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G = C23.4D20order 320 = 26·5

4th non-split extension by C23 of D20 acting via D20/C5=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.4D20, C4.D4.D5, (C2×D4).6D10, (C2×C20).14D4, C23.D54C4, C23.4(C4×D5), (C22×Dic5)⋊2C4, (C22×C10).13D4, C54(C23.D4), C23⋊Dic5.3C2, C10.33(C23⋊C4), (D4×C10).171C22, C23.18D10.4C2, C22.13(D10⋊C4), C2.13(C23.1D10), (C2×C4).2(C5⋊D4), (C22×C10).4(C2×C4), (C5×C4.D4).1C2, (C2×C10).70(C22⋊C4), SmallGroup(320,34)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C23.4D20
C1C5C10C2×C10C2×C20D4×C10C23.18D10 — C23.4D20
C5C10C2×C10C22×C10 — C23.4D20
C1C2C22C2×D4C4.D4

Generators and relations for C23.4D20
 G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=ca=ac, ab=ba, dad-1=abc, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=acd19 >

Subgroups: 318 in 68 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2 [×3], C4 [×4], C22, C22 [×4], C5, C8, C2×C4, C2×C4 [×4], D4, C23 [×2], C10, C10 [×3], C22⋊C4 [×3], C4⋊C4, M4(2), C22×C4, C2×D4, Dic5 [×3], C20, C2×C10, C2×C10 [×4], C23⋊C4, C4.D4, C22.D4, C40, C2×Dic5 [×4], C2×C20, C5×D4, C22×C10 [×2], C23.D4, C10.D4, C23.D5, C23.D5 [×2], C5×M4(2), C22×Dic5, D4×C10, C23⋊Dic5, C5×C4.D4, C23.18D10, C23.4D20
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D10, C23⋊C4, C4×D5, D20, C5⋊D4, C23.D4, D10⋊C4, C23.1D10, C23.4D20

Smallest permutation representation of C23.4D20
On 80 points
Generators in S80
(1 70)(3 52)(4 24)(5 74)(7 56)(8 28)(9 78)(11 60)(12 32)(13 42)(15 64)(16 36)(17 46)(19 68)(20 40)(21 50)(23 72)(25 54)(27 76)(29 58)(31 80)(33 62)(35 44)(37 66)(39 48)(41 61)(45 65)(49 69)(53 73)(57 77)
(1 50)(2 71)(3 52)(4 73)(5 54)(6 75)(7 56)(8 77)(9 58)(10 79)(11 60)(12 41)(13 62)(14 43)(15 64)(16 45)(17 66)(18 47)(19 68)(20 49)(21 70)(22 51)(23 72)(24 53)(25 74)(26 55)(27 76)(28 57)(29 78)(30 59)(31 80)(32 61)(33 42)(34 63)(35 44)(36 65)(37 46)(38 67)(39 48)(40 69)
(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 64 50 35)(2 63 22 43)(3 33 72 42)(4 12)(5 60 54 31)(6 59 26 79)(7 29 76 78)(9 56 58 27)(10 55 30 75)(11 25 80 74)(13 52 62 23)(14 51 34 71)(15 21 44 70)(16 40)(17 48 66 19)(18 47 38 67)(20 36)(24 32)(37 68 46 39)(41 53)(45 49)(57 77)(61 73)(65 69)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F5A5B8A8B10A10B10C10D10E10F10G10H20A20B20C20D40A···40H
order12222444444558810101010101010102020202040···40
size112444202040404022882244888844448···8

35 irreducible representations

dim11111122222224448
type++++++++++-
imageC1C2C2C2C4C4D4D4D5D10C5⋊D4C4×D5D20C23⋊C4C23.D4C23.1D10C23.4D20
kernelC23.4D20C23⋊Dic5C5×C4.D4C23.18D10C23.D5C22×Dic5C2×C20C22×C10C4.D4C2×D4C2×C4C23C23C10C5C2C1
# reps11112211224441242

Matrix representation of C23.4D20 in GL6(𝔽41)

4000000
0400000
00163200
0012500
0000259
00004016
,
100000
010000
0025900
00401600
0000259
00004016
,
100000
010000
0040000
0004000
0000400
0000040
,
1990000
3200000
006212620
0010354015
0026203520
004015316
,
090000
900000
00001632
00002425
001000
00404000

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,1,0,0,0,0,32,25,0,0,0,0,0,0,25,40,0,0,0,0,9,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,25,40,0,0,0,0,9,16,0,0,0,0,0,0,25,40,0,0,0,0,9,16],[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],[19,32,0,0,0,0,9,0,0,0,0,0,0,0,6,10,26,40,0,0,21,35,20,15,0,0,26,40,35,31,0,0,20,15,20,6],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,40,0,0,16,24,0,0,0,0,32,25,0,0] >;

C23.4D20 in GAP, Magma, Sage, TeX

C_2^3._4D_{20}
% in TeX

G:=Group("C2^3.4D20");
// GroupNames label

G:=SmallGroup(320,34);
// by ID

G=gap.SmallGroup(320,34);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,422,184,346,297,851,12550]);
// Polycyclic

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

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