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## G = C23⋊2Dic10order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C23⋊2Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C23.D5 — C23⋊2Dic10
 Lower central C5 — C2×C10 — C23⋊2Dic10
 Upper central C1 — C22 — C2×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 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4L 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 4 4 4 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + - + + + + - + image C1 C2 C2 C2 C2 Q8 D5 D10 D10 D10 Dic10 2+ 1+4 D4⋊6D10 kernel C23⋊2Dic10 Dic5.14D4 C20.48D4 C2×C23.D5 C10×C22⋊C4 C22×C10 C2×C22⋊C4 C22⋊C4 C22×C4 C24 C23 C10 C2 # reps 1 8 4 2 1 4 2 8 4 2 16 2 8

Matrix representation of C232Dic10 in GL6(𝔽41)

 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 40 0 0 0 0 1 0 0 0 0 0 0 0 0 18 0 0 0 0 23 0 0 0 0 0 0 0 0 25 0 0 0 0 16 0
,
 1 30 0 0 0 0 30 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 40 0 0 0 0 0 0 40 0 0

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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