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## G = (C2×C8)⋊14D4order 128 = 27

### 10th 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)⋊14D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — (C2×C8)⋊14D4
 Lower central C1 — C2 — C2×C4 — (C2×C8)⋊14D4
 Upper central C1 — C22 — C2×C4○D4 — (C2×C8)⋊14D4
 Jennings C1 — C2 — C2 — C2×C4 — (C2×C8)⋊14D4

Generators and relations for (C2×C8)⋊14D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, dad=ab4, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 476 in 242 conjugacy classes, 100 normal (24 characteristic)
C1, C2 [×3], 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], C2.D8 [×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×C2.D8, C87D4 [×2], C8.18D4 [×2], C8⋊D4 [×4], C22.31C24 [×2], C2×C8○D4, C2×C4○D8, (C2×C8)⋊14D4
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○D8, Q8○D8, (C2×C8)⋊14D4

Smallest permutation representation of (C2×C8)⋊14D4
On 64 points
Generators in S64
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 62)(34 63)(35 64)(36 57)(37 58)(38 59)(39 60)(40 61)
(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 45 51 58)(2 44 52 57)(3 43 53 64)(4 42 54 63)(5 41 55 62)(6 48 56 61)(7 47 49 60)(8 46 50 59)(9 18 31 35)(10 17 32 34)(11 24 25 33)(12 23 26 40)(13 22 27 39)(14 21 28 38)(15 20 29 37)(16 19 30 36)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 37)(18 36)(19 35)(20 34)(21 33)(22 40)(23 39)(24 38)(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)| (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,62)(34,63)(35,64)(36,57)(37,58)(38,59)(39,60)(40,61), (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), (1,45,51,58)(2,44,52,57)(3,43,53,64)(4,42,54,63)(5,41,55,62)(6,48,56,61)(7,47,49,60)(8,46,50,59)(9,18,31,35)(10,17,32,34)(11,24,25,33)(12,23,26,40)(13,22,27,39)(14,21,28,38)(15,20,29,37)(16,19,30,36), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,37)(18,36)(19,35)(20,34)(21,33)(22,40)(23,39)(24,38)(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( (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,62)(34,63)(35,64)(36,57)(37,58)(38,59)(39,60)(40,61), (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), (1,45,51,58)(2,44,52,57)(3,43,53,64)(4,42,54,63)(5,41,55,62)(6,48,56,61)(7,47,49,60)(8,46,50,59)(9,18,31,35)(10,17,32,34)(11,24,25,33)(12,23,26,40)(13,22,27,39)(14,21,28,38)(15,20,29,37)(16,19,30,36), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,37)(18,36)(19,35)(20,34)(21,33)(22,40)(23,39)(24,38)(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([(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,41),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,62),(34,63),(35,64),(36,57),(37,58),(38,59),(39,60),(40,61)], [(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)], [(1,45,51,58),(2,44,52,57),(3,43,53,64),(4,42,54,63),(5,41,55,62),(6,48,56,61),(7,47,49,60),(8,46,50,59),(9,18,31,35),(10,17,32,34),(11,24,25,33),(12,23,26,40),(13,22,27,39),(14,21,28,38),(15,20,29,37),(16,19,30,36)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(17,37),(18,36),(19,35),(20,34),(21,33),(22,40),(23,39),(24,38),(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 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 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4○D4 D4○D8 Q8○D8 kernel (C2×C8)⋊14D4 C23.36D4 C2×C2.D8 C8⋊7D4 C8.18D4 C8⋊D4 C22.31C24 C2×C8○D4 C2×C4○D8 C2×C8 C2×D4 C2×Q8 C2×C4 C2 C2 # reps 1 2 1 2 2 4 2 1 1 4 3 1 4 2 2

Matrix representation of (C2×C8)⋊14D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 1 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 14 0 0 0 0 3 3 0 0 0 0 0 0 3 14 0 0 0 0 3 3
,
 2 16 0 0 0 0 5 15 0 0 0 0 0 0 5 12 1 16 0 0 12 12 16 16 0 0 1 16 12 5 0 0 16 16 5 5
,
 1 0 0 0 0 0 4 16 0 0 0 0 0 0 3 14 0 0 0 0 14 14 0 0 0 0 0 0 3 14 0 0 0 0 14 14

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,14,3],[2,5,0,0,0,0,16,15,0,0,0,0,0,0,5,12,1,16,0,0,12,12,16,16,0,0,1,16,12,5,0,0,16,16,5,5],[1,4,0,0,0,0,0,16,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,0,0,0,3,14,0,0,0,0,14,14] >;

(C2×C8)⋊14D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes_{14}D_4
% in TeX

G:=Group("(C2xC8):14D4");
// GroupNames label

G:=SmallGroup(128,1793);
// by ID

G=gap.SmallGroup(128,1793);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,521,248,2804,172]);
// 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^-1,d*c*d=c^-1>;
// generators/relations

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