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G = (C2xC20):15D4order 320 = 26·5

11st semidirect product of C2xC20 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2xC20):15D4, (C2xD4):43D10, (C2xQ8):32D10, D10:6(C4oD4), C20:2D4:46C2, C20.265(C2xD4), (C22xC4):29D10, C23:D10:34C2, D10:3Q8:47C2, (D4xC10):47C22, C4:Dic5:80C22, (Q8xC10):39C22, (C2xC20).651C23, (C2xC10).314C24, (C22xC20):42C22, C5:8(C22.19C24), (C4xDic5):60C22, C10.164(C22xD4), C23.D5:65C22, C10.D4:76C22, C22.323(C23xD5), C23.137(C22xD5), C23.18D10:34C2, C23.21D10:38C2, (C22xC10).240C23, (C2xDic5).303C23, (C23xD5).127C22, (C22xD5).253C23, D10:C4.159C22, (C22xDic5).259C22, (C2xC4oD4):6D5, (D5xC22xC4):7C2, (C10xC4oD4):6C2, (C4xC5:D4):60C2, (C2xC4):14(C5:D4), C2.103(D5xC4oD4), (C2xC10).80(C2xD4), C4.100(C2xC5:D4), C10.215(C2xC4oD4), (C2xC4xD5).333C22, C22.23(C2xC5:D4), C2.37(C22xC5:D4), (C2xC4).639(C22xD5), (C2xC5:D4).148C22, SmallGroup(320,1500)

Series: Derived Chief Lower central Upper central

C1C2xC10 — (C2xC20):15D4
C1C5C10C2xC10C22xD5C23xD5D5xC22xC4 — (C2xC20):15D4
C5C2xC10 — (C2xC20):15D4
C1C2xC4C2xC4oD4

Generators and relations for (C2xC20):15D4
 G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, cac-1=ab10, ad=da, cbc-1=dbd=b9, dcd=c-1 >

Subgroups: 1118 in 330 conjugacy classes, 115 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C23, D5, C10, C10, C10, C42, C22:C4, C4:C4, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, C2xC10, C42:C2, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C23xC4, C2xC4oD4, C4xD5, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C2xC20, C5xD4, C5xQ8, C22xD5, C22xD5, C22xC10, C22xC10, C22.19C24, C4xDic5, C10.D4, C4:Dic5, D10:C4, C23.D5, C2xC4xD5, C2xC4xD5, C22xDic5, C2xC5:D4, C22xC20, C22xC20, D4xC10, D4xC10, Q8xC10, C5xC4oD4, C23xD5, C23.21D10, C4xC5:D4, C23.18D10, C23:D10, C20:2D4, D10:3Q8, D5xC22xC4, C10xC4oD4, (C2xC20):15D4
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C4oD4, C24, D10, C22xD4, C2xC4oD4, C5:D4, C22xD5, C22.19C24, C2xC5:D4, C23xD5, D5xC4oD4, C22xC5:D4, (C2xC20):15D4

Smallest permutation representation of (C2xC20):15D4
On 80 points
Generators in S80
(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 79 27 47)(2 68 28 56)(3 77 29 45)(4 66 30 54)(5 75 31 43)(6 64 32 52)(7 73 33 41)(8 62 34 50)(9 71 35 59)(10 80 36 48)(11 69 37 57)(12 78 38 46)(13 67 39 55)(14 76 40 44)(15 65 21 53)(16 74 22 42)(17 63 23 51)(18 72 24 60)(19 61 25 49)(20 70 26 58)
(1 11)(2 20)(3 9)(4 18)(5 7)(6 16)(8 14)(10 12)(13 19)(15 17)(21 23)(22 32)(24 30)(25 39)(26 28)(27 37)(29 35)(31 33)(34 40)(36 38)(41 75)(42 64)(43 73)(44 62)(45 71)(46 80)(47 69)(48 78)(49 67)(50 76)(51 65)(52 74)(53 63)(54 72)(55 61)(56 70)(57 79)(58 68)(59 77)(60 66)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B10A···10F10G···10R20A···20H20I···20T
order12222222222244444444444444445510···1010···1020···2020···20
size1111224410101010111122441010101020202020222···24···42···24···4

68 irreducible representations

dim11111111122222224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D5C4oD4D10D10D10C5:D4D5xC4oD4
kernel(C2xC20):15D4C23.21D10C4xC5:D4C23.18D10C23:D10C20:2D4D10:3Q8D5xC22xC4C10xC4oD4C2xC20C2xC4oD4D10C22xC4C2xD4C2xQ8C2xC4C2
# reps114222211428662168

Matrix representation of (C2xC20):15D4 in GL4(F41) generated by

1000
0100
0010
004040
,
1100
5600
00320
00032
,
20300
32100
004039
0011
,
6700
363500
00400
0011
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,40,0,0,0,40],[1,5,0,0,1,6,0,0,0,0,32,0,0,0,0,32],[20,3,0,0,3,21,0,0,0,0,40,1,0,0,39,1],[6,36,0,0,7,35,0,0,0,0,40,1,0,0,0,1] >;

(C2xC20):15D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})\rtimes_{15}D_4
% in TeX

G:=Group("(C2xC20):15D4");
// GroupNames label

G:=SmallGroup(320,1500);
// by ID

G=gap.SmallGroup(320,1500);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,570,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^10,a*d=d*a,c*b*c^-1=d*b*d=b^9,d*c*d=c^-1>;
// generators/relations

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x
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Z
F
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