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## G = (C2×C8)⋊13D4order 128 = 27

### 9th 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 — C2×C4 — (C2×C8)⋊13D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — (C2×C8)⋊13D4
 Lower central C1 — C2 — C2×C4 — (C2×C8)⋊13D4
 Upper central C1 — C22 — C2×C4○D4 — (C2×C8)⋊13D4
 Jennings C1 — C2 — C2 — C2×C4 — (C2×C8)⋊13D4

Generators and relations for (C2×C8)⋊13D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, dad=ab4, cbc-1=dbd=b3, dcd=c-1 >

Subgroups: 476 in 242 conjugacy classes, 100 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×14], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×18], Q8 [×6], C23, C23 [×2], C23 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×6], C2×C8 [×4], M4(2) [×6], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×4], C2×D4 [×6], C2×Q8, C2×Q8 [×2], C4○D4 [×12], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×4], C2×C4⋊C4 [×2], C4⋊D4 [×4], C4⋊D4 [×6], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×2], C8○D4 [×4], C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×4], C2×C4○D4, C2×C4○D4 [×2], C23.36D4 [×2], C2×C4.Q8, C88D4 [×4], C82D4 [×2], C8.D4 [×2], C22.31C24 [×2], C2×C8○D4, C2×C4○D8, (C2×C8)⋊13D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D4○SD16 [×2], (C2×C8)⋊13D4

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4○D4 D4○SD16 kernel (C2×C8)⋊13D4 C23.36D4 C2×C4.Q8 C8⋊8D4 C8⋊2D4 C8.D4 C22.31C24 C2×C8○D4 C2×C4○D8 C2×C8 C2×D4 C2×Q8 C2×C4 C2 # reps 1 2 1 4 2 2 2 1 1 4 3 1 4 4

Matrix representation of (C2×C8)⋊13D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 2 0 0 1 0 1 1 0 0 16 1 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 7 0 0 0 0 5 7 0 0 0 0 0 5 12 5 0 0 12 5 12 12
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 8 3 0 2 0 0 0 10 16 1 0 0 10 6 0 9 0 0 10 7 10 16
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 1 16 0 0 0 0 0 0 1 0 0 0 16 0 0 16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,16,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,2,1,16,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,12,0,0,7,7,5,5,0,0,0,0,12,12,0,0,0,0,5,12],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,8,0,10,10,0,0,3,10,6,7,0,0,0,16,0,10,0,0,2,1,9,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,16,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

(C2×C8)⋊13D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes_{13}D_4
% in TeX

G:=Group("(C2xC8):13D4");
// GroupNames label

G:=SmallGroup(128,1792);
// by ID

G=gap.SmallGroup(128,1792);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,521,248,2804,172]);
// Polycyclic

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

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