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G = C24.48D10order 320 = 26·5

6th non-split extension by C24 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.48D10, C23.44D20, (C23×D5)⋊8C4, (C22×C4)⋊1D10, C52(C243C4), (D5×C24).1C2, C23.51(C4×D5), C10.37C22≀C2, D104(C22⋊C4), (C22×C20)⋊1C22, (C22×C10).67D4, C22.43(C2×D20), C22.100(D4×D5), C2.2(C23⋊D10), C2.4(C22⋊D20), (C22×D5).124D4, C23.52(C5⋊D4), C223(D10⋊C4), (C23×C10).38C22, (C22×Dic5)⋊2C22, (C23×D5).99C22, C23.282(C22×D5), (C22×C10).329C23, (C2×C22⋊C4)⋊2D5, (C10×C22⋊C4)⋊2C2, (C2×C23.D5)⋊2C2, C2.28(D5×C22⋊C4), (C2×D10⋊C4)⋊3C2, C22.126(C2×C4×D5), (C2×C10)⋊4(C22⋊C4), C2.9(C2×D10⋊C4), (C2×C10).321(C2×D4), C10.77(C2×C22⋊C4), C22.50(C2×C5⋊D4), (C22×C10).121(C2×C4), (C2×C10).209(C22×C4), (C22×D5).104(C2×C4), SmallGroup(320,582)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.48D10
C1C5C10C2×C10C22×C10C23×D5D5×C24 — C24.48D10
C5C2×C10 — C24.48D10
C1C23C2×C22⋊C4

Generators and relations for C24.48D10
 G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=b, ab=ba, dad-1=eae-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 2318 in 506 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×4], C22 [×3], C22 [×8], C22 [×76], C5, C2×C4 [×12], C23, C23 [×6], C23 [×80], D5 [×8], C10, C10 [×6], C10 [×4], C22⋊C4 [×12], C22×C4 [×2], C22×C4 [×2], C24, C24 [×18], Dic5 [×2], C20 [×2], D10 [×8], D10 [×56], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2×C22⋊C4, C2×C22⋊C4 [×5], C25, C2×Dic5 [×6], C2×C20 [×6], C22×D5 [×12], C22×D5 [×64], C22×C10, C22×C10 [×6], C22×C10 [×4], C243C4, D10⋊C4 [×8], C23.D5 [×2], C5×C22⋊C4 [×2], C22×Dic5 [×2], C22×C20 [×2], C23×D5 [×6], C23×D5 [×12], C23×C10, C2×D10⋊C4 [×4], C2×C23.D5, C10×C22⋊C4, D5×C24, C24.48D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, D5, C22⋊C4 [×12], C22×C4, C2×D4 [×6], D10 [×3], C2×C22⋊C4 [×3], C22≀C2 [×4], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C243C4, D10⋊C4 [×4], C2×C4×D5, C2×D20, D4×D5 [×4], C2×C5⋊D4, D5×C22⋊C4 [×2], C22⋊D20 [×2], C2×D10⋊C4, C23⋊D10 [×2], C24.48D10

Smallest permutation representation of C24.48D10
On 80 points
Generators in S80
(1 22)(2 62)(3 24)(4 64)(5 26)(6 66)(7 28)(8 68)(9 30)(10 70)(11 32)(12 72)(13 34)(14 74)(15 36)(16 76)(17 38)(18 78)(19 40)(20 80)(21 52)(23 54)(25 56)(27 58)(29 60)(31 42)(33 44)(35 46)(37 48)(39 50)(41 69)(43 71)(45 73)(47 75)(49 77)(51 79)(53 61)(55 63)(57 65)(59 67)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 41)
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 80)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)
(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 31 71 52)(2 51 72 30)(3 29 73 50)(4 49 74 28)(5 27 75 48)(6 47 76 26)(7 25 77 46)(8 45 78 24)(9 23 79 44)(10 43 80 22)(11 21 61 42)(12 41 62 40)(13 39 63 60)(14 59 64 38)(15 37 65 58)(16 57 66 36)(17 35 67 56)(18 55 68 34)(19 33 69 54)(20 53 70 32)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S4A4B4C4D4E4F4G4H5A5B10A···10N10O···10V20A···20P
order12···222222···2444444445510···1010···1020···20
size11···1222210···10444420202020222···24···44···4

68 irreducible representations

dim111111222222224
type++++++++++++
imageC1C2C2C2C2C4D4D4D5D10D10C4×D5D20C5⋊D4D4×D5
kernelC24.48D10C2×D10⋊C4C2×C23.D5C10×C22⋊C4D5×C24C23×D5C22×D5C22×C10C2×C22⋊C4C22×C4C24C23C23C23C22
# reps141118842428888

Matrix representation of C24.48D10 in GL6(𝔽41)

4000000
0400000
0040000
0004000
0000400
000001
,
4000000
0400000
001000
000100
000010
000001
,
100000
010000
001000
000100
0000400
0000040
,
21240000
38180000
0012100
00374000
000001
000010
,
23240000
36180000
0012100
0004000
0000040
0000400

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[21,38,0,0,0,0,24,18,0,0,0,0,0,0,1,37,0,0,0,0,21,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[23,36,0,0,0,0,24,18,0,0,0,0,0,0,1,0,0,0,0,0,21,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;

C24.48D10 in GAP, Magma, Sage, TeX

C_2^4._{48}D_{10}
% in TeX

G:=Group("C2^4.48D10");
// GroupNames label

G:=SmallGroup(320,582);
// by ID

G=gap.SmallGroup(320,582);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*c=c*b,b*d=d*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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