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G = C36.9D4order 288 = 25·32

9th non-split extension by C36 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C36.9D4, (C2×C4).Dic9, (C2×C36).1C4, (C2×C4).4D18, (C2×Q8).3D9, (C6×Q8).9S3, (C2×C12).47D6, (Q8×C18).1C2, C4.14(C9⋊D4), C12.7(C3⋊D4), C92(C4.10D4), (C2×C12).2Dic3, C4.Dic9.4C2, (C2×C36).25C22, C3.(C12.10D4), C22.4(C2×Dic9), C18.16(C22⋊C4), C2.6(C18.D4), C6.17(C6.D4), (C2×C18).31(C2×C4), (C2×C6).35(C2×Dic3), SmallGroup(288,42)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C36.9D4
C1C3C9C18C36C2×C36C4.Dic9 — C36.9D4
C9C18C2×C18 — C36.9D4
C1C2C2×C4C2×Q8

Generators and relations for C36.9D4
 G = < a,b,c | a36=1, b4=a18, c2=a9, bab-1=a-1, cac-1=a17, cbc-1=a9b3 >

Subgroups: 148 in 57 conjugacy classes, 30 normal (16 characteristic)
C1, C2, C2, C3, C4 [×2], C4 [×2], C22, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C9, C12 [×2], C12 [×2], C2×C6, M4(2) [×2], C2×Q8, C18, C18, C3⋊C8 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C4.10D4, C36 [×2], C36 [×2], C2×C18, C4.Dic3 [×2], C6×Q8, C9⋊C8 [×2], C2×C36, C2×C36 [×2], Q8×C9 [×2], C12.10D4, C4.Dic9 [×2], Q8×C18, C36.9D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, D9, C2×Dic3, C3⋊D4 [×2], C4.10D4, Dic9 [×2], D18, C6.D4, C2×Dic9, C9⋊D4 [×2], C12.10D4, C18.D4, C36.9D4

Smallest permutation representation of C36.9D4
On 144 points
Generators in S144
(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)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(1 118 28 127 19 136 10 109)(2 117 29 126 20 135 11 144)(3 116 30 125 21 134 12 143)(4 115 31 124 22 133 13 142)(5 114 32 123 23 132 14 141)(6 113 33 122 24 131 15 140)(7 112 34 121 25 130 16 139)(8 111 35 120 26 129 17 138)(9 110 36 119 27 128 18 137)(37 102 64 75 55 84 46 93)(38 101 65 74 56 83 47 92)(39 100 66 73 57 82 48 91)(40 99 67 108 58 81 49 90)(41 98 68 107 59 80 50 89)(42 97 69 106 60 79 51 88)(43 96 70 105 61 78 52 87)(44 95 71 104 62 77 53 86)(45 94 72 103 63 76 54 85)
(1 92 10 101 19 74 28 83)(2 73 11 82 20 91 29 100)(3 90 12 99 21 108 30 81)(4 107 13 80 22 89 31 98)(5 88 14 97 23 106 32 79)(6 105 15 78 24 87 33 96)(7 86 16 95 25 104 34 77)(8 103 17 76 26 85 35 94)(9 84 18 93 27 102 36 75)(37 110 46 119 55 128 64 137)(38 127 47 136 56 109 65 118)(39 144 48 117 57 126 66 135)(40 125 49 134 58 143 67 116)(41 142 50 115 59 124 68 133)(42 123 51 132 60 141 69 114)(43 140 52 113 61 122 70 131)(44 121 53 130 62 139 71 112)(45 138 54 111 63 120 72 129)

G:=sub<Sym(144)| (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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,118,28,127,19,136,10,109)(2,117,29,126,20,135,11,144)(3,116,30,125,21,134,12,143)(4,115,31,124,22,133,13,142)(5,114,32,123,23,132,14,141)(6,113,33,122,24,131,15,140)(7,112,34,121,25,130,16,139)(8,111,35,120,26,129,17,138)(9,110,36,119,27,128,18,137)(37,102,64,75,55,84,46,93)(38,101,65,74,56,83,47,92)(39,100,66,73,57,82,48,91)(40,99,67,108,58,81,49,90)(41,98,68,107,59,80,50,89)(42,97,69,106,60,79,51,88)(43,96,70,105,61,78,52,87)(44,95,71,104,62,77,53,86)(45,94,72,103,63,76,54,85), (1,92,10,101,19,74,28,83)(2,73,11,82,20,91,29,100)(3,90,12,99,21,108,30,81)(4,107,13,80,22,89,31,98)(5,88,14,97,23,106,32,79)(6,105,15,78,24,87,33,96)(7,86,16,95,25,104,34,77)(8,103,17,76,26,85,35,94)(9,84,18,93,27,102,36,75)(37,110,46,119,55,128,64,137)(38,127,47,136,56,109,65,118)(39,144,48,117,57,126,66,135)(40,125,49,134,58,143,67,116)(41,142,50,115,59,124,68,133)(42,123,51,132,60,141,69,114)(43,140,52,113,61,122,70,131)(44,121,53,130,62,139,71,112)(45,138,54,111,63,120,72,129)>;

