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## G = C2×C8⋊9D4order 128 = 27

### Direct product of C2 and C8⋊9D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C8⋊9D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C23×C8 — C2×C8⋊9D4
 Lower central C1 — C22 — C2×C8⋊9D4
 Upper central C1 — C22×C4 — C2×C8⋊9D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8⋊9D4

Generators and relations for C2×C89D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b5, dcd=c-1 >

Subgroups: 420 in 282 conjugacy classes, 156 normal (36 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×10], C22 [×22], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×12], C2×C4 [×24], D4 [×8], C23, C23 [×8], C23 [×10], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×12], C2×C8 [×14], M4(2) [×8], C22×C4 [×5], C22×C4 [×8], C22×C4 [×8], C2×D4 [×4], C2×D4 [×4], C24 [×2], C8⋊C4 [×4], C22⋊C8 [×8], C4⋊C8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C22×C8 [×4], C22×C8 [×4], C22×C8 [×4], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4 [×2], C22×D4, C2×C8⋊C4, C2×C22⋊C8 [×2], C2×C4⋊C8, C89D4 [×8], C2×C4×D4, C23×C8, C22×M4(2), C2×C89D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], M4(2) [×4], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C2×M4(2) [×6], C8○D4 [×2], C23×C4, C22×D4, C2×C4○D4, C89D4 [×4], C2×C4×D4, C22×M4(2), C2×C8○D4, C2×C89D4

Smallest permutation representation of C2×C89D4
On 64 points
Generators in S64
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 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 34 47 63)(2 39 48 60)(3 36 41 57)(4 33 42 62)(5 38 43 59)(6 35 44 64)(7 40 45 61)(8 37 46 58)(9 49 22 30)(10 54 23 27)(11 51 24 32)(12 56 17 29)(13 53 18 26)(14 50 19 31)(15 55 20 28)(16 52 21 25)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 52)(26 49)(27 54)(28 51)(29 56)(30 53)(31 50)(32 55)(33 58)(34 63)(35 60)(36 57)(37 62)(38 59)(39 64)(40 61)(42 46)(44 48)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 4A ··· 4H 4I 4J 4K 4L 4M ··· 4R 8A ··· 8P 8Q ··· 8X order 1 2 ··· 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 1 ··· 1 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 M4(2) C8○D4 kernel C2×C8⋊9D4 C2×C8⋊C4 C2×C22⋊C8 C2×C4⋊C8 C8⋊9D4 C2×C4×D4 C23×C8 C22×M4(2) C2×C22⋊C4 C2×C4⋊C4 C4×D4 C22×D4 C2×C8 C2×C4 C23 C22 # reps 1 1 2 1 8 1 1 1 4 2 8 2 4 4 8 8

Matrix representation of C2×C89D4 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 16 0 0 0 0 0 0 16
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 15 0 0 0 0 11 14
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 3 16
,
 1 0 0 0 0 0 0 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 3 16

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,11,0,0,0,0,15,14],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,3,0,0,0,0,0,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,3,0,0,0,0,0,16] >;

C2×C89D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_9D_4
% in TeX

G:=Group("C2xC8:9D4");
// GroupNames label

G:=SmallGroup(128,1659);
// by ID

G=gap.SmallGroup(128,1659);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,124]);
// 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^5,d*c*d=c^-1>;
// generators/relations

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