metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36.20D4, Q8.12D18, C36.16C23, D36.10C22, Dic18.10C22, (C2×Q8)⋊4D9, (Q8×C18)⋊2C2, C9⋊Q16⋊5C2, C9⋊C8.3C22, Q8⋊2D9⋊5C2, (C2×C12).64D6, (C2×C4).17D18, C18.55(C2×D4), (C2×C18).43D4, C9⋊4(C8.C22), C4.Dic9⋊7C2, (C6×Q8).13S3, (C3×Q8).55D6, C4.17(C9⋊D4), C4.16(C22×D9), D36⋊5C2.5C2, (Q8×C9).7C22, C12.17(C3⋊D4), (C2×C36).42C22, C12.55(C22×S3), C3.(Q8.11D6), C22.11(C9⋊D4), C2.19(C2×C9⋊D4), C6.103(C2×C3⋊D4), (C2×C6).82(C3⋊D4), SmallGroup(288,153)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.C23
G = < a,b,c,d | a36=b2=1, c2=d2=a18, bab=a-1, ac=ca, dad-1=a19, bc=cb, dbd-1=a9b, dcd-1=a18c >
Subgroups: 356 in 90 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C9, Dic3, C12, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, D9, C18, C18, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8, C3×Q8, C8.C22, Dic9, C36, C36, D18, C2×C18, C4.Dic3, Q8⋊2S3, C3⋊Q16, C4○D12, C6×Q8, C9⋊C8, Dic18, C4×D9, D36, C9⋊D4, C2×C36, C2×C36, Q8×C9, Q8×C9, Q8.11D6, C4.Dic9, C9⋊Q16, Q8⋊2D9, D36⋊5C2, Q8×C18, C36.C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, C8.C22, D18, C2×C3⋊D4, C9⋊D4, C22×D9, Q8.11D6, C2×C9⋊D4, C36.C23
►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)
(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(37 44)(38 43)(39 42)(40 41)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(106 108)(109 118)(110 117)(111 116)(112 115)(113 114)(119 144)(120 143)(121 142)(122 141)(123 140)(124 139)(125 138)(126 137)(127 136)(128 135)(129 134)(130 133)(131 132)
(1 107 19 89)(2 108 20 90)(3 73 21 91)(4 74 22 92)(5 75 23 93)(6 76 24 94)(7 77 25 95)(8 78 26 96)(9 79 27 97)(10 80 28 98)(11 81 29 99)(12 82 30 100)(13 83 31 101)(14 84 32 102)(15 85 33 103)(16 86 34 104)(17 87 35 105)(18 88 36 106)(37 110 55 128)(38 111 56 129)(39 112 57 130)(40 113 58 131)(41 114 59 132)(42 115 60 133)(43 116 61 134)(44 117 62 135)(45 118 63 136)(46 119 64 137)(47 120 65 138)(48 121 66 139)(49 122 67 140)(50 123 68 141)(51 124 69 142)(52 125 70 143)(53 126 71 144)(54 127 72 109)
(1 72 19 54)(2 55 20 37)(3 38 21 56)(4 57 22 39)(5 40 23 58)(6 59 24 41)(7 42 25 60)(8 61 26 43)(9 44 27 62)(10 63 28 45)(11 46 29 64)(12 65 30 47)(13 48 31 66)(14 67 32 49)(15 50 33 68)(16 69 34 51)(17 52 35 70)(18 71 36 53)(73 129 91 111)(74 112 92 130)(75 131 93 113)(76 114 94 132)(77 133 95 115)(78 116 96 134)(79 135 97 117)(80 118 98 136)(81 137 99 119)(82 120 100 138)(83 139 101 121)(84 122 102 140)(85 141 103 123)(86 124 104 142)(87 143 105 125)(88 126 106 144)(89 109 107 127)(90 128 108 110)
