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G = C20.32D6order 240 = 24·3·5

11st non-split extension by C20 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.32D6, C155M4(2), C12.32D10, C60.32C22, D10.1Dic3, Dic5.1Dic3, C3⋊C84D5, C33(C8⋊D5), (C6×D5).2C4, (C4×D5).2S3, C153C811C2, C4.25(S3×D5), C6.11(C4×D5), C30.25(C2×C4), (D5×C12).3C2, C53(C4.Dic3), C2.3(D5×Dic3), (C3×Dic5).2C4, C10.9(C2×Dic3), (C5×C3⋊C8)⋊6C2, SmallGroup(240,10)

Series: Derived Chief Lower central Upper central

C1C30 — C20.32D6
C1C5C15C30C60D5×C12 — C20.32D6
C15C30 — C20.32D6
C1C4

Generators and relations for C20.32D6
 G = < a,b,c | a20=b6=1, c2=a15, bab-1=cac-1=a9, cbc-1=a10b-1 >

10C2
5C22
5C4
10C6
2D5
3C8
5C2×C4
15C8
5C12
5C2×C6
2C3×D5
15M4(2)
5C3⋊C8
5C2×C12
3C40
3C52C8
5C4.Dic3
3C8⋊D5

Smallest permutation representation of C20.32D6
On 120 points
Generators in S120
(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)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 103 67)(2 112 68 10 104 76)(3 101 69 19 105 65)(4 110 70 8 106 74)(5 119 71 17 107 63)(6 108 72)(7 117 73 15 109 61)(9 115 75 13 111 79)(11 113 77)(12 102 78 20 114 66)(14 120 80 18 116 64)(16 118 62)(21 83 54 35 89 48)(22 92 55 24 90 57)(23 81 56 33 91 46)(25 99 58 31 93 44)(26 88 59 40 94 53)(27 97 60 29 95 42)(28 86 41 38 96 51)(30 84 43 36 98 49)(32 82 45 34 100 47)(37 87 50 39 85 52)
(1 33 16 28 11 23 6 38)(2 22 17 37 12 32 7 27)(3 31 18 26 13 21 8 36)(4 40 19 35 14 30 9 25)(5 29 20 24 15 39 10 34)(41 113 56 108 51 103 46 118)(42 102 57 117 52 112 47 107)(43 111 58 106 53 101 48 116)(44 120 59 115 54 110 49 105)(45 109 60 104 55 119 50 114)(61 87 76 82 71 97 66 92)(62 96 77 91 72 86 67 81)(63 85 78 100 73 95 68 90)(64 94 79 89 74 84 69 99)(65 83 80 98 75 93 70 88)

G:=sub<Sym(120)| (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,103,67)(2,112,68,10,104,76)(3,101,69,19,105,65)(4,110,70,8,106,74)(5,119,71,17,107,63)(6,108,72)(7,117,73,15,109,61)(9,115,75,13,111,79)(11,113,77)(12,102,78,20,114,66)(14,120,80,18,116,64)(16,118,62)(21,83,54,35,89,48)(22,92,55,24,90,57)(23,81,56,33,91,46)(25,99,58,31,93,44)(26,88,59,40,94,53)(27,97,60,29,95,42)(28,86,41,38,96,51)(30,84,43,36,98,49)(32,82,45,34,100,47)(37,87,50,39,85,52), (1,33,16,28,11,23,6,38)(2,22,17,37,12,32,7,27)(3,31,18,26,13,21,8,36)(4,40,19,35,14,30,9,25)(5,29,20,24,15,39,10,34)(41,113,56,108,51,103,46,118)(42,102,57,117,52,112,47,107)(43,111,58,106,53,101,48,116)(44,120,59,115,54,110,49,105)(45,109,60,104,55,119,50,114)(61,87,76,82,71,97,66,92)(62,96,77,91,72,86,67,81)(63,85,78,100,73,95,68,90)(64,94,79,89,74,84,69,99)(65,83,80,98,75,93,70,88)>;

