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## G = C23.Dic10order 320 = 26·5

### 6th non-split extension by C23 of Dic10 acting via Dic10/C10=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C23.Dic10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C2×C4.Dic5 — C23.Dic10
 Lower central C5 — C10 — C20 — C23.Dic10
 Upper central C1 — C4 — C22×C4 — C2×M4(2)

Generators and relations for C23.Dic10
G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=bcd10, ab=ba, eae-1=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd19 >

Subgroups: 238 in 102 conjugacy classes, 59 normal (39 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C23, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C20, C2×C10, C2×C10, C8.C4, C2×M4(2), C2×M4(2), C52C8, C52C8, C40, C2×C20, C22×C10, M4(2).C4, C2×C52C8, C2×C52C8, C4.Dic5, C4.Dic5, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C20.53D4, C2×C4.Dic5, C10×M4(2), C23.Dic10
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, Dic10, C4×D5, C5⋊D4, C22×D5, M4(2).C4, C10.D4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×C10.D4, C23.Dic10

Smallest permutation representation of C23.Dic10
On 80 points
Generators in S80
(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(2 22)(4 24)(6 26)(8 28)(10 30)(12 32)(14 34)(16 36)(18 38)(20 40)(42 62)(44 64)(46 66)(48 68)(50 70)(52 72)(54 74)(56 76)(58 78)(60 80)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 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 71 31 61 21 51 11 41)(2 50 12 60 22 70 32 80)(3 49 33 79 23 69 13 59)(4 68 14 78 24 48 34 58)(5 67 35 57 25 47 15 77)(6 46 16 56 26 66 36 76)(7 45 37 75 27 65 17 55)(8 64 18 74 28 44 38 54)(9 63 39 53 29 43 19 73)(10 42 20 52 30 62 40 72)

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - - + + + - - image C1 C2 C2 C2 C4 D4 Q8 Q8 D5 D10 D10 Dic10 C4×D5 C5⋊D4 Dic10 M4(2).C4 C23.Dic10 kernel C23.Dic10 C20.53D4 C2×C4.Dic5 C10×M4(2) C4.Dic5 C2×C20 C2×C20 C22×C10 C2×M4(2) M4(2) C22×C4 C2×C4 C2×C4 C2×C4 C23 C5 C1 # reps 1 4 2 1 8 2 1 1 2 4 2 4 8 8 4 2 8

Matrix representation of C23.Dic10 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 40 0 0 0 0 1 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 0 10 0 0 8 0 0 0 0 0 0 36 0 0 4 0
,
 0 0 1 0 0 0 0 1 32 0 0 0 0 9 0 0
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,8,0,0,10,0,0,0,0,0,0,4,0,0,36,0],[0,0,32,0,0,0,0,9,1,0,0,0,0,1,0,0] >;

C23.Dic10 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,751);
// by ID

G=gap.SmallGroup(320,751);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,422,58,136,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=b*c*d^10,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^19>;
// generators/relations

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