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## G = C2×C8⋊7D4order 128 = 27

### Direct product of C2 and C8⋊7D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C8⋊7D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C23×C8 — C2×C8⋊7D4
 Lower central C1 — C2 — C2×C4 — C2×C8⋊7D4
 Upper central C1 — C23 — C23×C4 — C2×C8⋊7D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8⋊7D4

Generators and relations for C2×C87D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 692 in 308 conjugacy classes, 116 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×10], C22 [×32], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×18], D4 [×28], C23, C23 [×6], C23 [×20], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×8], C2×C8 [×10], D8 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×4], C2×D4 [×26], C24, C24 [×2], D4⋊C4 [×8], C2.D8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4⋊D4 [×8], C4⋊D4 [×4], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C2×D8 [×4], C2×D8 [×4], C23×C4, C22×D4 [×2], C22×D4 [×2], C2×D4⋊C4 [×2], C2×C2.D8, C87D4 [×8], C2×C4⋊D4 [×2], C23×C8, C22×D8, C2×C87D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D8 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×D8 [×6], C4○D8 [×2], C22×D4 [×2], C2×C4○D4, C87D4 [×4], C2×C4⋊D4, C22×D8, C2×C4○D8, C2×C87D4

Smallest permutation representation of C2×C87D4
On 64 points
Generators in S64
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 61)(18 62)(19 63)(20 64)(21 57)(22 58)(23 59)(24 60)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)
(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 42 9 19)(2 41 10 18)(3 48 11 17)(4 47 12 24)(5 46 13 23)(6 45 14 22)(7 44 15 21)(8 43 16 20)(25 50 60 34)(26 49 61 33)(27 56 62 40)(28 55 63 39)(29 54 64 38)(30 53 57 37)(31 52 58 36)(32 51 59 35)
(1 51)(2 50)(3 49)(4 56)(5 55)(6 54)(7 53)(8 52)(9 35)(10 34)(11 33)(12 40)(13 39)(14 38)(15 37)(16 36)(17 61)(18 60)(19 59)(20 58)(21 57)(22 64)(23 63)(24 62)(25 41)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A ··· 4H 4I 4J 4K 4L 8A ··· 8P order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 8 8 8 8 2 ··· 2 8 8 8 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4○D4 D8 C4○D8 kernel C2×C8⋊7D4 C2×D4⋊C4 C2×C2.D8 C8⋊7D4 C2×C4⋊D4 C23×C8 C22×D8 C2×C8 C22×C4 C24 C2×C4 C23 C22 # reps 1 2 1 8 2 1 1 4 3 1 4 8 8

Matrix representation of C2×C87D4 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 16 0 0 0 0 0 0 16
,
 14 3 0 0 0 0 14 14 0 0 0 0 0 0 14 3 0 0 0 0 14 14 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 2 0 0 0 0 16 1
,
 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 0 0 0 0 0 1 16

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

C2×C87D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_7D_4
% in TeX

G:=Group("C2xC8:7D4");
// GroupNames label

G:=SmallGroup(128,1780);
// by ID

G=gap.SmallGroup(128,1780);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,2804,172]);
// Polycyclic

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

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