direct product, metacyclic, nilpotent (class 2), monomial
Aliases: C32×C4⋊C4, C12⋊3C12, C22.3C62, C62.37C22, C4⋊(C3×C12), (C3×C12)⋊7C4, C6.5(C3×Q8), (C3×C6).8Q8, C2.2(C6×C12), (C6×C12).3C2, C2.(Q8×C32), (C3×C6).40D4, C6.19(C3×D4), (C2×C12).13C6, C6.14(C2×C12), C2.2(D4×C32), (C2×C4).1(C3×C6), (C2×C6).20(C2×C6), (C3×C6).36(C2×C4), SmallGroup(144,103)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C4⋊C4
G = < a,b,c,d | a3=b3=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 90 in 78 conjugacy classes, 66 normal (14 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C32, C12, C12, C2×C6, C4⋊C4, C3×C6, C2×C12, C3×C12, C3×C12, C62, C3×C4⋊C4, C6×C12, C6×C12, C32×C4⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C32, C12, C2×C6, C4⋊C4, C3×C6, C2×C12, C3×D4, C3×Q8, C3×C12, C62, C3×C4⋊C4, C6×C12, D4×C32, Q8×C32, C32×C4⋊C4
►Regular action on 144 points
Generators in S144
(1 76 30)(2 73 31)(3 74 32)(4 75 29)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(9 54 46)(10 55 47)(11 56 48)(12 53 45)(17 50 42)(18 51 43)(19 52 44)(20 49 41)(25 70 62)(26 71 63)(27 72 64)(28 69 61)(33 66 58)(34 67 59)(35 68 60)(36 65 57)(37 102 94)(38 103 95)(39 104 96)(40 101 93)(77 124 85)(78 121 86)(79 122 87)(80 123 88)(81 116 89)(82 113 90)(83 114 91)(84 115 92)(97 144 136)(98 141 133)(99 142 134)(100 143 135)(105 140 132)(106 137 129)(107 138 130)(108 139 131)(109 125 117)(110 126 118)(111 127 119)(112 128 120)
(1 26 59)(2 27 60)(3 28 57)(4 25 58)(5 9 42)(6 10 43)(7 11 44)(8 12 41)(13 46 50)(14 47 51)(15 48 52)(16 45 49)(17 21 54)(18 22 55)(19 23 56)(20 24 53)(29 62 66)(30 63 67)(31 64 68)(32 61 65)(33 75 70)(34 76 71)(35 73 72)(36 74 69)(37 141 106)(38 142 107)(39 143 108)(40 144 105)(77 112 116)(78 109 113)(79 110 114)(80 111 115)(81 85 120)(82 86 117)(83 87 118)(84 88 119)(89 124 128)(90 121 125)(91 122 126)(92 123 127)(93 97 132)(94 98 129)(95 99 130)(96 100 131)(101 136 140)(102 133 137)(103 134 138)(104 135 139)
(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)
(1 110 6 94)(2 109 7 93)(3 112 8 96)(4 111 5 95)(9 99 25 115)(10 98 26 114)(11 97 27 113)(12 100 28 116)(13 103 29 119)(14 102 30 118)(15 101 31 117)(16 104 32 120)(17 107 33 123)(18 106 34 122)(19 105 35 121)(20 108 36 124)(21 38 75 127)(22 37 76 126)(23 40 73 125)(24 39 74 128)(41 131 57 77)(42 130 58 80)(43 129 59 79)(44 132 60 78)(45 135 61 81)(46 134 62 84)(47 133 63 83)(48 136 64 82)(49 139 65 85)(50 138 66 88)(51 137 67 87)(52 140 68 86)(53 143 69 89)(54 142 70 92)(55 141 71 91)(56 144 72 90)
