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G = (C2×D20)⋊25C4order 320 = 26·5

10th semidirect product of C2×D20 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×D20)⋊25C4, (C2×C4).48D20, (C2×C20).145D4, C42⋊C25D5, (C2×Dic10)⋊24C4, C22⋊C4.83D10, C22.15(C2×D20), C20.96(C22⋊C4), (C22×C4).114D10, C23.1D107C2, C4.53(D10⋊C4), C23.73(C22×D5), C55(C23.C23), C23.D5.78C22, C23.21D1014C2, (C22×C20).155C22, (C22×C10).112C23, (C2×C4×D5)⋊3C4, (C2×C4).47(C4×D5), C22.19(C2×C4×D5), (C2×C4○D20).9C2, (C2×C20).268(C2×C4), (C5×C42⋊C2)⋊5C2, (C2×C10).462(C2×D4), (C2×C4).46(C5⋊D4), C10.89(C2×C22⋊C4), (C2×Dic5).4(C2×C4), (C22×D5).4(C2×C4), C22.28(C2×C5⋊D4), C2.21(C2×D10⋊C4), (C2×C5⋊D4).92C22, (C2×C10).114(C22×C4), (C5×C22⋊C4).94C22, SmallGroup(320,633)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (C2×D20)⋊25C4
C1C5C10C2×C10C22×C10C2×C5⋊D4C2×C4○D20 — (C2×D20)⋊25C4
C5C10C2×C10 — (C2×D20)⋊25C4
C1C4C22×C4C42⋊C2

Generators and relations for (C2×D20)⋊25C4
 G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, dcd-1=ac=ca, dad-1=ab10, cbc=b-1, bd=db >

Subgroups: 638 in 158 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×6], C22 [×3], C22 [×5], C5, C2×C4 [×6], C2×C4 [×10], D4 [×6], Q8 [×2], C23, C23 [×2], D5 [×2], C10, C10 [×3], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×4], C20 [×4], C20 [×2], D10 [×4], C2×C10 [×3], C2×C10, C23⋊C4 [×4], C42⋊C2, C42⋊C2, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×6], C2×C20 [×2], C22×D5 [×2], C22×C10, C23.C23, C4×Dic5, C4⋊Dic5, C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, C23.1D10 [×4], C23.21D10, C5×C42⋊C2, C2×C4○D20, (C2×D20)⋊25C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.C23, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, (C2×D20)⋊25C4

Smallest permutation representation of (C2×D20)⋊25C4
On 80 points
Generators in S80
(1 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 61)(57 62)(58 63)(59 64)(60 65)
(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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 80)(19 79)(20 78)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 60)(29 59)(30 58)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)
(1 63 26 58)(2 64 27 59)(3 65 28 60)(4 66 29 41)(5 67 30 42)(6 68 31 43)(7 69 32 44)(8 70 33 45)(9 71 34 46)(10 72 35 47)(11 73 36 48)(12 74 37 49)(13 75 38 50)(14 76 39 51)(15 77 40 52)(16 78 21 53)(17 79 22 54)(18 80 23 55)(19 61 24 56)(20 62 25 57)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J···4O5A5B10A···10F10G10H10I10J20A···20H20I···20AB
order12222224444444444···45510···101010101020···2020···20
size11222202011222444420···20222···244442···24···4

62 irreducible representations

dim11111111222222244
type++++++++++
imageC1C2C2C2C2C4C4C4D4D5D10D10C4×D5D20C5⋊D4C23.C23(C2×D20)⋊25C4
kernel(C2×D20)⋊25C4C23.1D10C23.21D10C5×C42⋊C2C2×C4○D20C2×Dic10C2×C4×D5C2×D20C2×C20C42⋊C2C22⋊C4C22×C4C2×C4C2×C4C2×C4C5C1
# reps14111242424288828

Matrix representation of (C2×D20)⋊25C4 in GL4(𝔽41) generated by

24402440
117117
00171
004024
,
93000
111400
00930
001114
,
38383838
173173
6633
7352438
,
171821
40242032
7392440
234117
G:=sub<GL(4,GF(41))| [24,1,0,0,40,17,0,0,24,1,17,40,40,17,1,24],[9,11,0,0,30,14,0,0,0,0,9,11,0,0,30,14],[38,17,6,7,38,3,6,35,38,17,3,24,38,3,3,38],[17,40,7,2,1,24,39,34,8,20,24,1,21,32,40,17] >;

(C2×D20)⋊25C4 in GAP, Magma, Sage, TeX

(C_2\times D_{20})\rtimes_{25}C_4
% in TeX

G:=Group("(C2xD20):25C4");
// GroupNames label

G:=SmallGroup(320,633);
// by ID

G=gap.SmallGroup(320,633);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,422,58,1123,438,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,d*c*d^-1=a*c=c*a,d*a*d^-1=a*b^10,c*b*c=b^-1,b*d=d*b>;
// generators/relations

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