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G = (C2×C10)⋊8D8order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C10)⋊8D8, (C5×D4)⋊13D4, D45(C5⋊D4), C55(C22⋊D8), C10.72(C2×D8), (C22×D4)⋊1D5, C207D425C2, C223(D4⋊D5), (C2×C20).300D4, C20.205(C2×D4), C10.71C22≀C2, (C2×D4).198D10, D4⋊Dic538C2, (C2×D20)⋊14C22, C4⋊Dic521C22, C20.55D415C2, (C2×C20).472C23, (C22×C4).149D10, (C22×C10).196D4, C2.4(C242D5), C23.85(C5⋊D4), C10.102(C8⋊C22), (D4×C10).240C22, C2.22(D4.D10), (C22×C20).197C22, (D4×C2×C10)⋊1C2, (C2×D4⋊D5)⋊23C2, C2.26(C2×D4⋊D5), C4.58(C2×C5⋊D4), (C2×C52C8)⋊10C22, (C2×C10).553(C2×D4), (C2×C4).83(C5⋊D4), (C2×C4).558(C22×D5), C22.216(C2×C5⋊D4), SmallGroup(320,844)

Series: Derived Chief Lower central Upper central

C1C2×C20 — (C2×C10)⋊8D8
C1C5C10C2×C10C2×C20C2×D20C2×D4⋊D5 — (C2×C10)⋊8D8
C5C10C2×C20 — (C2×C10)⋊8D8
C1C22C22×C4C22×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 [×3], C2 [×7], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×21], C5, C8 [×2], C2×C4 [×2], C2×C4 [×3], D4 [×4], D4 [×10], C23, C23 [×11], D5, C10 [×3], C10 [×6], C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4 [×2], C2×D4 [×7], C24, Dic5, C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×18], C22⋊C8, D4⋊C4 [×2], C4⋊D4, C2×D8 [×2], C22×D4, C52C8 [×2], D20 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C5×D4 [×6], C22×D5, C22×C10, C22×C10 [×10], C22⋊D8, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, D4⋊D5 [×4], C2×D20, C2×C5⋊D4, C22×C20, D4×C10 [×2], D4×C10 [×5], C23×C10, C20.55D4, D4⋊Dic5 [×2], C207D4, C2×D4⋊D5 [×2], D4×C2×C10, (C2×C10)⋊8D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, D8 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×D8, C8⋊C22, C5⋊D4 [×6], C22×D5, C22⋊D8, D4⋊D5 [×2], C2×C5⋊D4 [×3], 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 11 72 33 63 41 52)(2 24 12 71 34 62 42 51)(3 23 13 80 35 61 43 60)(4 22 14 79 36 70 44 59)(5 21 15 78 37 69 45 58)(6 30 16 77 38 68 46 57)(7 29 17 76 39 67 47 56)(8 28 18 75 40 66 48 55)(9 27 19 74 31 65 49 54)(10 26 20 73 32 64 50 53)
(1 57)(2 56)(3 55)(4 54)(5 53)(6 52)(7 51)(8 60)(9 59)(10 58)(11 68)(12 67)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 70)(20 69)(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), (1,25,11,72,33,63,41,52)(2,24,12,71,34,62,42,51)(3,23,13,80,35,61,43,60)(4,22,14,79,36,70,44,59)(5,21,15,78,37,69,45,58)(6,30,16,77,38,68,46,57)(7,29,17,76,39,67,47,56)(8,28,18,75,40,66,48,55)(9,27,19,74,31,65,49,54)(10,26,20,73,32,64,50,53), (1,57)(2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,60)(9,59)(10,58)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,70)(20,69)(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), (1,25,11,72,33,63,41,52)(2,24,12,71,34,62,42,51)(3,23,13,80,35,61,43,60)(4,22,14,79,36,70,44,59)(5,21,15,78,37,69,45,58)(6,30,16,77,38,68,46,57)(7,29,17,76,39,67,47,56)(8,28,18,75,40,66,48,55)(9,27,19,74,31,65,49,54)(10,26,20,73,32,64,50,53), (1,57)(2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,60)(9,59)(10,58)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,70)(20,69)(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)], [(1,25,11,72,33,63,41,52),(2,24,12,71,34,62,42,51),(3,23,13,80,35,61,43,60),(4,22,14,79,36,70,44,59),(5,21,15,78,37,69,45,58),(6,30,16,77,38,68,46,57),(7,29,17,76,39,67,47,56),(8,28,18,75,40,66,48,55),(9,27,19,74,31,65,49,54),(10,26,20,73,32,64,50,53)], [(1,57),(2,56),(3,55),(4,54),(5,53),(6,52),(7,51),(8,60),(9,59),(10,58),(11,68),(12,67),(13,66),(14,65),(15,64),(16,63),(17,62),(18,61),(19,70),(20,69),(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 2A2B2C2D2E2F2G2H2I2J4A4B4C4D5A5B8A8B8C8D10A···10N10O···10AD20A···20H
order12222222222444455888810···1010···1020···20
size1111224444402244022202020202···24···44···4

59 irreducible representations

dim1111112222222222444
type+++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5D8D10D10C5⋊D4C5⋊D4C5⋊D4C8⋊C22D4⋊D5D4.D10
kernel(C2×C10)⋊8D8C20.55D4D4⋊Dic5C207D4C2×D4⋊D5D4×C2×C10C2×C20C5×D4C22×C10C22×D4C2×C10C22×C4C2×D4C2×C4D4C23C10C22C2
# reps11212114124244164144

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

1000
0100
0010
00140
,
1000
0100
0040
00731
,
17100
40000
002826
00313
,
17100
402400
001315
003828
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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