metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36⋊4D4, C4⋊1D36, C42⋊6D9, C12.38D12, (C4×C36)⋊4C2, (C2×D36)⋊1C2, C9⋊1(C4⋊1D4), (C4×C12).6S3, C18.3(C2×D4), C2.5(C2×D36), C3.(C4⋊D12), (C2×C4).78D18, C6.32(C2×D12), (C2×C12).369D6, (C2×C36).87C22, (C2×C18).15C23, (C22×D9).1C22, C22.36(C22×D9), (C2×C6).172(C22×S3), SmallGroup(288,84)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36⋊4D4
G = < a,b,c | a36=b4=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 1020 in 162 conjugacy classes, 56 normal (10 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C22, C22 [×12], S3 [×4], C6 [×3], C2×C4 [×3], D4 [×12], C23 [×4], C9, C12 [×6], D6 [×12], C2×C6, C42, C2×D4 [×6], D9 [×4], C18 [×3], D12 [×12], C2×C12 [×3], C22×S3 [×4], C4⋊1D4, C36 [×6], D18 [×12], C2×C18, C4×C12, C2×D12 [×6], D36 [×12], C2×C36 [×3], C22×D9 [×4], C4⋊D12, C4×C36, C2×D36 [×6], C36⋊4D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D9, D12 [×6], C22×S3, C4⋊1D4, D18 [×3], C2×D12 [×3], D36 [×6], C22×D9, C4⋊D12, C2×D36 [×3], C36⋊4D4
►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)
(1 124 55 87)(2 125 56 88)(3 126 57 89)(4 127 58 90)(5 128 59 91)(6 129 60 92)(7 130 61 93)(8 131 62 94)(9 132 63 95)(10 133 64 96)(11 134 65 97)(12 135 66 98)(13 136 67 99)(14 137 68 100)(15 138 69 101)(16 139 70 102)(17 140 71 103)(18 141 72 104)(19 142 37 105)(20 143 38 106)(21 144 39 107)(22 109 40 108)(23 110 41 73)(24 111 42 74)(25 112 43 75)(26 113 44 76)(27 114 45 77)(28 115 46 78)(29 116 47 79)(30 117 48 80)(31 118 49 81)(32 119 50 82)(33 120 51 83)(34 121 52 84)(35 122 53 85)(36 123 54 86)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 72)(21 71)(22 70)(23 69)(24 68)(25 67)(26 66)(27 65)(28 64)(29 63)(30 62)(31 61)(32 60)(33 59)(34 58)(35 57)(36 56)(73 101)(74 100)(75 99)(76 98)(77 97)(78 96)(79 95)(80 94)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(102 108)(103 107)(104 106)(109 139)(110 138)(111 137)(112 136)(113 135)(114 134)(115 133)(116 132)(117 131)(118 130)(119 129)(120 128)(121 127)(122 126)(123 125)(140 144)(141 143)
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), (1,124,55,87)(2,125,56,88)(3,126,57,89)(4,127,58,90)(5,128,59,91)(6,129,60,92)(7,130,61,93)(8,131,62,94)(9,132,63,95)(10,133,64,96)(11,134,65,97)(12,135,66,98)(13,136,67,99)(14,137,68,100)(15,138,69,101)(16,139,70,102)(17,140,71,103)(18,141,72,104)(19,142,37,105)(20,143,38,106)(21,144,39,107)(22,109,40,108)(23,110,41,73)(24,111,42,74)(25,112,43,75)(26,113,44,76)(27,114,45,77)(28,115,46,78)(29,116,47,79)(30,117,48,80)(31,118,49,81)(32,119,50,82)(33,120,51,83)(34,121,52,84)(35,122,53,85)(36,123,54,86), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,60)(33,59)(34,58)(35,57)(36,56)(73,101)(74,100)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(102,108)(103,107)(104,106)(109,139)(110,138)(111,137)(112,136)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(140,144)(141,143)>;