G:=sub<Sym(144)| (1,76,30)(2,73,31)(3,74,32)(4,75,29)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(9,54,46)(10,55,47)(11,56,48)(12,53,45)(17,50,42)(18,51,43)(19,52,44)(20,49,41)(25,70,62)(26,71,63)(27,72,64)(28,69,61)(33,66,58)(34,67,59)(35,68,60)(36,65,57)(37,102,94)(38,103,95)(39,104,96)(40,101,93)(77,124,85)(78,121,86)(79,122,87)(80,123,88)(81,116,89)(82,113,90)(83,114,91)(84,115,92)(97,144,136)(98,141,133)(99,142,134)(100,143,135)(105,140,132)(106,137,129)(107,138,130)(108,139,131)(109,125,117)(110,126,118)(111,127,119)(112,128,120), (1,26,59)(2,27,60)(3,28,57)(4,25,58)(5,9,42)(6,10,43)(7,11,44)(8,12,41)(13,46,50)(14,47,51)(15,48,52)(16,45,49)(17,21,54)(18,22,55)(19,23,56)(20,24,53)(29,62,66)(30,63,67)(31,64,68)(32,61,65)(33,75,70)(34,76,71)(35,73,72)(36,74,69)(37,141,106)(38,142,107)(39,143,108)(40,144,105)(77,112,116)(78,109,113)(79,110,114)(80,111,115)(81,85,120)(82,86,117)(83,87,118)(84,88,119)(89,124,128)(90,121,125)(91,122,126)(92,123,127)(93,97,132)(94,98,129)(95,99,130)(96,100,131)(101,136,140)(102,133,137)(103,134,138)(104,135,139), (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), (1,110,6,94)(2,109,7,93)(3,112,8,96)(4,111,5,95)(9,99,25,115)(10,98,26,114)(11,97,27,113)(12,100,28,116)(13,103,29,119)(14,102,30,118)(15,101,31,117)(16,104,32,120)(17,107,33,123)(18,106,34,122)(19,105,35,121)(20,108,36,124)(21,38,75,127)(22,37,76,126)(23,40,73,125)(24,39,74,128)(41,131,57,77)(42,130,58,80)(43,129,59,79)(44,132,60,78)(45,135,61,81)(46,134,62,84)(47,133,63,83)(48,136,64,82)(49,139,65,85)(50,138,66,88)(51,137,67,87)(52,140,68,86)(53,143,69,89)(54,142,70,92)(55,141,71,91)(56,144,72,90)>;
G:=Group( (1,76,30)(2,73,31)(3,74,32)(4,75,29)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(9,54,46)(10,55,47)(11,56,48)(12,53,45)(17,50,42)(18,51,43)(19,52,44)(20,49,41)(25,70,62)(26,71,63)(27,72,64)(28,69,61)(33,66,58)(34,67,59)(35,68,60)(36,65,57)(37,102,94)(38,103,95)(39,104,96)(40,101,93)(77,124,85)(78,121,86)(79,122,87)(80,123,88)(81,116,89)(82,113,90)(83,114,91)(84,115,92)(97,144,136)(98,141,133)(99,142,134)(100,143,135)(105,140,132)(106,137,129)(107,138,130)(108,139,131)(109,125,117)(110,126,118)(111,127,119)(112,128,120), (1,26,59)(2,27,60)(3,28,57)(4,25,58)(5,9,42)(6,10,43)(7,11,44)(8,12,41)(13,46,50)(14,47,51)(15,48,52)(16,45,49)(17,21,54)(18,22,55)(19,23,56)(20,24,53)(29,62,66)(30,63,67)(31,64,68)(32,61,65)(33,75,70)(34,76,71)(35,73,72)(36,74,69)(37,141,106)(38,142,107)(39,143,108)(40,144,105)(77,112,116)(78,109,113)(79,110,114)(80,111,115)(81,85,120)(82,86,117)(83,87,118)(84,88,119)(89,124,128)(90,121,125)(91,122,126)(92,123,127)(93,97,132)(94,98,129)(95,99,130)(96,100,131)(101,136,140)(102,133,137)(103,134,138)(104,135,139), (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), (1,110,6,94)(2,109,7,93)(3,112,8,96)(4,111,5,95)(9,99,25,115)(10,98,26,114)(11,97,27,113)(12,100,28,116)(13,103,29,119)(14,102,30,118)(15,101,31,117)(16,104,32,120)(17,107,33,123)(18,106,34,122)(19,105,35,121)(20,108,36,124)(21,38,75,127)(22,37,76,126)(23,40,73,125)(24,39,74,128)(41,131,57,77)(42,130,58,80)(43,129,59,79)(44,132,60,78)(45,135,61,81)(46,134,62,84)(47,133,63,83)(48,136,64,82)(49,139,65,85)(50,138,66,88)(51,137,67,87)(52,140,68,86)(53,143,69,89)(54,142,70,92)(55,141,71,91)(56,144,72,90) );
