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## G = C2×D40⋊C2order 320 = 26·5

### Direct product of C2 and D40⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D40⋊C2
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×D4×D5 — C2×D40⋊C2
 Lower central C5 — C10 — C20 — C2×D40⋊C2
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for C2×D40⋊C2
G = < a,b,c,d | a2=b40=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b11, cd=dc >

Subgroups: 1374 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×4], C22, C22 [×24], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×10], D4 [×2], D4 [×15], Q8 [×2], Q8, C23 [×12], D5 [×6], C10, C10 [×2], C10 [×2], C2×C8, C2×C8, M4(2) [×4], D8 [×8], SD16 [×4], SD16 [×4], C22×C4 [×2], C2×D4, C2×D4 [×10], C2×Q8, C4○D4 [×6], C24, Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×18], C2×C10, C2×C10 [×4], C2×M4(2), C2×D8 [×2], C2×SD16, C2×SD16, C8⋊C22 [×8], C22×D4, C2×C4○D4, C52C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×4], D20 [×4], D20 [×6], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C5×Q8 [×2], C5×Q8, C22×D5, C22×D5 [×10], C22×C10, C2×C8⋊C22, C8⋊D5 [×4], D40 [×4], C2×C52C8, D4⋊D5 [×4], Q8⋊D5 [×4], C2×C40, C5×SD16 [×4], C2×C4×D5, C2×C4×D5, C2×D20 [×2], C2×D20, D4×D5 [×4], D4×D5 [×2], Q82D5 [×4], Q82D5 [×2], C2×C5⋊D4, D4×C10, Q8×C10, C23×D5, C2×C8⋊D5, C2×D40, D40⋊C2 [×8], C2×D4⋊D5, C2×Q8⋊D5, C10×SD16, C2×D4×D5, C2×Q82D5, C2×D40⋊C2
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8⋊C22 [×2], C22×D4, C22×D5 [×7], C2×C8⋊C22, D4×D5 [×2], C23×D5, D40⋊C2 [×2], C2×D4×D5, C2×D40⋊C2

Smallest permutation representation of C2×D40⋊C2
On 80 points
Generators in S80
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(72 80)(73 79)(74 78)(75 77)
(1 11)(2 22)(3 33)(5 15)(6 26)(7 37)(9 19)(10 30)(13 23)(14 34)(17 27)(18 38)(21 31)(25 35)(29 39)(41 51)(42 62)(43 73)(45 55)(46 66)(47 77)(49 59)(50 70)(53 63)(54 74)(57 67)(58 78)(61 71)(65 75)(69 79)

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 D10 C8⋊C22 D4×D5 D4×D5 D40⋊C2 kernel C2×D40⋊C2 C2×C8⋊D5 C2×D40 D40⋊C2 C2×D4⋊D5 C2×Q8⋊D5 C10×SD16 C2×D4×D5 C2×Q8⋊2D5 C4×D5 C2×Dic5 C22×D5 C2×SD16 C2×C8 SD16 C2×D4 C2×Q8 C10 C4 C22 C2 # reps 1 1 1 8 1 1 1 1 1 2 1 1 2 2 8 2 2 2 2 2 8

Matrix representation of C2×D40⋊C2 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 7 35 0 0 0 0 7 0 0 0 0 0 0 0 26 26 26 26 0 0 15 34 15 34 0 0 28 28 0 0 0 0 13 24 0 0
,
 40 1 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 39 0 0 40 0 39 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 2 0 0 0 0 1 0 2 0 0 0 0 40 0 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[7,7,0,0,0,0,35,0,0,0,0,0,0,0,26,15,28,13,0,0,26,34,28,24,0,0,26,15,0,0,0,0,26,34,0,0],[40,0,0,0,0,0,1,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,39,0,1,0,0,39,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,2,0,40,0,0,0,0,2,0,40] >;

C2×D40⋊C2 in GAP, Magma, Sage, TeX

C_2\times D_{40}\rtimes C_2
% in TeX

G:=Group("C2xD40:C2");
// GroupNames label

G:=SmallGroup(320,1431);
// by ID

G=gap.SmallGroup(320,1431);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,1123,185,136,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^40=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^11,c*d=d*c>;
// generators/relations

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