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## G = C2×C8⋊6D4order 128 = 27

### Direct product of C2 and C8⋊6D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C8⋊6D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C22×M4(2) — C2×C8⋊6D4
 Lower central C1 — C22 — C2×C8⋊6D4
 Upper central C1 — C22×C4 — C2×C8⋊6D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8⋊6D4

Generators and relations for C2×C86D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 420 in 276 conjugacy classes, 156 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×6], C4 [×6], C22, C22 [×6], C22 [×20], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×16], C2×C4 [×22], D4 [×8], C23, C23 [×4], C23 [×12], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×12], C2×C8 [×6], M4(2) [×16], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×4], C2×D4 [×4], C24 [×2], C4×C8 [×4], C22⋊C8 [×8], C4⋊C8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C22×C8 [×2], C22×C8 [×2], C2×M4(2) [×8], C2×M4(2) [×8], C23×C4 [×2], C22×D4, C2×C4×C8, C2×C22⋊C8 [×2], C2×C4⋊C8, C86D4 [×8], C2×C4×D4, C22×M4(2) [×2], C2×C86D4
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, C86D4 [×4], C2×C4×D4, C22×M4(2), C2×C8○D4, C2×C86D4

Smallest permutation representation of C2×C86D4
On 64 points
Generators in S64
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(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 58 42 38)(2 59 43 39)(3 60 44 40)(4 61 45 33)(5 62 46 34)(6 63 47 35)(7 64 48 36)(8 57 41 37)(9 29 18 52)(10 30 19 53)(11 31 20 54)(12 32 21 55)(13 25 22 56)(14 26 23 49)(15 27 24 50)(16 28 17 51)
(1 38)(2 35)(3 40)(4 37)(5 34)(6 39)(7 36)(8 33)(9 56)(10 53)(11 50)(12 55)(13 52)(14 49)(15 54)(16 51)(17 28)(18 25)(19 30)(20 27)(21 32)(22 29)(23 26)(24 31)(41 61)(42 58)(43 63)(44 60)(45 57)(46 62)(47 59)(48 64)

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 M4(2) C4○D4 C8○D4 kernel C2×C8⋊6D4 C2×C4×C8 C2×C22⋊C8 C2×C4⋊C8 C8⋊6D4 C2×C4×D4 C22×M4(2) C2×C22⋊C4 C2×C4⋊C4 C4×D4 C22×D4 C2×C8 C2×C4 C2×C4 C22 # reps 1 1 2 1 8 1 2 4 2 8 2 4 8 4 8

Matrix representation of C2×C86D4 in GL5(𝔽17)

 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 4 7 0 0 0 8 13 0 0 0 0 0 13 0 0 0 0 0 13
,
 1 0 0 0 0 0 8 14 0 0 0 16 9 0 0 0 0 0 0 13 0 0 0 13 0
,
 16 0 0 0 0 0 9 5 0 0 0 1 8 0 0 0 0 0 0 13 0 0 0 4 0

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,4,8,0,0,0,7,13,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,8,16,0,0,0,14,9,0,0,0,0,0,0,13,0,0,0,13,0],[16,0,0,0,0,0,9,1,0,0,0,5,8,0,0,0,0,0,0,4,0,0,0,13,0] >;

C2×C86D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_6D_4
% in TeX

G:=Group("C2xC8:6D4");
// GroupNames label

G:=SmallGroup(128,1660);
// by ID

G=gap.SmallGroup(128,1660);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,723,184,124]);
// 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,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

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