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), (2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(37,44)(38,43)(39,42)(40,41)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(106,108)(109,118)(110,117)(111,116)(112,115)(113,114)(119,144)(120,143)(121,142)(122,141)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132), (1,107,19,89)(2,108,20,90)(3,73,21,91)(4,74,22,92)(5,75,23,93)(6,76,24,94)(7,77,25,95)(8,78,26,96)(9,79,27,97)(10,80,28,98)(11,81,29,99)(12,82,30,100)(13,83,31,101)(14,84,32,102)(15,85,33,103)(16,86,34,104)(17,87,35,105)(18,88,36,106)(37,110,55,128)(38,111,56,129)(39,112,57,130)(40,113,58,131)(41,114,59,132)(42,115,60,133)(43,116,61,134)(44,117,62,135)(45,118,63,136)(46,119,64,137)(47,120,65,138)(48,121,66,139)(49,122,67,140)(50,123,68,141)(51,124,69,142)(52,125,70,143)(53,126,71,144)(54,127,72,109), (1,72,19,54)(2,55,20,37)(3,38,21,56)(4,57,22,39)(5,40,23,58)(6,59,24,41)(7,42,25,60)(8,61,26,43)(9,44,27,62)(10,63,28,45)(11,46,29,64)(12,65,30,47)(13,48,31,66)(14,67,32,49)(15,50,33,68)(16,69,34,51)(17,52,35,70)(18,71,36,53)(73,129,91,111)(74,112,92,130)(75,131,93,113)(76,114,94,132)(77,133,95,115)(78,116,96,134)(79,135,97,117)(80,118,98,136)(81,137,99,119)(82,120,100,138)(83,139,101,121)(84,122,102,140)(85,141,103,123)(86,124,104,142)(87,143,105,125)(88,126,106,144)(89,109,107,127)(90,128,108,110)>;
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), (2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(37,44)(38,43)(39,42)(40,41)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(106,108)(109,118)(110,117)(111,116)(112,115)(113,114)(119,144)(120,143)(121,142)(122,141)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132), (1,107,19,89)(2,108,20,90)(3,73,21,91)(4,74,22,92)(5,75,23,93)(6,76,24,94)(7,77,25,95)(8,78,26,96)(9,79,27,97)(10,80,28,98)(11,81,29,99)(12,82,30,100)(13,83,31,101)(14,84,32,102)(15,85,33,103)(16,86,34,104)(17,87,35,105)(18,88,36,106)(37,110,55,128)(38,111,56,129)(39,112,57,130)(40,113,58,131)(41,114,59,132)(42,115,60,133)(43,116,61,134)(44,117,62,135)(45,118,63,136)(46,119,64,137)(47,120,65,138)(48,121,66,139)(49,122,67,140)(50,123,68,141)(51,124,69,142)(52,125,70,143)(53,126,71,144)(54,127,72,109), (1,72,19,54)(2,55,20,37)(3,38,21,56)(4,57,22,39)(5,40,23,58)(6,59,24,41)(7,42,25,60)(8,61,26,43)(9,44,27,62)(10,63,28,45)(11,46,29,64)(12,65,30,47)(13,48,31,66)(14,67,32,49)(15,50,33,68)(16,69,34,51)(17,52,35,70)(18,71,36,53)(73,129,91,111)(74,112,92,130)(75,131,93,113)(76,114,94,132)(77,133,95,115)(78,116,96,134)(79,135,97,117)(80,118,98,136)(81,137,99,119)(82,120,100,138)(83,139,101,121)(84,122,102,140)(85,141,103,123)(86,124,104,142)(87,143,105,125)(88,126,106,144)(89,109,107,127)(90,128,108,110) );
