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## G = C4.12D20order 160 = 25·5

### 4th non-split extension by C4 of D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C4.12D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×Dic10 — C4.12D20
 Lower central C5 — C10 — C2×C10 — C4.12D20
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C4.12D20
G = < a,b,c | a20=1, b4=c2=a10, bab-1=cac-1=a-1, cbc-1=a5b3 >

Smallest permutation representation of C4.12D20
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 71 6 66 11 61 16 76)(2 70 7 65 12 80 17 75)(3 69 8 64 13 79 18 74)(4 68 9 63 14 78 19 73)(5 67 10 62 15 77 20 72)(21 43 36 48 31 53 26 58)(22 42 37 47 32 52 27 57)(23 41 38 46 33 51 28 56)(24 60 39 45 34 50 29 55)(25 59 40 44 35 49 30 54)
(1 43 11 53)(2 42 12 52)(3 41 13 51)(4 60 14 50)(5 59 15 49)(6 58 16 48)(7 57 17 47)(8 56 18 46)(9 55 19 45)(10 54 20 44)(21 66 31 76)(22 65 32 75)(23 64 33 74)(24 63 34 73)(25 62 35 72)(26 61 36 71)(27 80 37 70)(28 79 38 69)(29 78 39 68)(30 77 40 67)

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

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

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

C4.12D20 is a maximal subgroup of
M4(2).19D10  D20.2D4  D5×C4.10D4  D20.7D4  D4.9D20  D4.10D20  C8.20D20  C8.24D20  M4(2).31D10  D4.3D20  D4.5D20  M4(2).13D10  D20.38D4  M4(2).16D10  D20.40D4  C12.6D20  C60.31D4  C4.D60
C4.12D20 is a maximal quotient of
C42.2D10  (C2×Dic5)⋊C8  C20.47D8  C20.2D8  M4(2)⋊Dic5  C12.6D20  C60.31D4  C4.D60

31 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 10A 10B 10C 10D 20A 20B 20C 20D 20E 20F 40A ··· 40H order 1 2 2 4 4 4 4 5 5 8 8 8 8 10 10 10 10 20 20 20 20 20 20 40 ··· 40 size 1 1 2 2 2 20 20 2 2 4 4 20 20 2 2 4 4 2 2 2 2 4 4 4 ··· 4

31 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + - - image C1 C2 C2 C2 C4 D4 D5 D10 D20 C5⋊D4 C4×D5 C4.10D4 C4.12D20 kernel C4.12D20 C4.Dic5 C5×M4(2) C2×Dic10 C2×Dic5 C20 M4(2) C2×C4 C4 C4 C22 C5 C1 # reps 1 1 1 1 4 2 2 2 4 4 4 1 4

Matrix representation of C4.12D20 in GL4(𝔽41) generated by

 25 39 0 0 2 13 0 0 0 0 25 39 0 0 2 13
,
 0 0 14 37 0 0 39 27 2 15 0 0 27 39 0 0
,
 14 37 0 0 39 27 0 0 0 0 14 37 0 0 39 27
G:=sub<GL(4,GF(41))| [25,2,0,0,39,13,0,0,0,0,25,2,0,0,39,13],[0,0,2,27,0,0,15,39,14,39,0,0,37,27,0,0],[14,39,0,0,37,27,0,0,0,0,14,39,0,0,37,27] >;

C4.12D20 in GAP, Magma, Sage, TeX

C_4._{12}D_{20}
% in TeX

G:=Group("C4.12D20");
// GroupNames label

G:=SmallGroup(160,31);
// by ID

G=gap.SmallGroup(160,31);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,121,31,362,86,297,4613]);
// Polycyclic

G:=Group<a,b,c|a^20=1,b^4=c^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^5*b^3>;
// generators/relations

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