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G = C4:C4:26D10order 320 = 26·5

9th semidirect product of C4:C4 and D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4:C4:26D10, (C4xD5):13D4, (C2xQ8):15D10, C4.186(D4xD5), C22:Q8:26D5, D10:4(C4oD4), C4:D20:23C2, C20:7D4:35C2, D10.76(C2xD4), C20.231(C2xD4), D20:8C4:24C2, C22:D20:15C2, (Q8xC10):6C22, D10:3Q8:13C2, (C2xD20):24C22, C4:Dic5:35C22, C22:C4.56D10, C10.73(C22xD4), Dic5:4D4:14C2, (C2xC20).598C23, (C2xC10).171C24, Dic5.119(C2xD4), C5:5(C22.19C24), C22:1(Q8:2D5), (C4xDic5):27C22, (C22xC4).372D10, D10.13D4:15C2, D10:C4:19C22, C10.D4:17C22, (C2xDic5).86C23, C23.188(C22xD5), C22.192(C23xD5), (C22xC20).251C22, (C22xC10).199C23, (C22xD5).203C23, (C23xD5).121C22, (C22xDic5).248C22, C2.46(C2xD4xD5), (D5xC22xC4):5C2, C2.48(D5xC4oD4), (C2xC4xD5):17C22, C4:C4:7D5:24C2, (C5xC22:Q8):7C2, (C2xC10):6(C4oD4), (C5xC4:C4):18C22, (C2xQ8:2D5):5C2, C10.160(C2xC4oD4), C2.16(C2xQ8:2D5), (C2xC4).46(C22xD5), (C2xC5:D4).38C22, (C5xC22:C4).26C22, SmallGroup(320,1299)

Series: Derived Chief Lower central Upper central

C1C2xC10 — C4:C4:26D10
C1C5C10C2xC10C22xD5C23xD5D5xC22xC4 — C4:C4:26D10
C5C2xC10 — C4:C4:26D10
C1C22C22:Q8

Generators and relations for C4:C4:26D10
 G = < a,b,c,d | a4=b4=c10=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b-1, dbd=a2b, dcd=c-1 >

Subgroups: 1294 in 330 conjugacy classes, 109 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, D5, C10, C10, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C24, Dic5, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, C2xC10, C42:C2, C4xD4, C22wrC2, C4:D4, C22:Q8, C22:Q8, C22.D4, C23xC4, C2xC4oD4, C4xD5, C4xD5, D20, C2xDic5, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C2xC20, C5xQ8, C22xD5, C22xD5, C22xD5, C22xC10, C22.19C24, C4xDic5, C10.D4, C4:Dic5, D10:C4, C5xC22:C4, C5xC4:C4, C5xC4:C4, C2xC4xD5, C2xC4xD5, C2xC4xD5, C2xD20, C2xD20, Q8:2D5, C22xDic5, C2xC5:D4, C22xC20, Q8xC10, C23xD5, Dic5:4D4, C22:D20, C4:C4:7D5, D20:8C4, D10.13D4, C4:D20, C20:7D4, D10:3Q8, C5xC22:Q8, D5xC22xC4, C2xQ8:2D5, C4:C4:26D10
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C4oD4, C24, D10, C22xD4, C2xC4oD4, C22xD5, C22.19C24, D4xD5, Q8:2D5, C23xD5, C2xD4xD5, C2xQ8:2D5, D5xC4oD4, C4:C4:26D10

Smallest permutation representation of C4:C4:26D10
On 80 points
Generators in S80
(1 53 13 43)(2 54 14 44)(3 55 15 45)(4 56 16 46)(5 57 17 47)(6 58 18 48)(7 59 19 49)(8 60 20 50)(9 51 11 41)(10 52 12 42)(21 77 26 72)(22 78 27 73)(23 79 28 74)(24 80 29 75)(25 71 30 76)(31 65 36 70)(32 66 37 61)(33 67 38 62)(34 68 39 63)(35 69 40 64)
(1 33 6 27)(2 23 7 39)(3 35 8 29)(4 25 9 31)(5 37 10 21)(11 36 16 30)(12 26 17 32)(13 38 18 22)(14 28 19 34)(15 40 20 24)(41 65 46 71)(42 77 47 61)(43 67 48 73)(44 79 49 63)(45 69 50 75)(51 70 56 76)(52 72 57 66)(53 62 58 78)(54 74 59 68)(55 64 60 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)(2 9)(3 8)(4 7)(5 6)(11 14)(12 13)(15 20)(16 19)(17 18)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 40)(30 39)(41 44)(42 43)(45 50)(46 49)(47 48)(51 54)(52 53)(55 60)(56 59)(57 58)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 80)(70 79)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B10A···10F10G10H10I10J20A···20H20I···20P
order12222222222244444444444444445510···101010101020···2020···20
size11112210101010202022224444555510102020222···244444···48···8

56 irreducible representations

dim11111111111122222222444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D5C4oD4C4oD4D10D10D10D10D4xD5Q8:2D5D5xC4oD4
kernelC4:C4:26D10Dic5:4D4C22:D20C4:C4:7D5D20:8C4D10.13D4C4:D20C20:7D4D10:3Q8C5xC22:Q8D5xC22xC4C2xQ8:2D5C4xD5C22:Q8D10C2xC10C22:C4C4:C4C22xC4C2xQ8C4C22C2
# reps12212211111142444622444

Matrix representation of C4:C4:26D10 in GL6(F41)

100000
010000
0032000
0032900
000010
000001
,
4000000
0400000
0013900
0014000
000001
0000400
,
34350000
700000
0040000
0004000
0000400
000001
,
710000
34340000
0040000
0040100
0000400
0000040

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,32,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[34,7,0,0,0,0,35,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[7,34,0,0,0,0,1,34,0,0,0,0,0,0,40,40,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

C4:C4:26D10 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes_{26}D_{10}
% in TeX

G:=Group("C4:C4:26D10");
// GroupNames label

G:=SmallGroup(320,1299);
// by ID

G=gap.SmallGroup(320,1299);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,1123,794,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

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