direct product, metabelian, nilpotent (class 2), monomial
Aliases: C4○D4×C3×C6, D4⋊3C62, Q8⋊4C62, C23.14C62, C62.156C23, (C6×D4)⋊16C6, (C2×C4)⋊5C62, (C6×Q8)⋊17C6, C4.8(C2×C62), (C22×C12)⋊17C6, (C6×C12)⋊38C22, (C3×C6).70C24, C6.23(C23×C6), C12.62(C22×C6), C22.1(C2×C62), C2.3(C22×C62), (C3×C12).196C23, (D4×C32)⋊30C22, (C2×C62).90C22, (Q8×C32)⋊27C22, (C2×C6×C12)⋊20C2, (D4×C3×C6)⋊25C2, (Q8×C3×C6)⋊20C2, (C2×D4)⋊7(C3×C6), (C2×Q8)⋊8(C3×C6), (C2×C12)⋊16(C2×C6), (C3×D4)⋊12(C2×C6), (C22×C4)⋊8(C3×C6), (C3×Q8)⋊13(C2×C6), (C22×C6).55(C2×C6), (C2×C6).11(C22×C6), SmallGroup(288,1021)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D4×C3×C6
G = < a,b,c,d,e | a3=b6=c4=e2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >
Subgroups: 564 in 492 conjugacy classes, 420 normal (12 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, C12, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×C6, C3×C6, C3×C6, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4○D4, C3×C12, C62, C62, C62, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C6×C12, C6×C12, D4×C32, Q8×C32, C2×C62, C6×C4○D4, C2×C6×C12, D4×C3×C6, Q8×C3×C6, C32×C4○D4, C4○D4×C3×C6
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C4○D4, C24, C3×C6, C22×C6, C2×C4○D4, C62, C3×C4○D4, C23×C6, C2×C62, C6×C4○D4, C32×C4○D4, C22×C62, C4○D4×C3×C6
►On 144 points
Generators in S144
(1 40 35)(2 41 36)(3 42 31)(4 37 32)(5 38 33)(6 39 34)(7 17 28)(8 18 29)(9 13 30)(10 14 25)(11 15 26)(12 16 27)(19 143 122)(20 144 123)(21 139 124)(22 140 125)(23 141 126)(24 142 121)(43 56 64)(44 57 65)(45 58 66)(46 59 61)(47 60 62)(48 55 63)(49 75 70)(50 76 71)(51 77 72)(52 78 67)(53 73 68)(54 74 69)(79 92 100)(80 93 101)(81 94 102)(82 95 97)(83 96 98)(84 91 99)(85 111 106)(86 112 107)(87 113 108)(88 114 103)(89 109 104)(90 110 105)(115 128 136)(116 129 137)(117 130 138)(118 131 133)(119 132 134)(120 127 135)
(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 53 10 55)(2 54 11 56)(3 49 12 57)(4 50 7 58)(5 51 8 59)(6 52 9 60)(13 62 39 78)(14 63 40 73)(15 64 41 74)(16 65 42 75)(17 66 37 76)(18 61 38 77)(19 99 138 109)(20 100 133 110)(21 101 134 111)(22 102 135 112)(23 97 136 113)(24 98 137 114)(25 48 35 68)(26 43 36 69)(27 44 31 70)(28 45 32 71)(29 46 33 72)(30 47 34 67)(79 118 105 144)(80 119 106 139)(81 120 107 140)(82 115 108 141)(83 116 103 142)(84 117 104 143)(85 124 93 132)(86 125 94 127)(87 126 95 128)(88 121 96 129)(89 122 91 130)(90 123 92 131)
