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G = D36.C4order 288 = 25·32

The non-split extension by D36 of C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D36.C4, C24.45D6, C8.12D18, Dic18.C4, M4(2)⋊5D9, C72.12C22, C36.39C23, C9⋊D4.C4, (C8×D9)⋊8C2, C92(C8○D4), C4.5(C4×D9), C8⋊D96C2, C12.12(C4×S3), C36.13(C2×C4), D18.2(C2×C4), (C2×C4).47D18, (C2×C12).52D6, C3.(D12.C4), C9⋊C8.12C22, C22.1(C4×D9), (C9×M4(2))⋊4C2, Dic9.4(C2×C4), C4.39(C22×D9), D365C2.3C2, (C2×C36).30C22, C18.16(C22×C4), (C4×D9).16C22, (C3×M4(2)).4S3, C12.200(C22×S3), (C2×C9⋊C8)⋊3C2, C6.55(S3×C2×C4), C2.17(C2×C4×D9), (C2×C6).9(C4×S3), (C2×C18).6(C2×C4), SmallGroup(288,117)

Series: Derived Chief Lower central Upper central

C1C18 — D36.C4
C1C3C9C18C36C4×D9D365C2 — D36.C4
C9C18 — D36.C4
C1C4M4(2)

Generators and relations for D36.C4
 G = < a,b,c | a36=b2=1, c4=a18, bab=a-1, cac-1=a19, cbc-1=a18b >

Subgroups: 332 in 93 conjugacy classes, 46 normal (28 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6, C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, C9, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C2×C8 [×3], M4(2), M4(2) [×2], C4○D4, D9 [×2], C18, C18, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, C8○D4, Dic9 [×2], C36 [×2], D18 [×2], C2×C18, S3×C8 [×2], C8⋊S3 [×2], C2×C3⋊C8, C3×M4(2), C4○D12, C9⋊C8 [×2], C72 [×2], Dic18, C4×D9 [×2], D36, C9⋊D4 [×2], C2×C36, D12.C4, C8×D9 [×2], C8⋊D9 [×2], C2×C9⋊C8, C9×M4(2), D365C2, D36.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, D9, C4×S3 [×2], C22×S3, C8○D4, D18 [×3], S3×C2×C4, C4×D9 [×2], C22×D9, D12.C4, C2×C4×D9, D36.C4

Smallest permutation representation of D36.C4
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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 15)(28 36)(29 35)(30 34)(31 33)(37 61)(38 60)(39 59)(40 58)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(62 72)(63 71)(64 70)(65 69)(66 68)(73 79)(74 78)(75 77)(80 108)(81 107)(82 106)(83 105)(84 104)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(109 135)(110 134)(111 133)(112 132)(113 131)(114 130)(115 129)(116 128)(117 127)(118 126)(119 125)(120 124)(121 123)(136 144)(137 143)(138 142)(139 141)
(1 118 81 45 19 136 99 63)(2 137 82 64 20 119 100 46)(3 120 83 47 21 138 101 65)(4 139 84 66 22 121 102 48)(5 122 85 49 23 140 103 67)(6 141 86 68 24 123 104 50)(7 124 87 51 25 142 105 69)(8 143 88 70 26 125 106 52)(9 126 89 53 27 144 107 71)(10 109 90 72 28 127 108 54)(11 128 91 55 29 110 73 37)(12 111 92 38 30 129 74 56)(13 130 93 57 31 112 75 39)(14 113 94 40 32 131 76 58)(15 132 95 59 33 114 77 41)(16 115 96 42 34 133 78 60)(17 134 97 61 35 116 79 43)(18 117 98 44 36 135 80 62)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E6A6B8A8B8C8D8E8F8G8H8I8J9A9B9C12A12B12C18A18B18C18D18E18F24A24B24C24D36A···36F36G36H36I72A···72L
order122223444446688888888889991212121818181818182424242436···3636363672···72
size1121818211218182422229999181822222422244444442···24444···4

60 irreducible representations

dim1111111112222222222244
type++++++++++++
imageC1C2C2C2C2C2C4C4C4S3D6D6D9C4×S3C4×S3C8○D4D18D18C4×D9C4×D9D12.C4D36.C4
kernelD36.C4C8×D9C8⋊D9C2×C9⋊C8C9×M4(2)D365C2Dic18D36C9⋊D4C3×M4(2)C24C2×C12M4(2)C12C2×C6C9C8C2×C4C4C22C3C1
# reps1221112241213224636626

Matrix representation of D36.C4 in GL4(𝔽73) generated by

703100
422800
00051
00100
,
72100
0100
00720
0001
,
27000
02700
0001
00270
G:=sub<GL(4,GF(73))| [70,42,0,0,31,28,0,0,0,0,0,10,0,0,51,0],[72,0,0,0,1,1,0,0,0,0,72,0,0,0,0,1],[27,0,0,0,0,27,0,0,0,0,0,27,0,0,1,0] >;

D36.C4 in GAP, Magma, Sage, TeX

D_{36}.C_4
% in TeX

G:=Group("D36.C4");
// GroupNames label

G:=SmallGroup(288,117);
// by ID

G=gap.SmallGroup(288,117);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,219,58,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^36=b^2=1,c^4=a^18,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^18*b>;
// generators/relations

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