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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,103,67)(2,112,68,10,104,76)(3,101,69,19,105,65)(4,110,70,8,106,74)(5,119,71,17,107,63)(6,108,72)(7,117,73,15,109,61)(9,115,75,13,111,79)(11,113,77)(12,102,78,20,114,66)(14,120,80,18,116,64)(16,118,62)(21,83,54,35,89,48)(22,92,55,24,90,57)(23,81,56,33,91,46)(25,99,58,31,93,44)(26,88,59,40,94,53)(27,97,60,29,95,42)(28,86,41,38,96,51)(30,84,43,36,98,49)(32,82,45,34,100,47)(37,87,50,39,85,52), (1,33,16,28,11,23,6,38)(2,22,17,37,12,32,7,27)(3,31,18,26,13,21,8,36)(4,40,19,35,14,30,9,25)(5,29,20,24,15,39,10,34)(41,113,56,108,51,103,46,118)(42,102,57,117,52,112,47,107)(43,111,58,106,53,101,48,116)(44,120,59,115,54,110,49,105)(45,109,60,104,55,119,50,114)(61,87,76,82,71,97,66,92)(62,96,77,91,72,86,67,81)(63,85,78,100,73,95,68,90)(64,94,79,89,74,84,69,99)(65,83,80,98,75,93,70,88) );

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),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,103,67),(2,112,68,10,104,76),(3,101,69,19,105,65),(4,110,70,8,106,74),(5,119,71,17,107,63),(6,108,72),(7,117,73,15,109,61),(9,115,75,13,111,79),(11,113,77),(12,102,78,20,114,66),(14,120,80,18,116,64),(16,118,62),(21,83,54,35,89,48),(22,92,55,24,90,57),(23,81,56,33,91,46),(25,99,58,31,93,44),(26,88,59,40,94,53),(27,97,60,29,95,42),(28,86,41,38,96,51),(30,84,43,36,98,49),(32,82,45,34,100,47),(37,87,50,39,85,52)], [(1,33,16,28,11,23,6,38),(2,22,17,37,12,32,7,27),(3,31,18,26,13,21,8,36),(4,40,19,35,14,30,9,25),(5,29,20,24,15,39,10,34),(41,113,56,108,51,103,46,118),(42,102,57,117,52,112,47,107),(43,111,58,106,53,101,48,116),(44,120,59,115,54,110,49,105),(45,109,60,104,55,119,50,114),(61,87,76,82,71,97,66,92),(62,96,77,91,72,86,67,81),(63,85,78,100,73,95,68,90),(64,94,79,89,74,84,69,99),(65,83,80,98,75,93,70,88)])

C20.32D6 is a maximal subgroup of
S3×C8⋊D5  C40⋊D6  C40.54D6  C40.34D6  D5×C4.Dic3  D20.3Dic3  D20.2Dic3  Dic103D6  C60.8C23  D1210D10  D20.9D6  D20⋊D6  D20.13D6  D12.27D10  C60.39C23
C20.32D6 is a maximal quotient of
C30.21C42  C60.93D4  C60.13Q8

42 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B6A6B6C8A8B8C8D10A10B12A12B12C12D15A15B20A20B20C20D30A30B40A···40H60A60B60C60D
order1223444556668888101012121212151520202020303040···4060606060
size111021110222101066303022221010442222446···64444

42 irreducible representations

dim1111112222222222444
type++++++-+-++-
imageC1C2C2C2C4C4S3D5Dic3D6Dic3M4(2)D10C4×D5C4.Dic3C8⋊D5S3×D5D5×Dic3C20.32D6
kernelC20.32D6C5×C3⋊C8C153C8D5×C12C3×Dic5C6×D5C4×D5C3⋊C8Dic5C20D10C15C12C6C5C3C4C2C1
# reps1111221211122448224

Matrix representation of C20.32D6 in GL4(𝔽241) generated by

5124000
1000
00640
00064
,
1000
5124000
00150
00016
,
1000
5124000
000225
0040
G:=sub<GL(4,GF(241))| [51,1,0,0,240,0,0,0,0,0,64,0,0,0,0,64],[1,51,0,0,0,240,0,0,0,0,15,0,0,0,0,16],[1,51,0,0,0,240,0,0,0,0,0,4,0,0,225,0] >;

C20.32D6 in GAP, Magma, Sage, TeX

C_{20}._{32}D_6
% in TeX

G:=Group("C20.32D6");
// GroupNames label

G:=SmallGroup(240,10);
// by ID

G=gap.SmallGroup(240,10);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,31,50,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^20=b^6=1,c^2=a^15,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^10*b^-1>;
// generators/relations

Export

Subgroup lattice of C20.32D6 in TeX

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