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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,118,28,127,19,136,10,109)(2,117,29,126,20,135,11,144)(3,116,30,125,21,134,12,143)(4,115,31,124,22,133,13,142)(5,114,32,123,23,132,14,141)(6,113,33,122,24,131,15,140)(7,112,34,121,25,130,16,139)(8,111,35,120,26,129,17,138)(9,110,36,119,27,128,18,137)(37,102,64,75,55,84,46,93)(38,101,65,74,56,83,47,92)(39,100,66,73,57,82,48,91)(40,99,67,108,58,81,49,90)(41,98,68,107,59,80,50,89)(42,97,69,106,60,79,51,88)(43,96,70,105,61,78,52,87)(44,95,71,104,62,77,53,86)(45,94,72,103,63,76,54,85), (1,92,10,101,19,74,28,83)(2,73,11,82,20,91,29,100)(3,90,12,99,21,108,30,81)(4,107,13,80,22,89,31,98)(5,88,14,97,23,106,32,79)(6,105,15,78,24,87,33,96)(7,86,16,95,25,104,34,77)(8,103,17,76,26,85,35,94)(9,84,18,93,27,102,36,75)(37,110,46,119,55,128,64,137)(38,127,47,136,56,109,65,118)(39,144,48,117,57,126,66,135)(40,125,49,134,58,143,67,116)(41,142,50,115,59,124,68,133)(42,123,51,132,60,141,69,114)(43,140,52,113,61,122,70,131)(44,121,53,130,62,139,71,112)(45,138,54,111,63,120,72,129) );

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),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,118,28,127,19,136,10,109),(2,117,29,126,20,135,11,144),(3,116,30,125,21,134,12,143),(4,115,31,124,22,133,13,142),(5,114,32,123,23,132,14,141),(6,113,33,122,24,131,15,140),(7,112,34,121,25,130,16,139),(8,111,35,120,26,129,17,138),(9,110,36,119,27,128,18,137),(37,102,64,75,55,84,46,93),(38,101,65,74,56,83,47,92),(39,100,66,73,57,82,48,91),(40,99,67,108,58,81,49,90),(41,98,68,107,59,80,50,89),(42,97,69,106,60,79,51,88),(43,96,70,105,61,78,52,87),(44,95,71,104,62,77,53,86),(45,94,72,103,63,76,54,85)], [(1,92,10,101,19,74,28,83),(2,73,11,82,20,91,29,100),(3,90,12,99,21,108,30,81),(4,107,13,80,22,89,31,98),(5,88,14,97,23,106,32,79),(6,105,15,78,24,87,33,96),(7,86,16,95,25,104,34,77),(8,103,17,76,26,85,35,94),(9,84,18,93,27,102,36,75),(37,110,46,119,55,128,64,137),(38,127,47,136,56,109,65,118),(39,144,48,117,57,126,66,135),(40,125,49,134,58,143,67,116),(41,142,50,115,59,124,68,133),(42,123,51,132,60,141,69,114),(43,140,52,113,61,122,70,131),(44,121,53,130,62,139,71,112),(45,138,54,111,63,120,72,129)])

51 conjugacy classes

class 1 2A2B 3 4A4B4C4D6A6B6C8A8B8C8D9A9B9C12A···12F18A···18I36A···36R
order12234444666888899912···1218···1836···36
size11222244222363636362224···42···24···4

51 irreducible representations

dim1111222222222444
type+++++-++-+-
imageC1C2C2C4S3D4Dic3D6D9C3⋊D4Dic9D18C9⋊D4C4.10D4C12.10D4C36.9D4
kernelC36.9D4C4.Dic9Q8×C18C2×C36C6×Q8C36C2×C12C2×C12C2×Q8C12C2×C4C2×C4C4C9C3C1
# reps12141221346312126

Matrix representation of C36.9D4 in GL6(𝔽73)

41650000
0570000
000800
0065000
0000064
000090
,
51670000
44220000
000089
0000965
0064800
008900
,
51570000
44220000
000010
000001
000100
0072000

G:=sub<GL(6,GF(73))| [41,0,0,0,0,0,65,57,0,0,0,0,0,0,0,65,0,0,0,0,8,0,0,0,0,0,0,0,0,9,0,0,0,0,64,0],[51,44,0,0,0,0,67,22,0,0,0,0,0,0,0,0,64,8,0,0,0,0,8,9,0,0,8,9,0,0,0,0,9,65,0,0],[51,44,0,0,0,0,57,22,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C36.9D4 in GAP, Magma, Sage, TeX

C_{36}._9D_4
% in TeX

G:=Group("C36.9D4");
// GroupNames label

G:=SmallGroup(288,42);
// by ID

G=gap.SmallGroup(288,42);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,120,219,100,675,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^36=1,b^4=a^18,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^17,c*b*c^-1=a^9*b^3>;
// generators/relations

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