G=PermutationGroup([[(1,76,30),(2,73,31),(3,74,32),(4,75,29),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(9,54,46),(10,55,47),(11,56,48),(12,53,45),(17,50,42),(18,51,43),(19,52,44),(20,49,41),(25,70,62),(26,71,63),(27,72,64),(28,69,61),(33,66,58),(34,67,59),(35,68,60),(36,65,57),(37,102,94),(38,103,95),(39,104,96),(40,101,93),(77,124,85),(78,121,86),(79,122,87),(80,123,88),(81,116,89),(82,113,90),(83,114,91),(84,115,92),(97,144,136),(98,141,133),(99,142,134),(100,143,135),(105,140,132),(106,137,129),(107,138,130),(108,139,131),(109,125,117),(110,126,118),(111,127,119),(112,128,120)], [(1,26,59),(2,27,60),(3,28,57),(4,25,58),(5,9,42),(6,10,43),(7,11,44),(8,12,41),(13,46,50),(14,47,51),(15,48,52),(16,45,49),(17,21,54),(18,22,55),(19,23,56),(20,24,53),(29,62,66),(30,63,67),(31,64,68),(32,61,65),(33,75,70),(34,76,71),(35,73,72),(36,74,69),(37,141,106),(38,142,107),(39,143,108),(40,144,105),(77,112,116),(78,109,113),(79,110,114),(80,111,115),(81,85,120),(82,86,117),(83,87,118),(84,88,119),(89,124,128),(90,121,125),(91,122,126),(92,123,127),(93,97,132),(94,98,129),(95,99,130),(96,100,131),(101,136,140),(102,133,137),(103,134,138),(104,135,139)], [(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)], [(1,110,6,94),(2,109,7,93),(3,112,8,96),(4,111,5,95),(9,99,25,115),(10,98,26,114),(11,97,27,113),(12,100,28,116),(13,103,29,119),(14,102,30,118),(15,101,31,117),(16,104,32,120),(17,107,33,123),(18,106,34,122),(19,105,35,121),(20,108,36,124),(21,38,75,127),(22,37,76,126),(23,40,73,125),(24,39,74,128),(41,131,57,77),(42,130,58,80),(43,129,59,79),(44,132,60,78),(45,135,61,81),(46,134,62,84),(47,133,63,83),(48,136,64,82),(49,139,65,85),(50,138,66,88),(51,137,67,87),(52,140,68,86),(53,143,69,89),(54,142,70,92),(55,141,71,91),(56,144,72,90)]])
C32×C4⋊C4 is a maximal subgroup of
C12.9Dic6 C12.10Dic6 C62.113D4 C62.114D4 C62.231C23 C12⋊2Dic6 C62.233C23 C62.234C23 C62.236C23 C62.237C23 C62.238C23 C12⋊3D12 C62.240C23 C12.31D12 C62.242C23 D4×C3×C12 Q8×C3×C12
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4A | ··· | 4F | 6A | ··· | 6X | 12A | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D4 | Q8 | C3×D4 | C3×Q8 |
| kernel | C32×C4⋊C4 | C6×C12 | C3×C4⋊C4 | C3×C12 | C2×C12 | C12 | C3×C6 | C3×C6 | C6 | C6 |
| # reps | 1 | 3 | 8 | 4 | 24 | 32 | 1 | 1 | 8 | 8 |
Matrix representation of C32×C4⋊C4 ►in GL5(𝔽13)
| 3 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 1 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 9 | 1 |
| 0 | 0 | 0 | 11 | 4 |
G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,1,0,0,0,11,1],[12,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,9,11,0,0,0,1,4] >;
C32×C4⋊C4 in GAP, Magma, Sage, TeX
C_3^2\times C_4\rtimes C_4
% in TeX
G:=Group("C3^2xC4:C4"); // GroupNames label
G:=SmallGroup(144,103);
// by ID
G=gap.SmallGroup(144,103);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-2,432,457,223]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations