direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C36, C42⋊7C18, C4⋊C4⋊7C18, C3.(D4×C12), C4⋊1(C2×C36), C36⋊6(C2×C4), (D4×C12).C3, (C4×C36)⋊11C2, C2.3(D4×C18), C6.66(C6×D4), C22⋊C4⋊6C18, (C4×C12).16C6, C22⋊2(C2×C36), (C22×C4)⋊4C18, (C22×C36)⋊4C2, (C3×D4).4C12, (C2×D4).7C18, (C6×D4).26C6, C18.66(C2×D4), C12.86(C3×D4), C12.33(C2×C12), (D4×C18).14C2, C2.4(C22×C36), C18.39(C4○D4), C6.32(C22×C12), (C2×C18).73C23, (C22×C12).13C6, C23.12(C2×C18), C18.32(C22×C4), (C2×C36).120C22, C22.6(C22×C18), (C22×C18).26C22, (C9×C4⋊C4)⋊16C2, (C2×C18)⋊4(C2×C4), C2.2(C9×C4○D4), (C3×C4⋊C4).25C6, (C2×C6).9(C2×C12), C6.39(C3×C4○D4), (C9×C22⋊C4)⋊14C2, (C2×C4).18(C2×C18), (C2×C12).81(C2×C6), (C3×C22⋊C4).15C6, (C2×C6).78(C22×C6), (C22×C6).44(C2×C6), SmallGroup(288,168)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C36
G = < a,b,c | a36=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 186 in 141 conjugacy classes, 96 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C9, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C18, C18, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C4×D4, C36, C36, C2×C18, C2×C18, C2×C18, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C2×C36, C2×C36, C2×C36, D4×C9, C22×C18, D4×C12, C4×C36, C9×C22⋊C4, C9×C4⋊C4, C22×C36, D4×C18, D4×C36
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C9, C12, C2×C6, C22×C4, C2×D4, C4○D4, C18, C2×C12, C3×D4, C22×C6, C4×D4, C36, C2×C18, C22×C12, C6×D4, C3×C4○D4, C2×C36, D4×C9, C22×C18, D4×C12, C22×C36, D4×C18, C9×C4○D4, D4×C36
►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)
(1 127 80 69)(2 128 81 70)(3 129 82 71)(4 130 83 72)(5 131 84 37)(6 132 85 38)(7 133 86 39)(8 134 87 40)(9 135 88 41)(10 136 89 42)(11 137 90 43)(12 138 91 44)(13 139 92 45)(14 140 93 46)(15 141 94 47)(16 142 95 48)(17 143 96 49)(18 144 97 50)(19 109 98 51)(20 110 99 52)(21 111 100 53)(22 112 101 54)(23 113 102 55)(24 114 103 56)(25 115 104 57)(26 116 105 58)(27 117 106 59)(28 118 107 60)(29 119 108 61)(30 120 73 62)(31 121 74 63)(32 122 75 64)(33 123 76 65)(34 124 77 66)(35 125 78 67)(36 126 79 68)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(73 138)(74 139)(75 140)(76 141)(77 142)(78 143)(79 144)(80 109)(81 110)(82 111)(83 112)(84 113)(85 114)(86 115)(87 116)(88 117)(89 118)(90 119)(91 120)(92 121)(93 122)(94 123)(95 124)(96 125)(97 126)(98 127)(99 128)(100 129)(101 130)(102 131)(103 132)(104 133)(105 134)(106 135)(107 136)(108 137)
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), (1,127,80,69)(2,128,81,70)(3,129,82,71)(4,130,83,72)(5,131,84,37)(6,132,85,38)(7,133,86,39)(8,134,87,40)(9,135,88,41)(10,136,89,42)(11,137,90,43)(12,138,91,44)(13,139,92,45)(14,140,93,46)(15,141,94,47)(16,142,95,48)(17,143,96,49)(18,144,97,50)(19,109,98,51)(20,110,99,52)(21,111,100,53)(22,112,101,54)(23,113,102,55)(24,114,103,56)(25,115,104,57)(26,116,105,58)(27,117,106,59)(28,118,107,60)(29,119,108,61)(30,120,73,62)(31,121,74,63)(32,122,75,64)(33,123,76,65)(34,124,77,66)(35,125,78,67)(36,126,79,68), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(73,138)(74,139)(75,140)(76,141)(77,142)(78,143)(79,144)(80,109)(81,110)(82,111)(83,112)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119)(91,120)(92,121)(93,122)(94,123)(95,124)(96,125)(97,126)(98,127)(99,128)(100,129)(101,130)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)>;
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), (1,127,80,69)(2,128,81,70)(3,129,82,71)(4,130,83,72)(5,131,84,37)(6,132,85,38)(7,133,86,39)(8,134,87,40)(9,135,88,41)(10,136,89,42)(11,137,90,43)(12,138,91,44)(13,139,92,45)(14,140,93,46)(15,141,94,47)(16,142,95,48)(17,143,96,49)(18,144,97,50)(19,109,98,51)(20,110,99,52)(21,111,100,53)(22,112,101,54)(23,113,102,55)(24,114,103,56)(25,115,104,57)(26,116,105,58)(27,117,106,59)(28,118,107,60)(29,119,108,61)(30,120,73,62)(31,121,74,63)(32,122,75,64)(33,123,76,65)(34,124,77,66)(35,125,78,67)(36,126,79,68), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(73,138)(74,139)(75,140)(76,141)(77,142)(78,143)(79,144)(80,109)(81,110)(82,111)(83,112)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119)(91,120)(92,121)(93,122)(94,123)(95,124)(96,125)(97,126)(98,127)(99,128)(100,129)(101,130)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137) );
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)], [(1,127,80,69),(2,128,81,70),(3,129,82,71),(4,130,83,72),(5,131,84,37),(6,132,85,38),(7,133,86,39),(8,134,87,40),(9,135,88,41),(10,136,89,42),(11,137,90,43),(12,138,91,44),(13,139,92,45),(14,140,93,46),(15,141,94,47),(16,142,95,48),(17,143,96,49),(18,144,97,50),(19,109,98,51),(20,110,99,52),(21,111,100,53),(22,112,101,54),(23,113,102,55),(24,114,103,56),(25,115,104,57),(26,116,105,58),(27,117,106,59),(28,118,107,60),(29,119,108,61),(30,120,73,62),(31,121,74,63),(32,122,75,64),(33,123,76,65),(34,124,77,66),(35,125,78,67),(36,126,79,68)], [(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,61),(12,62),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,37),(24,38),(25,39),(26,40),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(73,138),(74,139),(75,140),(76,141),(77,142),(78,143),(79,144),(80,109),(81,110),(82,111),(83,112),(84,113),(85,114),(86,115),(87,116),(88,117),(89,118),(90,119),(91,120),(92,121),(93,122),(94,123),(95,124),(96,125),(97,126),(98,127),(99,128),(100,129),(101,130),(102,131),(103,132),(104,133),(105,134),(106,135),(107,136),(108,137)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6F | 6G | ··· | 6N | 9A | ··· | 9F | 12A | ··· | 12H | 12I | ··· | 12X | 18A | ··· | 18R | 18S | ··· | 18AP | 36A | ··· | 36X | 36Y | ··· | 36BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C9 | C12 | C18 | C18 | C18 | C18 | C18 | C36 | D4 | C4○D4 | C3×D4 | C3×C4○D4 | D4×C9 | C9×C4○D4 |
| kernel | D4×C36 | C4×C36 | C9×C22⋊C4 | C9×C4⋊C4 | C22×C36 | D4×C18 | D4×C12 | D4×C9 | C4×C12 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C6×D4 | C4×D4 | C3×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C36 | C18 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 8 | 2 | 4 | 2 | 4 | 2 | 6 | 16 | 6 | 12 | 6 | 12 | 6 | 48 | 2 | 2 | 4 | 4 | 12 | 12 |
Matrix representation of D4×C36 ►in GL4(𝔽37) generated by
| 30 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 36 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 31 |
| 0 | 0 | 12 | 21 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 31 |
| 0 | 0 | 24 | 21 |
G:=sub<GL(4,GF(37))| [30,0,0,0,0,8,0,0,0,0,27,0,0,0,0,27],[36,0,0,0,0,1,0,0,0,0,16,12,0,0,31,21],[1,0,0,0,0,1,0,0,0,0,16,24,0,0,31,21] >;
D4×C36 in GAP, Magma, Sage, TeX
D_4\times C_{36} % in TeX
G:=Group("D4xC36"); // GroupNames label
G:=SmallGroup(288,168);
// by ID
G=gap.SmallGroup(288,168);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,365,268,360]);
// Polycyclic
G:=Group<a,b,c|a^36=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations