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G = C2×C23.1D10order 320 = 26·5

Direct product of C2 and C23.1D10

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C23.1D10, C23.15D20, C24.11D10, (C23×D5)⋊4C4, C103(C23⋊C4), C22⋊C436D10, C23.17(C4×D5), (C22×Dic5)⋊5C4, (C22×C10).66D4, C22.14(C2×D20), C23.74(C5⋊D4), C23.D542C22, C23.72(C22×D5), (C23×C10).37C22, (C22×C10).111C23, C22.44(D10⋊C4), C55(C2×C23⋊C4), (C2×C5⋊D4)⋊8C4, (C2×C22⋊C4)⋊1D5, C22.18(C2×C4×D5), (C10×C22⋊C4)⋊1C2, (C2×Dic5)⋊2(C2×C4), (C22×D5)⋊2(C2×C4), (C2×C23.D5)⋊1C2, (C2×C10).433(C2×D4), C2.8(C2×D10⋊C4), C10.76(C2×C22⋊C4), (C22×C5⋊D4).1C2, C22.26(C2×C5⋊D4), (C5×C22⋊C4)⋊44C22, (C2×C5⋊D4).91C22, (C22×C10).120(C2×C4), (C2×C10).113(C22×C4), (C2×C10).120(C22⋊C4), SmallGroup(320,581)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C23.1D10
C1C5C10C2×C10C22×C10C2×C5⋊D4C22×C5⋊D4 — C2×C23.1D10
C5C10C2×C10 — C2×C23.1D10
C1C22C24C2×C22⋊C4

Generators and relations for C2×C23.1D10
 G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=b, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 926 in 210 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×6], C22 [×7], C22 [×18], C5, C2×C4 [×12], D4 [×8], C23 [×7], C23 [×8], D5 [×2], C10, C10 [×2], C10 [×6], C22⋊C4 [×2], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×8], C24, C24, Dic5 [×4], C20 [×2], D10 [×8], C2×C10 [×7], C2×C10 [×10], C23⋊C4 [×4], C2×C22⋊C4, C2×C22⋊C4, C22×D4, C2×Dic5 [×2], C2×Dic5 [×6], C5⋊D4 [×8], C2×C20 [×4], C22×D5 [×2], C22×D5 [×4], C22×C10 [×7], C22×C10 [×2], C2×C23⋊C4, C23.D5 [×2], C23.D5, C5×C22⋊C4 [×2], C5×C22⋊C4, C22×Dic5, C22×Dic5, C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×C20, C23×D5, C23×C10, C23.1D10 [×4], C2×C23.D5, C10×C22⋊C4, C22×C5⋊D4, C2×C23.1D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C23⋊C4 [×2], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C2×C23⋊C4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C23.1D10 [×2], C2×D10⋊C4, C2×C23.1D10

Smallest permutation representation of C2×C23.1D10
On 80 points
Generators in S80
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)(41 77)(42 78)(43 79)(44 80)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)
(1 11)(2 73)(3 13)(4 75)(5 15)(6 77)(7 17)(8 79)(9 19)(10 61)(12 63)(14 65)(16 67)(18 69)(20 71)(21 31)(22 49)(23 33)(24 51)(25 35)(26 53)(27 37)(28 55)(29 39)(30 57)(32 59)(34 41)(36 43)(38 45)(40 47)(42 52)(44 54)(46 56)(48 58)(50 60)(62 72)(64 74)(66 76)(68 78)(70 80)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 61)(21 58)(22 59)(23 60)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)
(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 11 20)(2 80 73 70)(3 8 13 79)(4 17 75 7)(5 67 15 16)(6 76 77 66)(9 63 19 12)(10 72 61 62)(14 68 65 78)(18 64 69 74)(21 43 31 36)(22 52 49 42)(23 24 33 51)(25 59 35 32)(26 48 53 58)(27 40 37 47)(28 29 55 39)(30 44 57 54)(34 60 41 50)(38 56 45 46)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A4B4C4D4E···4J5A5B10A···10N10O···10V20A···20P
order12222···22244444···45510···1010···1020···20
size11112···22020444420···20222···24···44···4

62 irreducible representations

dim11111111222222244
type+++++++++++
imageC1C2C2C2C2C4C4C4D4D5D10D10C4×D5D20C5⋊D4C23⋊C4C23.1D10
kernelC2×C23.1D10C23.1D10C2×C23.D5C10×C22⋊C4C22×C5⋊D4C22×Dic5C2×C5⋊D4C23×D5C22×C10C2×C22⋊C4C22⋊C4C24C23C23C23C10C2
# reps14111242424288828

Matrix representation of C2×C23.1D10 in GL6(𝔽41)

4000000
0400000
0040000
0004000
0000400
0000040
,
100000
010000
0040900
000100
00399040
00232400
,
100000
010000
0040000
0004000
0000400
0000040
,
2230000
38220000
0090320
00004040
00040320
000090
,
3190000
19380000
0090320
0020401
00040320
0023090

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,39,2,0,0,9,1,9,32,0,0,0,0,0,40,0,0,0,0,40,0],[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],[22,38,0,0,0,0,3,22,0,0,0,0,0,0,9,0,0,0,0,0,0,0,40,0,0,0,32,40,32,9,0,0,0,40,0,0],[3,19,0,0,0,0,19,38,0,0,0,0,0,0,9,2,0,23,0,0,0,0,40,0,0,0,32,40,32,9,0,0,0,1,0,0] >;

C2×C23.1D10 in GAP, Magma, Sage, TeX

C_2\times C_2^3._1D_{10}
% in TeX

G:=Group("C2xC2^3.1D10");
// GroupNames label

G:=SmallGroup(320,581);
// by ID

G=gap.SmallGroup(320,581);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,58,1123,438,12550]);
// Polycyclic

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

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