(1 94 10 86)(2 95 11 87)(3 96 12 88)(4 91 7 89)(5 92 8 90)(6 93 9 85)(13 111 39 101)(14 112 40 102)(15 113 41 97)(16 114 42 98)(17 109 37 99)(18 110 38 100)(19 76 138 66)(20 77 133 61)(21 78 134 62)(22 73 135 63)(23 74 136 64)(24 75 137 65)(25 107 35 81)(26 108 36 82)(27 103 31 83)(28 104 32 84)(29 105 33 79)(30 106 34 80)(43 141 69 115)(44 142 70 116)(45 143 71 117)(46 144 72 118)(47 139 67 119)(48 140 68 120)(49 129 57 121)(50 130 58 122)(51 131 59 123)(52 132 60 124)(53 127 55 125)(54 128 56 126)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 86)(8 87)(9 88)(10 89)(11 90)(12 85)(13 114)(14 109)(15 110)(16 111)(17 112)(18 113)(19 63)(20 64)(21 65)(22 66)(23 61)(24 62)(25 104)(26 105)(27 106)(28 107)(29 108)(30 103)(31 80)(32 81)(33 82)(34 83)(35 84)(36 79)(37 102)(38 97)(39 98)(40 99)(41 100)(42 101)(43 144)(44 139)(45 140)(46 141)(47 142)(48 143)(49 132)(50 127)(51 128)(52 129)(53 130)(54 131)(55 122)(56 123)(57 124)(58 125)(59 126)(60 121)(67 116)(68 117)(69 118)(70 119)(71 120)(72 115)(73 138)(74 133)(75 134)(76 135)(77 136)(78 137)
G:=sub<Sym(144)| (1,40,35)(2,41,36)(3,42,31)(4,37,32)(5,38,33)(6,39,34)(7,17,28)(8,18,29)(9,13,30)(10,14,25)(11,15,26)(12,16,27)(19,143,122)(20,144,123)(21,139,124)(22,140,125)(23,141,126)(24,142,121)(43,56,64)(44,57,65)(45,58,66)(46,59,61)(47,60,62)(48,55,63)(49,75,70)(50,76,71)(51,77,72)(52,78,67)(53,73,68)(54,74,69)(79,92,100)(80,93,101)(81,94,102)(82,95,97)(83,96,98)(84,91,99)(85,111,106)(86,112,107)(87,113,108)(88,114,103)(89,109,104)(90,110,105)(115,128,136)(116,129,137)(117,130,138)(118,131,133)(119,132,134)(120,127,135), (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), (1,53,10,55)(2,54,11,56)(3,49,12,57)(4,50,7,58)(5,51,8,59)(6,52,9,60)(13,62,39,78)(14,63,40,73)(15,64,41,74)(16,65,42,75)(17,66,37,76)(18,61,38,77)(19,99,138,109)(20,100,133,110)(21,101,134,111)(22,102,135,112)(23,97,136,113)(24,98,137,114)(25,48,35,68)(26,43,36,69)(27,44,31,70)(28,45,32,71)(29,46,33,72)(30,47,34,67)(79,118,105,144)(80,119,106,139)(81,120,107,140)(82,115,108,141)(83,116,103,142)(84,117,104,143)(85,124,93,132)(86,125,94,127)(87,126,95,128)(88,121,96,129)(89,122,91,130)(90,123,92,131), (1,94,10,86)(2,95,11,87)(3,96,12,88)(4,91,7,89)(5,92,8,90)(6,93,9,85)(13,111,39,101)(14,112,40,102)(15,113,41,97)(16,114,42,98)(17,109,37,99)(18,110,38,100)(19,76,138,66)(20,77,133,61)(21,78,134,62)(22,73,135,63)(23,74,136,64)(24,75,137,65)(25,107,35,81)(26,108,36,82)(27,103,31,83)(28,104,32,84)(29,105,33,79)(30,106,34,80)(43,141,69,115)(44,142,70,116)(45,143,71,117)(46,144,72,118)(47,139,67,119)(48,140,68,120)(49,129,57,121)(50,130,58,122)(51,131,59,123)(52,132,60,124)(53,127,55,125)(54,128,56,126), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,86)(8,87)(9,88)(10,89)(11,90)(12,85)(13,114)(14,109)(15,110)(16,111)(17,112)(18,113)(19,63)(20,64)(21,65)(22,66)(23,61)(24,62)(25,104)(26,105)(27,106)(28,107)(29,108)(30,103)(31,80)(32,81)(33,82)(34,83)(35,84)(36,79)(37,102)(38,97)(39,98)(40,99)(41,100)(42,101)(43,144)(44,139)(45,140)(46,141)(47,142)(48,143)(49,132)(50,127)(51,128)(52,129)(53,130)(54,131)(55,122)(56,123)(57,124)(58,125)(59,126)(60,121)(67,116)(68,117)(69,118)(70,119)(71,120)(72,115)(73,138)(74,133)(75,134)(76,135)(77,136)(78,137)>;
