋊C4 C2×C4⋊1D4 C22×D8 C2×C8 C22×C4 C2×D4 C2×C4 C2×C4 C22 # reps 1 1 1 2 1 2 4 2 6 8 2 2

Matrix representation of (C2×C4)⋊2D8 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
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 16 0 0 0 0 2 16 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 3 3 0 0 0 0 14 3 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 16
,
 14 14 0 0 0 0 14 3 0 0 0 0 0 0 16 0 0 0 0 0 15 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,2,0,0,0,0,16,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[3,14,0,0,0,0,3,3,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[14,14,0,0,0,0,14,3,0,0,0,0,0,0,16,15,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

(C2×C4)⋊2D8 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_2D_8
% in TeX

G:=Group("(C2xC4):2D8");
// GroupNames label

G:=SmallGroup(128,743);
// by ID

G=gap.SmallGroup(128,743);
# by ID

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

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

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