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G = C232Dic10order 320 = 26·5

2nd semidirect product of C23 and Dic10 acting via Dic10/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C232Dic10, C24.23D10, C10.12+ 1+4, (C22×C10)⋊5Q8, C4⋊Dic53C22, C51(C232Q8), C20.48D43C2, C10.6(C22×Q8), (C2×C10).27C24, C22⋊C4.86D10, C2.6(D46D10), (C2×C20).127C23, (C2×Dic10)⋊2C22, (C22×C4).169D10, C10.D41C22, (C2×Dic5).8C23, C22.5(C2×Dic10), C2.8(C22×Dic10), C22.69(C23×D5), Dic5.14D41C2, (C23×C10).53C22, (C22×C20).71C22, C23.144(C22×D5), C23.D5.85C22, (C22×C10).119C23, (C22×Dic5).76C22, (C2×C10).49(C2×Q8), (C2×C22⋊C4).18D5, (C10×C22⋊C4).18C2, (C2×C4).133(C22×D5), (C2×C23.D5).22C2, (C5×C22⋊C4).97C22, SmallGroup(320,1155)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C232Dic10
C1C5C10C2×C10C2×Dic5C22×Dic5C2×C23.D5 — C232Dic10
C5C2×C10 — C232Dic10
C1C22C2×C22⋊C4

Generators and relations for C232Dic10
 G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=d10, ab=ba, dad-1=ac=ca, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 782 in 242 conjugacy classes, 111 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×12], C22, C22 [×6], C22 [×10], C5, C2×C4 [×4], C2×C4 [×14], Q8 [×4], C23, C23 [×6], C23 [×2], C10, C10 [×2], C10 [×6], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×4], C2×Q8 [×4], C24, Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×6], C2×C10 [×10], C2×C22⋊C4, C2×C22⋊C4 [×2], C22⋊Q8 [×12], Dic10 [×4], C2×Dic5 [×8], C2×Dic5 [×4], C2×C20 [×4], C2×C20 [×2], C22×C10, C22×C10 [×6], C22×C10 [×2], C232Q8, C10.D4 [×8], C4⋊Dic5 [×4], C23.D5 [×8], C5×C22⋊C4 [×4], C2×Dic10 [×4], C22×Dic5 [×4], C22×C20 [×2], C23×C10, Dic5.14D4 [×8], C20.48D4 [×4], C2×C23.D5 [×2], C10×C22⋊C4, C232Dic10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, 2+ 1+4 [×2], Dic10 [×4], C22×D5 [×7], C232Q8, C2×Dic10 [×6], C23×D5, C22×Dic10, D46D10 [×2], C232Dic10

Smallest permutation representation of C232Dic10
On 80 points
Generators in S80
(2 23)(4 25)(6 27)(8 29)(10 31)(12 33)(14 35)(16 37)(18 39)(20 21)(42 76)(44 78)(46 80)(48 62)(50 64)(52 66)(54 68)(56 70)(58 72)(60 74)
(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 71)(58 72)(59 73)(60 74)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 21)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 71)(58 72)(59 73)(60 74)
(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 51 11 41)(2 50 12 60)(3 49 13 59)(4 48 14 58)(5 47 15 57)(6 46 16 56)(7 45 17 55)(8 44 18 54)(9 43 19 53)(10 42 20 52)(21 66 31 76)(22 65 32 75)(23 64 33 74)(24 63 34 73)(25 62 35 72)(26 61 36 71)(27 80 37 70)(28 79 38 69)(29 78 39 68)(30 77 40 67)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4L5A5B10A···10N10O···10V20A···20P
order12222···244444···45510···1010···1020···20
size11112···2444420···20222···24···44···4

62 irreducible representations

dim1111122222244
type+++++-++++-+
imageC1C2C2C2C2Q8D5D10D10D10Dic102+ 1+4D46D10
kernelC232Dic10Dic5.14D4C20.48D4C2×C23.D5C10×C22⋊C4C22×C10C2×C22⋊C4C22⋊C4C22×C4C24C23C10C2
# reps18421428421628

Matrix representation of C232Dic10 in GL6(𝔽41)

4000000
0400000
001000
0004000
000010
0000040
,
100000
010000
001000
000100
0000400
0000040
,
100000
010000
0040000
0004000
0000400
0000040
,
0400000
100000
0001800
0023000
0000025
0000160
,
1300000
30400000
000010
000001
0040000
0004000

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,23,0,0,0,0,18,0,0,0,0,0,0,0,0,16,0,0,0,0,25,0],[1,30,0,0,0,0,30,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;

C232Dic10 in GAP, Magma, Sage, TeX

C_2^3\rtimes_2{\rm Dic}_{10}
% in TeX

G:=Group("C2^3:2Dic10");
// GroupNames label

G:=SmallGroup(320,1155);
// by ID

G=gap.SmallGroup(320,1155);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,758,675,570,80,12550]);
// Polycyclic

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

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