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G = C23.D10order 160 = 25·5

3rd non-split extension by C23 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.3D10, C4⋊Dic53C2, (C2×C4).27D10, C22⋊C4.2D5, C52(C422C2), (C4×Dic5)⋊10C2, C10.7(C4○D4), C2.9(C4○D20), C10.D48C2, (C2×C20).2C22, C23.D5.3C2, C2.7(D42D5), (C2×C10).20C23, (C22×C10).9C22, C22.40(C22×D5), (C2×Dic5).29C22, (C5×C22⋊C4).2C2, SmallGroup(160,100)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C23.D10
C1C5C10C2×C10C2×Dic5C4×Dic5 — C23.D10
C5C2×C10 — C23.D10
C1C22C22⋊C4

Generators and relations for C23.D10
 G = < a,b,c,d,e | a2=b2=c2=1, d10=b, e2=cb=bc, eae-1=ab=ba, dad-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d9 >

Subgroups: 160 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2 [×3], C2, C4 [×6], C22, C22 [×3], C5, C2×C4 [×2], C2×C4 [×4], C23, C10 [×3], C10, C42, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×3], Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×3], C422C2, C2×Dic5 [×4], C2×C20 [×2], C22×C10, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C23.D5 [×2], C5×C22⋊C4, C23.D10
Quotients: C1, C2 [×7], C22 [×7], C23, D5, C4○D4 [×3], D10 [×3], C422C2, C22×D5, C4○D20, D42D5 [×2], C23.D10

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

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

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

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

C23.D10 is a maximal subgroup of
C24.30D10  C24.31D10  C42.89D10  C42.93D10  C42.94D10  C42.98D10  C42.102D10  C42.104D10  C42.105D10  C42.106D10  C42.229D10  C42.113D10  C42.115D10  C42.118D10  C24.32D10  C24.35D10  C24.36D10  C4⋊C4.178D10  C10.342+ 1+4  C10.352+ 1+4  C10.362+ 1+4  C10.422+ 1+4  C10.432+ 1+4  C10.1152+ 1+4  C10.482+ 1+4  C10.152- 1+4  C10.202- 1+4  C10.212- 1+4  C10.222- 1+4  C10.232- 1+4  C10.582+ 1+4  C4⋊C4.197D10  C10.802- 1+4  C10.812- 1+4  C10.612+ 1+4  C10.622+ 1+4  C10.632+ 1+4  C10.642+ 1+4  C10.842- 1+4  C10.852- 1+4  C42.137D10  C42.139D10  C42.140D10  C4220D10  C4221D10  C42.234D10  C42.144D10  C42.159D10  C42.160D10  D5×C422C2  C4224D10  C42.162D10  C42.165D10  D6⋊C4.D5  C605C4⋊C2  D6⋊Dic5.C2  C5⋊(C423S3)  C23.13(S3×D5)  C23.14(S3×D5)  C23.8D30
C23.D10 is a maximal quotient of
C52(C425C4)  C10.51(C4×D4)  C2.(C4×D20)  C10.52(C4×D4)  (C2×Dic5).Q8  (C2×C20).28D4  (C2×C4).Dic10  (C22×C4).D10  C24.3D10  C24.4D10  C24.6D10  C24.8D10  C24.9D10  C23.14D20  D6⋊C4.D5  C605C4⋊C2  D6⋊Dic5.C2  C5⋊(C423S3)  C23.13(S3×D5)  C23.14(S3×D5)  C23.8D30

34 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I5A5B10A···10F10G10H10I10J20A···20H
order122224444444445510···101010101020···20
size11114224101010102020222···244444···4

34 irreducible representations

dim111111222224
type+++++++++-
imageC1C2C2C2C2C2D5C4○D4D10D10C4○D20D42D5
kernelC23.D10C4×Dic5C10.D4C4⋊Dic5C23.D5C5×C22⋊C4C22⋊C4C10C2×C4C23C2C2
# reps112121264284

Matrix representation of C23.D10 in GL4(𝔽41) generated by

1000
64000
0010
002540
,
40000
04000
0010
0001
,
1000
0100
00400
00040
,
8000
9500
00405
0001
,
303100
41100
00320
00032
G:=sub<GL(4,GF(41))| [1,6,0,0,0,40,0,0,0,0,1,25,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[8,9,0,0,0,5,0,0,0,0,40,0,0,0,5,1],[30,4,0,0,31,11,0,0,0,0,32,0,0,0,0,32] >;

C23.D10 in GAP, Magma, Sage, TeX

C_2^3.D_{10}
% in TeX

G:=Group("C2^3.D10");
// GroupNames label

G:=SmallGroup(160,100);
// by ID

G=gap.SmallGroup(160,100);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,55,506,188,4613]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^10=b,e^2=c*b=b*c,e*a*e^-1=a*b=b*a,d*a*d^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^9>;
// generators/relations

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