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G = C20.12C42order 320 = 26·5

5th non-split extension by C20 of C42 acting via C42/C4=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.12C42, D10.13C42, Dic5.11C42, (C2×C8)⋊9F5, (C8×F5)⋊6C2, C4.F55C4, C4⋊F5.2C4, (C2×C40)⋊10C4, C4.5(C4×F5), (C8×D5)⋊13C4, C8⋊F58C2, C8.34(C2×F5), C40.41(C2×C4), C22.F57C4, C22⋊F5.4C4, C22.4(C4×F5), D5.1(C8○D4), C4.50(C22×F5), C53(C82M4(2)), (C2×C10).18C42, C20.90(C22×C4), C10.13(C2×C42), D5⋊C8.18C22, (C8×D5).66C22, (C4×D5).87C23, (C4×F5).17C22, D5⋊M4(2).12C2, D10.33(C22×C4), Dic5.32(C22×C4), D10.C23.12C2, C5⋊C8.1(C2×C4), C2.14(C2×C4×F5), (D5×C2×C8).34C2, (C2×C52C8)⋊19C4, (C2×F5).3(C2×C4), C52C8.54(C2×C4), (C4×D5).66(C2×C4), (C2×C4).135(C2×F5), (C2×C20).146(C2×C4), (C2×C4×D5).402C22, (C22×D5).88(C2×C4), (C2×Dic5).126(C2×C4), SmallGroup(320,1056)

Series: Derived Chief Lower central Upper central

C1C10 — C20.12C42
C1C5C10D10C4×D5C4×F5D10.C23 — C20.12C42
C5C10 — C20.12C42
C1C8C2×C8

Generators and relations for C20.12C42
 G = < a,b,c | a20=b4=1, c4=a10, bab-1=a3, ac=ca, bc=cb >

Subgroups: 394 in 130 conjugacy classes, 66 normal (40 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×9], C23, D5 [×2], D5, C10, C10, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8 [×7], M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×2], C2×C10, C4×C8 [×2], C8⋊C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C52C8 [×2], C40 [×2], C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C22×D5, C82M4(2), C8×D5 [×4], C2×C52C8, C2×C40, D5⋊C8 [×2], C4.F5 [×2], C4×F5 [×2], C4⋊F5 [×2], C22.F5 [×2], C22⋊F5 [×2], C2×C4×D5, C8×F5 [×2], C8⋊F5 [×2], D5×C2×C8, D5⋊M4(2), D10.C23, C20.12C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], C22×C4 [×3], F5, C2×C42, C8○D4 [×2], C2×F5 [×3], C82M4(2), C4×F5 [×2], C22×F5, C2×C4×F5, C20.12C42

Smallest permutation representation of C20.12C42
On 80 points
Generators in S80
(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 63 11 73)(2 70 20 76)(3 77 9 79)(4 64 18 62)(5 71 7 65)(6 78 16 68)(8 72 14 74)(10 66 12 80)(13 67 19 69)(15 61 17 75)(21 55 23 49)(22 42 32 52)(24 56 30 58)(25 43 39 41)(26 50 28 44)(27 57 37 47)(29 51 35 53)(31 45 33 59)(34 46 40 48)(36 60 38 54)
(1 27 73 47 11 37 63 57)(2 28 74 48 12 38 64 58)(3 29 75 49 13 39 65 59)(4 30 76 50 14 40 66 60)(5 31 77 51 15 21 67 41)(6 32 78 52 16 22 68 42)(7 33 79 53 17 23 69 43)(8 34 80 54 18 24 70 44)(9 35 61 55 19 25 71 45)(10 36 62 56 20 26 72 46)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F···4N 5 8A8B8C8D8E8F8G8H8I8J8K···8T10A10B10C20A20B20C20D40A···40H
order122222444444···4588888888888···81010102020202040···40
size11255101125510···104111122555510···1044444444···4

56 irreducible representations

dim11111111111112444444
type+++++++++
imageC1C2C2C2C2C2C4C4C4C4C4C4C4C8○D4F5C2×F5C2×F5C4×F5C4×F5C20.12C42
kernelC20.12C42C8×F5C8⋊F5D5×C2×C8D5⋊M4(2)D10.C23C8×D5C2×C52C8C2×C40C4.F5C4⋊F5C22.F5C22⋊F5D5C2×C8C8C2×C4C4C22C1
# reps12211142244448121228

Matrix representation of C20.12C42 in GL4(𝔽41) generated by

347734
701414
2734270
0273427
,
9000
0009
0900
32323232
,
27000
02700
00270
00027
G:=sub<GL(4,GF(41))| [34,7,27,0,7,0,34,27,7,14,27,34,34,14,0,27],[9,0,0,32,0,0,9,32,0,0,0,32,0,9,0,32],[27,0,0,0,0,27,0,0,0,0,27,0,0,0,0,27] >;

C20.12C42 in GAP, Magma, Sage, TeX

C_{20}._{12}C_4^2
% in TeX

G:=Group("C20.12C4^2");
// GroupNames label

G:=SmallGroup(320,1056);
// by ID

G=gap.SmallGroup(320,1056);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c|a^20=b^4=1,c^4=a^10,b*a*b^-1=a^3,a*c=c*a,b*c=c*b>;
// generators/relations

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