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G = C2×D10.3Q8  order 320 = 26·5

Direct product of C2 and D10.3Q8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D10.3Q8, D10.14C42, (C22×C4)⋊5F5, (C22×F5)⋊3C4, (C22×C20)⋊10C4, D10.90(C2×D4), D10.16(C2×Q8), D10.33(C4⋊C4), (C23×F5).3C2, C23.62(C2×F5), C22.22(C4×F5), C10⋊(C2.C42), C10.19(C2×C42), (C2×C10).22C42, D5⋊(C2.C42), C22.28(C4⋊F5), (C22×D5).21Q8, (C22×Dic5)⋊16C4, D10.36(C22×C4), (C22×D5).143D4, D10.33(C22⋊C4), C22.49(C22×F5), C22.47(C22⋊F5), (C22×F5).16C22, (C22×D5).274C23, (C23×D5).132C22, (C2×C4×D5)⋊16C4, C2.5(C2×C4⋊F5), C2.19(C2×C4×F5), (C2×F5)⋊3(C2×C4), D5.2(C2×C4⋊C4), (C2×C4)⋊10(C2×F5), C5⋊(C2×C2.C42), (C2×C20)⋊11(C2×C4), C10.22(C2×C4⋊C4), C2.3(C2×C22⋊F5), C10.9(C2×C22⋊C4), D5.2(C2×C22⋊C4), (C2×C10).28(C4⋊C4), (D5×C22×C4).23C2, (C2×Dic5)⋊34(C2×C4), (C2×C4×D5).365C22, (C22×C10).69(C2×C4), (C2×C10).69(C22×C4), (C22×D5).89(C2×C4), (C2×C10).51(C22⋊C4), SmallGroup(320,1100)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C2×D10.3Q8
Chief series
C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C23×F5 — C2×D10.3Q8
Lower central
C5 — C10 — C2×D10.3Q8
Upper central
C1 — C23 — C22×C4

Generators and relations for C2×D10.3Q8
 G = < a,b,c,d,e | a2=b10=c2=d4=1, e2=b4cd2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b3, cd=dc, ece-1=b2c, ede-1=b5d-1 >

Subgroups: 1242 in 330 conjugacy classes, 124 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C10, C22×C4, C22×C4, C24, Dic5, C20, F5, D10, D10, C2×C10, C2×C10, C2.C42, C23×C4, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×F5, C2×F5, C22×D5, C22×D5, C22×C10, C2×C2.C42, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×C20, C22×F5, C22×F5, C23×D5, D10.3Q8, D5×C22×C4, C23×F5, C2×D10.3Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, F5, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×F5, C2×C2.C42, C4×F5, C4⋊F5, C22⋊F5, C22×F5, D10.3Q8, C2×C4×F5, C2×C4⋊F5, C2×C22⋊F5, C2×D10.3Q8

Smallest permutation representation of C2×D10.3Q8
►On 80 points
Generators in S80
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 37)(12 38)(13 39)(14 40)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 61)(28 62)(29 63)(30 64)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 71)(50 72)

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

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

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

56 conjugacy classes

class 1 2A···2G2H···2O4A4B4C4D4E···4X 5 10A···10G20A···20H
order12···22···244444···4510···1020···20
size11···15···5222210···1044···44···4

56 irreducible representations

dim1111111122444444
type+++++-++++
imageC1C2C2C2C4C4C4C4D4Q8F5C2×F5C2×F5C4×F5C4⋊F5C22⋊F5
kernelC2×D10.3Q8D10.3Q8D5×C22×C4C23×F5C2×C4×D5C22×Dic5C22×C20C22×F5C22×D5C22×D5C22×C4C2×C4C23C22C22C22
# reps14124221662121444

Matrix representation of C2×D10.3Q8 ►in GL8(𝔽41)

400000000
040000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
004000000
000400000
000000400
000000040
00001111
000040000
,
400000000
040000000
004000000
000400000
000040000
00001111
000000040
000000400
,
01000000
400000000
00920000
001320000
00009000
00000900
00000090
00000009
,
10000000
040000000
00900000
001320000
00000010
00001000
00000001
00000100

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,40,0,1,0,0,0,0,0,0,40,1,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,40,0,0,0,0,0,1,40,0],[0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,2,32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;
 

C2×D10.3Q8 in GAP, Magma, Sage, TeX 

C_2\times D_{10}._3Q_8
 
% in TeX
 
G:=Group("C2xD10.3Q8");
 
// GroupNames label
 
G:=SmallGroup(320,1100);
 
// by ID
 
G=gap.SmallGroup(320,1100);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,100,6278,1595]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^4=1,e^2=b^4*c*d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^3,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=b^5*d^-1>;
 
// generators/relations
 

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