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## G = (C2×C10)⋊8D8order 320 = 26·5

### 2nd semidirect product of C2×C10 and D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C2×C10)⋊8D8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — C2×D4⋊D5 — (C2×C10)⋊8D8
 Lower central C5 — C10 — C2×C20 — (C2×C10)⋊8D8
 Upper central C1 — C22 — C22×C4 — C22×D4

Generators and relations for (C2×C10)⋊8D8
G = < a,b,c,d | a2=b10=c8=d2=1, ab=ba, cac-1=dad=ab5, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 718 in 198 conjugacy classes, 51 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C22×D4, C52C8, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C22×C10, C22⋊D8, C2×C52C8, C4⋊Dic5, D10⋊C4, D4⋊D5, C2×D20, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×C10, C20.55D4, D4⋊Dic5, C207D4, C2×D4⋊D5, D4×C2×C10, (C2×C10)⋊8D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C22≀C2, C2×D8, C8⋊C22, C5⋊D4, C22×D5, C22⋊D8, D4⋊D5, C2×C5⋊D4, C2×D4⋊D5, D4.D10, C242D5, (C2×C10)⋊8D8

Smallest permutation representation of (C2×C10)⋊8D8
On 80 points
Generators in S80
(21 26)(22 27)(23 28)(24 29)(25 30)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)
(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 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)
(1 25 12 72 33 65 41 58)(2 24 13 71 34 64 42 57)(3 23 14 80 35 63 43 56)(4 22 15 79 36 62 44 55)(5 21 16 78 37 61 45 54)(6 30 17 77 38 70 46 53)(7 29 18 76 39 69 47 52)(8 28 19 75 40 68 48 51)(9 27 20 74 31 67 49 60)(10 26 11 73 32 66 50 59)
(1 53)(2 52)(3 51)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 61)(12 70)(13 69)(14 68)(15 67)(16 66)(17 65)(18 64)(19 63)(20 62)(21 50)(22 49)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 80)

G:=sub<Sym(80)| (21,26)(22,27)(23,28)(24,29)(25,30)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (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)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,25,12,72,33,65,41,58)(2,24,13,71,34,64,42,57)(3,23,14,80,35,63,43,56)(4,22,15,79,36,62,44,55)(5,21,16,78,37,61,45,54)(6,30,17,77,38,70,46,53)(7,29,18,76,39,69,47,52)(8,28,19,75,40,68,48,51)(9,27,20,74,31,67,49,60)(10,26,11,73,32,66,50,59), (1,53)(2,52)(3,51)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,61)(12,70)(13,69)(14,68)(15,67)(16,66)(17,65)(18,64)(19,63)(20,62)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,80)>;

G:=Group( (21,26)(22,27)(23,28)(24,29)(25,30)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (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)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,25,12,72,33,65,41,58)(2,24,13,71,34,64,42,57)(3,23,14,80,35,63,43,56)(4,22,15,79,36,62,44,55)(5,21,16,78,37,61,45,54)(6,30,17,77,38,70,46,53)(7,29,18,76,39,69,47,52)(8,28,19,75,40,68,48,51)(9,27,20,74,31,67,49,60)(10,26,11,73,32,66,50,59), (1,53)(2,52)(3,51)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,61)(12,70)(13,69)(14,68)(15,67)(16,66)(17,65)(18,64)(19,63)(20,62)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,80) );

G=PermutationGroup([[(21,26),(22,27),(23,28),(24,29),(25,30),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80)], [(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),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80)], [(1,25,12,72,33,65,41,58),(2,24,13,71,34,64,42,57),(3,23,14,80,35,63,43,56),(4,22,15,79,36,62,44,55),(5,21,16,78,37,61,45,54),(6,30,17,77,38,70,46,53),(7,29,18,76,39,69,47,52),(8,28,19,75,40,68,48,51),(9,27,20,74,31,67,49,60),(10,26,11,73,32,66,50,59)], [(1,53),(2,52),(3,51),(4,60),(5,59),(6,58),(7,57),(8,56),(9,55),(10,54),(11,61),(12,70),(13,69),(14,68),(15,67),(16,66),(17,65),(18,64),(19,63),(20,62),(21,50),(22,49),(23,48),(24,47),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,79),(32,78),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71),(40,80)]])

59 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 10A ··· 10N 10O ··· 10AD 20A ··· 20H order 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 4 4 4 4 40 2 2 4 40 2 2 20 20 20 20 2 ··· 2 4 ··· 4 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 D8 D10 D10 C5⋊D4 C5⋊D4 C5⋊D4 C8⋊C22 D4⋊D5 D4.D10 kernel (C2×C10)⋊8D8 C20.55D4 D4⋊Dic5 C20⋊7D4 C2×D4⋊D5 D4×C2×C10 C2×C20 C5×D4 C22×C10 C22×D4 C2×C10 C22×C4 C2×D4 C2×C4 D4 C23 C10 C22 C2 # reps 1 1 2 1 2 1 1 4 1 2 4 2 4 4 16 4 1 4 4

Matrix representation of (C2×C10)⋊8D8 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 40
,
 1 0 0 0 0 1 0 0 0 0 4 0 0 0 7 31
,
 17 1 0 0 40 0 0 0 0 0 28 26 0 0 3 13
,
 17 1 0 0 40 24 0 0 0 0 13 15 0 0 38 28
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,4,7,0,0,0,31],[17,40,0,0,1,0,0,0,0,0,28,3,0,0,26,13],[17,40,0,0,1,24,0,0,0,0,13,38,0,0,15,28] >;

(C2×C10)⋊8D8 in GAP, Magma, Sage, TeX

(C_2\times C_{10})\rtimes_8D_8
% in TeX

G:=Group("(C2xC10):8D8");
// GroupNames label

G:=SmallGroup(320,844);
// by ID

G=gap.SmallGroup(320,844);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^5,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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