G:=Group( (1,40,35)(2,41,36)(3,42,31)(4,37,32)(5,38,33)(6,39,34)(7,17,28)(8,18,29)(9,13,30)(10,14,25)(11,15,26)(12,16,27)(19,143,122)(20,144,123)(21,139,124)(22,140,125)(23,141,126)(24,142,121)(43,56,64)(44,57,65)(45,58,66)(46,59,61)(47,60,62)(48,55,63)(49,75,70)(50,76,71)(51,77,72)(52,78,67)(53,73,68)(54,74,69)(79,92,100)(80,93,101)(81,94,102)(82,95,97)(83,96,98)(84,91,99)(85,111,106)(86,112,107)(87,113,108)(88,114,103)(89,109,104)(90,110,105)(115,128,136)(116,129,137)(117,130,138)(118,131,133)(119,132,134)(120,127,135), (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), (1,53,10,55)(2,54,11,56)(3,49,12,57)(4,50,7,58)(5,51,8,59)(6,52,9,60)(13,62,39,78)(14,63,40,73)(15,64,41,74)(16,65,42,75)(17,66,37,76)(18,61,38,77)(19,99,138,109)(20,100,133,110)(21,101,134,111)(22,102,135,112)(23,97,136,113)(24,98,137,114)(25,48,35,68)(26,43,36,69)(27,44,31,70)(28,45,32,71)(29,46,33,72)(30,47,34,67)(79,118,105,144)(80,119,106,139)(81,120,107,140)(82,115,108,141)(83,116,103,142)(84,117,104,143)(85,124,93,132)(86,125,94,127)(87,126,95,128)(88,121,96,129)(89,122,91,130)(90,123,92,131), (1,94,10,86)(2,95,11,87)(3,96,12,88)(4,91,7,89)(5,92,8,90)(6,93,9,85)(13,111,39,101)(14,112,40,102)(15,113,41,97)(16,114,42,98)(17,109,37,99)(18,110,38,100)(19,76,138,66)(20,77,133,61)(21,78,134,62)(22,73,135,63)(23,74,136,64)(24,75,137,65)(25,107,35,81)(26,108,36,82)(27,103,31,83)(28,104,32,84)(29,105,33,79)(30,106,34,80)(43,141,69,115)(44,142,70,116)(45,143,71,117)(46,144,72,118)(47,139,67,119)(48,140,68,120)(49,129,57,121)(50,130,58,122)(51,131,59,123)(52,132,60,124)(53,127,55,125)(54,128,56,126), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,86)(8,87)(9,88)(10,89)(11,90)(12,85)(13,114)(14,109)(15,110)(16,111)(17,112)(18,113)(19,63)(20,64)(21,65)(22,66)(23,61)(24,62)(25,104)(26,105)(27,106)(28,107)(29,108)(30,103)(31,80)(32,81)(33,82)(34,83)(35,84)(36,79)(37,102)(38,97)(39,98)(40,99)(41,100)(42,101)(43,144)(44,139)(45,140)(46,141)(47,142)(48,143)(49,132)(50,127)(51,128)(52,129)(53,130)(54,131)(55,122)(56,123)(57,124)(58,125)(59,126)(60,121)(67,116)(68,117)(69,118)(70,119)(71,120)(72,115)(73,138)(74,133)(75,134)(76,135)(77,136)(78,137) );
G=PermutationGroup([[(1,40,35),(2,41,36),(3,42,31),(4,37,32),(5,38,33),(6,39,34),(7,17,28),(8,18,29),(9,13,30),(10,14,25),(11,15,26),(12,16,27),(19,143,122),(20,144,123),(21,139,124),(22,140,125),(23,141,126),(24,142,121),(43,56,64),(44,57,65),(45,58,66),(46,59,61),(47,60,62),(48,55,63),(49,75,70),(50,76,71),(51,77,72),(52,78,67),(53,73,68),(54,74,69),(79,92,100),(80,93,101),(81,94,102),(82,95,97),(83,96,98),(84,91,99),(85,111,106),(86,112,107),(87,113,108),(88,114,103),(89,109,104),(90,110,105),(115,128,136),(116,129,137),(117,130,138),(118,131,133),(119,132,134),(120,127,135)], [(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)], [(1,53,10,55),(2,54,11,56),