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)], [(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,20),(37,44),(38,43),(39,42),(40,41),(45,72),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,60),(58,59),(73,105),(74,104),(75,103),(76,102),(77,101),(78,100),(79,99),(80,98),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(106,108),(109,118),(110,117),(111,116),(112,115),(113,114),(119,144),(120,143),(121,142),(122,141),(123,140),(124,139),(125,138),(126,137),(127,136),(128,135),(129,134),(130,133),(131,132)], [(1,107,19,89),(2,108,20,90),(3,73,21,91),(4,74,22,92),(5,75,23,93),(6,76,24,94),(7,77,25,95),(8,78,26,96),(9,79,27,97),(10,80,28,98),(11,81,29,99),(12,82,30,100),(13,83,31,101),(14,84,32,102),(15,85,33,103),(16,86,34,104),(17,87,35,105),(18,88,36,106),(37,110,55,128),(38,111,56,129),(39,112,57,130),(40,113,58,131),(41,114,59,132),(42,115,60,133),(43,116,61,134),(44,117,62,135),(45,118,63,136),(46,119,64,137),(47,120,65,138),(48,121,66,139),(49,122,67,140),(50,123,68,141),(51,124,69,142),(52,125,70,143),(53,126,71,144),(54,127,72,109)], [(1,72,19,54),(2,55,20,37),(3,38,21,56),(4,57,22,39),(5,40,23,58),(6,59,24,41),(7,42,25,60),(8,61,26,43),(9,44,27,62),(10,63,28,45),(11,46,29,64),(12,65,30,47),(13,48,31,66),(14,67,32,49),(15,50,33,68),(16,69,34,51),(17,52,35,70),(18,71,36,53),(73,129,91,111),(74,112,92,130),(75,131,93,113),(76,114,94,132),(77,133,95,115),(78,116,96,134),(79,135,97,117),(80,118,98,136),(81,137,99,119),(82,120,100,138),(83,139,101,121),(84,122,102,140),(85,141,103,123),(86,124,104,142),(87,143,105,125),(88,126,106,144),(89,109,107,127),(90,128,108,110)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 36 | 2 | 2 | 2 | 4 | 4 | 36 | 2 | 2 | 2 | 36 | 36 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D9 | C3⋊D4 | C3⋊D4 | D18 | D18 | C9⋊D4 | C9⋊D4 | C8.C22 | Q8.11D6 | C36.C23 |
| kernel | C36.C23 | C4.Dic9 | C9⋊Q16 | Q8⋊2D9 | D36⋊5C2 | Q8×C18 | C6×Q8 | C36 | C2×C18 | C2×C12 | C3×Q8 | C2×Q8 | C12 | C2×C6 | C2×C4 | Q8 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 1 | 2 | 6 |
Matrix representation of C36.C23 ►in GL4(𝔽73) generated by
| 28 | 70 | 47 | 41 |
| 3 | 31 | 58 | 47 |
| 48 | 34 | 42 | 3 |
| 14 | 48 | 70 | 45 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 71 | 72 | 0 | 72 |
| 72 | 1 | 72 | 0 |
| 30 | 60 | 13 | 60 |
| 13 | 43 | 0 | 13 |
| 56 | 56 | 30 | 13 |
| 0 | 56 | 60 | 43 |
| 62 | 0 | 70 | 20 |
| 0 | 62 | 56 | 70 |
| 41 | 30 | 11 | 0 |
| 11 | 41 | 0 | 11 |
G:=sub<GL(4,GF(73))| [28,3,48,14,70,31,34,48,47,58,42,70,41,47,3,45],[1,72,71,72,0,72,72,1,0,0,0,72,0,0,72,0],[30,13,56,0,60,43,56,56,13,0,30,60,60,13,13,43],[62,0,41,11,0,62,30,41,70,56,11,0,20,70,0,11] >;
C36.C23 in GAP, Magma, Sage, TeX
C_{36}.C_2^3 % in TeX
G:=Group("C36.C2^3"); // GroupNames label
G:=SmallGroup(288,153);
// by ID
G=gap.SmallGroup(288,153);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,100,675,185,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^36=b^2=1,c^2=d^2=a^18,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^19,b*c=c*b,d*b*d^-1=a^9*b,d*c*d^-1=a^18*c>;
// generators/relations