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G = C23.21D10order 160 = 25·5

2nd non-split extension by C23 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.21D10, (C2×C20)⋊11C4, (C2×C4)⋊4Dic5, C20.58(C2×C4), C4⋊Dic517C2, (C22×C4).7D5, C55(C42⋊C2), (C4×Dic5)⋊15C2, C2.4(C4○D20), (C2×C4).102D10, C4.15(C2×Dic5), C23.D5.5C2, C10.16(C4○D4), C10.37(C22×C4), (C2×C10).44C23, (C22×C20).10C2, (C2×C20).93C22, C22.5(C2×Dic5), C2.5(C22×Dic5), C22.22(C22×D5), (C22×C10).36C22, (C2×Dic5).38C22, (C2×C10).55(C2×C4), SmallGroup(160,147)

Series: Derived Chief Lower central Upper central

C1C10 — C23.21D10
C1C5C10C2×C10C2×Dic5C4×Dic5 — C23.21D10
C5C10 — C23.21D10
C1C2×C4C22×C4

Generators and relations for C23.21D10
 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=d9 >

Subgroups: 168 in 76 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2, C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×2], C22×C20, C23.21D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, C22×C4, C4○D4 [×2], Dic5 [×4], D10 [×3], C42⋊C2, C2×Dic5 [×6], C22×D5, C4○D20 [×2], C22×Dic5, C23.21D10

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

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

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

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

C23.21D10 is a maximal subgroup of
C426Dic5  C421Dic5  C20.33C42  (C2×C40)⋊C4  C23.9D20  M4(2)⋊4Dic5  C408C4⋊C2  C23.10D20  C23.13D20  C20.47(C4⋊C4)  C20.76(C4⋊C4)  (C2×D20)⋊25C4  (C2×D4).D10  C10.(C4○D8)  C20.42C42  C20.65(C4⋊C4)  C23.22D20  C23.23D20  C20.51(C4⋊C4)  C23.46D20  C23.47D20  C20.37C42  C23.48D20  C23.20D20  (D4×C10)⋊18C4  (Q8×C10)⋊16C4  C20.(C2×D4)  (D4×C10)⋊21C4  (D4×C10)⋊22C4  C42.274D10  C4×C4○D20  C24.31D10  C10.12- 1+4  C10.82+ 1+4  C10.52- 1+4  C42.87D10  C42.88D10  C42.90D10  D5×C42⋊C2  C427D10  C42.102D10  C42.105D10  C42.106D10  C42.229D10  C42.117D10  C42.119D10  C10.362+ 1+4  C10.432+ 1+4  C10.452+ 1+4  C10.472+ 1+4  C10.152- 1+4  C10.212- 1+4  C10.232- 1+4  C10.242- 1+4  C10.802- 1+4  C10.1222+ 1+4  C24.72D10  C24.38D10  C24.41D10  C10.422- 1+4  C10.442- 1+4  C10.1052- 1+4  C4○D4×Dic5  C10.1062- 1+4  (C2×C20)⋊15D4  C10.1462+ 1+4  (S3×C20)⋊5C4  (S3×C20)⋊7C4  C23.26(S3×D5)  C23.26D30
C23.21D10 is a maximal quotient of
C42.6Dic5  C42.7Dic5  C424Dic5  C4×C4⋊Dic5  C429Dic5  C425Dic5  C24.8D10  C4⋊C45Dic5  C42.187D10  C4×C23.D5  C24.63D10  C24.64D10  (S3×C20)⋊5C4  (S3×C20)⋊7C4  C23.26(S3×D5)  C23.26D30

52 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G···4N5A5B10A···10N20A···20P
order1222224444444···45510···1020···20
size11112211112210···10222···22···2

52 irreducible representations

dim111111222222
type++++++-++
imageC1C2C2C2C2C4D5C4○D4Dic5D10D10C4○D20
kernelC23.21D10C4×Dic5C4⋊Dic5C23.D5C22×C20C2×C20C22×C4C10C2×C4C2×C4C23C2
# reps1222182484216

Matrix representation of C23.21D10 in GL3(𝔽41) generated by

4000
010
02040
,
4000
010
001
,
100
0400
0040
,
100
0360
03033
,
900
0416
02237
G:=sub<GL(3,GF(41))| [40,0,0,0,1,20,0,0,40],[40,0,0,0,1,0,0,0,1],[1,0,0,0,40,0,0,0,40],[1,0,0,0,36,30,0,0,33],[9,0,0,0,4,22,0,16,37] >;

C23.21D10 in GAP, Magma, Sage, TeX

C_2^3._{21}D_{10}
% in TeX

G:=Group("C2^3.21D10");
// GroupNames label

G:=SmallGroup(160,147);
// by ID

G=gap.SmallGroup(160,147);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,362,4613]);
// 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=d^9>;
// generators/relations

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