(3,49,12,57),(4,50,7,58),(5,51,8,59),(6,52,9,60),(13,62,39,78),(14,63,40,73),(15,64,41,74),(16,65,42,75),(17,66,37,76),(18,61,38,77),(19,99,138,109),(20,100,133,110),(21,101,134,111),(22,102,135,112),(23,97,136,113),(24,98,137,114),(25,48,35,68),(26,43,36,69),(27,44,31,70),(28,45,32,71),(29,46,33,72),(30,47,34,67),(79,118,105,144),(80,119,106,139),(81,120,107,140),(82,115,108,141),(83,116,103,142),(84,117,104,143),(85,124,93,132),(86,125,94,127),(87,126,95,128),(88,121,96,129),(89,122,91,130),(90,123,92,131)], [(1,94,10,86),(2,95,11,87),(3,96,12,88),(4,91,7,89),(5,92,8,90),(6,93,9,85),(13,111,39,101),(14,112,40,102),(15,113,41,97),(16,114,42,98),(17,109,37,99),(18,110,38,100),(19,76,138,66),(20,77,133,61),(21,78,134,62),(22,73,135,63),(23,74,136,64),(24,75,137,65),(25,107,35,81),(26,108,36,82),(27,103,31,83),(28,104,32,84),(29,105,33,79),(30,106,34,80),(43,141,69,115),(44,142,70,116),(45,143,71,117),(46,144,72,118),(47,139,67,119),(48,140,68,120),(49,129,57,121),(50,130,58,122),(51,131,59,123),(52,132,60,124),(53,127,55,125),(54,128,56,126)], [(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,86),(8,87),(9,88),(10,89),(11,90),(12,85),(13,114),(14,109),(15,110),(16,111),(17,112),(18,113),(19,63),(20,64),(21,65),(22,66),(23,61),(24,62),(25,104),(26,105),(27,106),(28,107),(29,108),(30,103),(31,80),(32,81),(33,82),(34,83),(35,84),(36,79),(37,102),(38,97),(39,98),(40,99),(41,100),(42,101),(43,144),(44,139),(45,140),(46,141),(47,142),(48,143),(49,132),(50,127),(51,128),(52,129),(53,130),(54,131),(55,122),(56,123),(57,124),(58,125),(59,126),(60,121),(67,116),(68,117),(69,118),(70,119),(71,120),(72,115),(73,138),(74,133),(75,134),(76,135),(77,136),(78,137)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | ··· | 6X | 6Y | ··· | 6BT | 12A | ··· | 12AF | 12AG | ··· | 12CB |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 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 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C4○D4 | C3×C4○D4 |
| kernel | C4○D4×C3×C6 | C2×C6×C12 | D4×C3×C6 | Q8×C3×C6 | C32×C4○D4 | C6×C4○D4 | C22×C12 | C6×D4 | C6×Q8 | C3×C4○D4 | C3×C6 | C6 |
| # reps | 1 | 3 | 3 | 1 | 8 | 8 | 24 | 24 | 8 | 64 | 4 | 32 |
Matrix representation of C4○D4×C3×C6 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 11 |
| 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[12,0,0,0,0,12,0,0,0,0,12,1,0,0,11,1],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,11,1] >;
C4○D4×C3×C6 in GAP, Magma, Sage, TeX
C_4\circ D_4\times C_3\times C_6
% in TeX
G:=Group("C4oD4xC3xC6"); // GroupNames label
G:=SmallGroup(288,1021);
// by ID
G=gap.SmallGroup(288,1021);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-2,2045,772]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=c^4=e^2=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d>;
// generators/relations