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G = C23.23D10order 160 = 25·5

4th non-split extension by C23 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.23D10, (C22×C4)⋊3D5, (C22×C20)⋊2C2, C10.42(C2×D4), (C2×C10).37D4, (C2×C4).65D10, C23.D56C2, D10⋊C42C2, C10.D43C2, C10.18(C4○D4), C2.18(C4○D20), (C2×C10).47C23, (C2×C20).78C22, C54(C22.D4), C22.9(C5⋊D4), (C22×D5).9C22, C22.55(C22×D5), (C22×C10).39C22, (C2×Dic5).15C22, C2.6(C2×C5⋊D4), (C2×C5⋊D4).6C2, SmallGroup(160,150)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C23.23D10
C1C5C10C2×C10C22×D5C2×C5⋊D4 — C23.23D10
C5C2×C10 — C23.23D10
C1C22C22×C4

Generators and relations for C23.23D10
 G = < a,b,c,d,e | a2=b2=c2=1, d10=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd9 >

Subgroups: 240 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×5], C22, C22 [×2], C22 [×5], C5, C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, D5, C10, C10 [×2], C10 [×2], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22.D4, C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C10.D4 [×2], D10⋊C4 [×2], C23.D5, C2×C5⋊D4, C22×C20, C23.23D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C5⋊D4 [×2], C22×D5, C4○D20 [×2], C2×C5⋊D4, C23.23D10

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

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

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

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

C23.23D10 is a maximal subgroup of
(C22×C4)⋊F5  C22⋊C4⋊D10  C42.277D10  C24.27D10  C24.31D10  C10.2- 1+4  C10.52- 1+4  C10.62- 1+4  C4210D10  C42.96D10  C42.104D10  C4216D10  C42.113D10  C42.114D10  C4217D10  C42.115D10  C42.116D10  C42.118D10  C10.422+ 1+4  C10.442+ 1+4  C10.482+ 1+4  C10.742- 1+4  C10.202- 1+4  C10.222- 1+4  C10.582+ 1+4  C10.262- 1+4  C10.792- 1+4  C4⋊C4.197D10  D5×C22.D4  C10.1202+ 1+4  C4⋊C428D10  C10.852- 1+4  C24.72D10  C248D10  C10.442- 1+4  C10.1042- 1+4  C10.1452+ 1+4  D6⋊Dic5⋊C2  D10⋊C4⋊S3  (C2×C30).D4  C10.(C2×D12)  C23.28D30
C23.23D10 is a maximal quotient of
C10.92(C4×D4)  (C2×C42)⋊D5  C24.9D10  C24.14D10  (C2×C10).40D8  C4⋊C4.228D10  C4⋊C4.230D10  C4⋊C4.231D10  (C2×C20).287D4  (C2×C20).288D4  (C2×C20).289D4  (C2×C20).290D4  C4⋊C4.233D10  C4⋊C4.236D10  C24.62D10  C24.63D10  C24.65D10  D6⋊Dic5⋊C2  D10⋊C4⋊S3  (C2×C30).D4  C10.(C2×D12)  C23.28D30

46 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B10A···10N20A···20P
order122222244444445510···1020···20
size111122202222202020222···22···2

46 irreducible representations

dim1111112222222
type++++++++++
imageC1C2C2C2C2C2D4D5C4○D4D10D10C5⋊D4C4○D20
kernelC23.23D10C10.D4D10⋊C4C23.D5C2×C5⋊D4C22×C20C2×C10C22×C4C10C2×C4C23C22C2
# reps12211122442816

Matrix representation of C23.23D10 in GL4(𝔽41) generated by

1000
0100
00241
004017
,
40000
04000
0010
0001
,
1000
0100
00400
00040
,
352600
382000
001919
00229
,
212600
242000
001919
00922
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,24,40,0,0,1,17],[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],[35,38,0,0,26,20,0,0,0,0,19,22,0,0,19,9],[21,24,0,0,26,20,0,0,0,0,19,9,0,0,19,22] >;

C23.23D10 in GAP, Magma, Sage, TeX

C_2^3._{23}D_{10}
% in TeX

G:=Group("C2^3.23D10");
// GroupNames label

G:=SmallGroup(160,150);
// by ID

G=gap.SmallGroup(160,150);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,218,86,4613]);
// Polycyclic

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

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