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G = C23.Dic10order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.6Dic10, M4(2).30D10, C20.68(C4⋊C4), (C2×C20).28Q8, C20.441(C2×D4), (C2×C20).483D4, (C2×C4).16Dic10, C4.Dic5.10C4, C20.53D413C2, (C22×C10).16Q8, C53(M4(2).C4), C20.127(C22×C4), (C2×C20).415C23, (C22×C4).134D10, (C2×M4(2)).16D5, C22.4(C2×Dic10), C4.21(C10.D4), (C10×M4(2)).27C2, C4.Dic5.41C22, (C22×C20).183C22, (C5×M4(2)).33C22, C22.17(C10.D4), C4.90(C2×C4×D5), C10.75(C2×C4⋊C4), C52C8.5(C2×C4), (C2×C4).49(C4×D5), C4.131(C2×C5⋊D4), (C2×C10).11(C2×Q8), (C2×C10).82(C4⋊C4), (C2×C20).276(C2×C4), (C2×C4).194(C5⋊D4), C2.19(C2×C10.D4), (C2×C4).511(C22×D5), (C2×C4.Dic5).24C2, (C2×C52C8).143C22, SmallGroup(320,751)

Series: Derived Chief Lower central Upper central

C1C20 — C23.Dic10
C1C5C10C20C2×C20C2×C52C8C2×C4.Dic5 — C23.Dic10
C5C10C20 — C23.Dic10
C1C4C22×C4C2×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 [×3], C4 [×4], C22 [×3], C22, C5, C8 [×8], C2×C4 [×6], C23, C10, C10 [×3], C2×C8 [×4], M4(2) [×2], M4(2) [×8], C22×C4, C20 [×4], C2×C10 [×3], C2×C10, C8.C4 [×4], C2×M4(2), C2×M4(2) [×2], C52C8 [×4], C52C8 [×2], C40 [×2], C2×C20 [×6], C22×C10, M4(2).C4, C2×C52C8 [×2], C2×C52C8, C4.Dic5 [×6], C4.Dic5, C2×C40, C5×M4(2) [×2], C5×M4(2), C22×C20, C20.53D4 [×4], C2×C4.Dic5 [×2], C10×M4(2), C23.Dic10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, M4(2).C4, C10.D4 [×4], 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)(41 61)(43 63)(45 65)(47 67)(49 69)(51 71)(53 73)(55 75)(57 77)(59 79)
(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 56 31 46 21 76 11 66)(2 75 12 45 22 55 32 65)(3 74 33 64 23 54 13 44)(4 53 14 63 24 73 34 43)(5 52 35 42 25 72 15 62)(6 71 16 41 26 51 36 61)(7 70 37 60 27 50 17 80)(8 49 18 59 28 69 38 79)(9 48 39 78 29 68 19 58)(10 67 20 77 30 47 40 57)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B8A8B8C8D8E···8L10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222444445588888···810···101010101020···202020202040···40
size112221122222444420···202···244442···244444···4

62 irreducible representations

dim11111222222222244
type+++++--+++--
imageC1C2C2C2C4D4Q8Q8D5D10D10Dic10C4×D5C5⋊D4Dic10M4(2).C4C23.Dic10
kernelC23.Dic10C20.53D4C2×C4.Dic5C10×M4(2)C4.Dic5C2×C20C2×C20C22×C10C2×M4(2)M4(2)C22×C4C2×C4C2×C4C2×C4C23C5C1
# reps14218211242488428

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

1000
0100
00400
00040
,
1000
04000
0010
00040
,
40000
04000
00400
00040
,
01000
8000
00036
0040
,
0010
0001
32000
0900
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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