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G = C233D20order 320 = 26·5

2nd semidirect product of C23 and D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C233D20, C24.29D10, C10.42+ 1+4, C207D43C2, (C22×C4)⋊9D10, C22⋊D203C2, C22⋊C443D10, (C2×D20)⋊3C22, (C22×C10)⋊10D4, C51(C233D4), C4⋊Dic55C22, C10.8(C22×D4), (C2×C10).37C24, (C22×C20)⋊8C22, C22.18(C2×D20), C2.10(C22×D20), (C23×D5)⋊4C22, C2.8(D46D10), D10⋊C41C22, (C2×C20).130C23, C22.D202C2, (C22×D5).9C23, C22.76(C23×D5), (C23×C10).63C22, (C2×Dic5).10C23, (C22×Dic5)⋊7C22, C23.148(C22×D5), (C22×C10).127C23, (C2×C22⋊C4)⋊16D5, (C22×C5⋊D4)⋊6C2, (C10×C22⋊C4)⋊15C2, (C2×C10).173(C2×D4), (C2×C5⋊D4)⋊36C22, (C5×C22⋊C4)⋊48C22, (C2×C4).136(C22×D5), SmallGroup(320,1165)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C233D20
C1C5C10C2×C10C22×D5C23×D5C22×C5⋊D4 — C233D20
C5C2×C10 — C233D20
C1C22C2×C22⋊C4

Generators and relations for C233D20
 G = < a,b,c,d,e | a2=b2=c2=d20=e2=1, ab=ba, dad-1=eae=ac=ca, ebe=bc=cb, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1550 in 346 conjugacy classes, 111 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C22, C22 [×6], C22 [×30], C5, C2×C4 [×4], C2×C4 [×10], D4 [×20], C23, C23 [×6], C23 [×14], D5 [×4], C10, C10 [×2], C10 [×6], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×2], C2×D4 [×20], C24, C24 [×2], Dic5 [×4], C20 [×4], D10 [×20], C2×C10, C2×C10 [×6], C2×C10 [×10], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], D20 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20 [×4], C2×C20 [×2], C22×D5 [×4], C22×D5 [×8], C22×C10, C22×C10 [×6], C22×C10 [×2], C233D4, C4⋊Dic5 [×4], D10⋊C4 [×8], C5×C22⋊C4 [×4], C2×D20 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×8], C2×C5⋊D4 [×8], C22×C20 [×2], C23×D5 [×2], C23×C10, C22⋊D20 [×4], C22.D20 [×4], C207D4 [×4], C10×C22⋊C4, C22×C5⋊D4 [×2], C233D20
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ 1+4 [×2], D20 [×4], C22×D5 [×7], C233D4, C2×D20 [×6], C23×D5, C22×D20, D46D10 [×2], C233D20

Smallest permutation representation of C233D20
On 80 points
Generators in S80
(1 40)(2 12)(3 22)(4 14)(5 24)(6 16)(7 26)(8 18)(9 28)(10 20)(11 30)(13 32)(15 34)(17 36)(19 38)(21 31)(23 33)(25 35)(27 37)(29 39)(41 61)(42 52)(43 63)(44 54)(45 65)(46 56)(47 67)(48 58)(49 69)(50 60)(51 71)(53 73)(55 75)(57 77)(59 79)(62 72)(64 74)(66 76)(68 78)(70 80)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(40 53)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(39 40)(41 74)(42 73)(43 72)(44 71)(45 70)(46 69)(47 68)(48 67)(49 66)(50 65)(51 64)(52 63)(53 62)(54 61)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A4B4C4D4E4F4G4H5A5B10A···10N10O···10V20A···20P
order12222···22222444444445510···1010···1020···20
size11112···220202020444420202020222···24···44···4

62 irreducible representations

dim11111122222244
type+++++++++++++
imageC1C2C2C2C2C2D4D5D10D10D10D202+ 1+4D46D10
kernelC233D20C22⋊D20C22.D20C207D4C10×C22⋊C4C22×C5⋊D4C22×C10C2×C22⋊C4C22⋊C4C22×C4C24C23C10C2
# reps144412428421628

Matrix representation of C233D20 in GL6(𝔽41)

100000
010000
001000
000100
0000400
0000040
,
100000
010000
00233500
0061800
00002335
0000618
,
100000
010000
0040000
0004000
0000400
0000040
,
0400000
100000
000061
0000400
00354000
001000
,
010000
100000
000001
000010
000100
001000

G:=sub<GL(6,GF(41))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,6,0,0,0,0,35,18,0,0,0,0,0,0,23,6,0,0,0,0,35,18],[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],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,35,1,0,0,0,0,40,0,0,0,6,40,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;

C233D20 in GAP, Magma, Sage, TeX

C_2^3\rtimes_3D_{20}
% in TeX

G:=Group("C2^3:3D20");
// GroupNames label

G:=SmallGroup(320,1165);
// by ID

G=gap.SmallGroup(320,1165);
# by ID

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

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

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