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), (1,124,55,87)(2,125,56,88)(3,126,57,89)(4,127,58,90)(5,128,59,91)(6,129,60,92)(7,130,61,93)(8,131,62,94)(9,132,63,95)(10,133,64,96)(11,134,65,97)(12,135,66,98)(13,136,67,99)(14,137,68,100)(15,138,69,101)(16,139,70,102)(17,140,71,103)(18,141,72,104)(19,142,37,105)(20,143,38,106)(21,144,39,107)(22,109,40,108)(23,110,41,73)(24,111,42,74)(25,112,43,75)(26,113,44,76)(27,114,45,77)(28,115,46,78)(29,116,47,79)(30,117,48,80)(31,118,49,81)(32,119,50,82)(33,120,51,83)(34,121,52,84)(35,122,53,85)(36,123,54,86), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,60)(33,59)(34,58)(35,57)(36,56)(73,101)(74,100)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(102,108)(103,107)(104,106)(109,139)(110,138)(111,137)(112,136)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(140,144)(141,143) );
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)], [(1,124,55,87),(2,125,56,88),(3,126,57,89),(4,127,58,90),(5,128,59,91),(6,129,60,92),(7,130,61,93),(8,131,62,94),(9,132,63,95),(10,133,64,96),(11,134,65,97),(12,135,66,98),(13,136,67,99),(14,137,68,100),(15,138,69,101),(16,139,70,102),(17,140,71,103),(18,141,72,104),(19,142,37,105),(20,143,38,106),(21,144,39,107),(22,109,40,108),(23,110,41,73),(24,111,42,74),(25,112,43,75),(26,113,44,76),(27,114,45,77),(28,115,46,78),(29,116,47,79),(30,117,48,80),(31,118,49,81),(32,119,50,82),(33,120,51,83),(34,121,52,84),(35,122,53,85),(36,123,54,86)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,48),(9,47),(10,46),(11,45),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,72),(21,71),(22,70),(23,69),(24,68),(25,67),(26,66),(27,65),(28,64),(29,63),(30,62),(31,61),(32,60),(33,59),(34,58),(35,57),(36,56),(73,101),(74,100),(75,99),(76,98),(77,97),(78,96),(79,95),(80,94),(81,93),(82,92),(83,91),(84,90),(85,89),(86,88),(102,108),(103,107),(104,106),(109,139),(110,138),(111,137),(112,136),(113,135),(114,134),(115,133),(116,132),(117,131),(118,130),(119,129),(120,128),(121,127),(122,126),(123,125),(140,144),(141,143)])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12L | 18A | ··· | 18I | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 36 | 36 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D4 | D6 | D9 | D12 | D18 | D36 |
| kernel | C36⋊4D4 | C4×C36 | C2×D36 | C4×C12 | C36 | C2×C12 | C42 | C12 | C2×C4 | C4 |
| # reps | 1 | 1 | 6 | 1 | 6 | 3 | 3 | 12 | 9 | 36 |
Matrix representation of C36⋊4D4 ►in GL6(𝔽37)
| 27 | 32 | 0 | 0 | 0 | 0 |
| 5 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 17 |
| 0 | 0 | 0 | 0 | 20 | 6 |
| 5 | 10 | 0 | 0 | 0 | 0 |
| 27 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 3 | 0 | 0 |
| 0 | 0 | 11 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(37))| [27,5,0,0,0,0,32,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,20,0,0,0,0,17,6],[5,27,0,0,0,0,10,32,0,0,0,0,0,0,15,11,0,0,0,0,3,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,1,0,0,0,0,0,1,0,0,0,0,0,0,36,10,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C36⋊4D4 in GAP, Magma, Sage, TeX
C_{36}\rtimes_4D_4 % in TeX
G:=Group("C36:4D4"); // GroupNames label
G:=SmallGroup(288,84);
// by ID
G=gap.SmallGroup(288,84);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^4=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations