direct product, metabelian, nilpotent (class 2), monomial
Aliases: C32×C42⋊C2, C122⋊3C2, C23.10C62, C62.286C23, (C4×C12)⋊8C6, (C2×C12)⋊9C12, (C6×C12)⋊19C4, C42⋊4(C3×C6), C4.11(C6×C12), C12.63(C2×C12), C22.5(C6×C12), C62.91(C2×C4), (C2×C4).10C62, (C22×C12).35C6, C6.38(C22×C12), C22.6(C2×C62), (C6×C12).292C22, (C2×C62).86C22, C4⋊C4⋊6(C3×C6), C2.3(C2×C6×C12), (C3×C4⋊C4)⋊15C6, (C2×C4)⋊4(C3×C12), (C2×C6×C12).24C2, C6.48(C3×C4○D4), (C32×C4⋊C4)⋊24C2, (C2×C6).32(C2×C12), C22⋊C4.3(C3×C6), (C22×C4).6(C3×C6), C2.1(C32×C4○D4), (C2×C12).173(C2×C6), (C3×C12).143(C2×C4), (C3×C22⋊C4).16C6, (C22×C6).51(C2×C6), (C2×C6).92(C22×C6), (C3×C6).165(C4○D4), (C3×C6).130(C22×C4), (C32×C22⋊C4).9C2, SmallGroup(288,814)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C42⋊C2
G = < a,b,c,d,e | a3=b3=c4=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=cd2, de=ed >
Subgroups: 276 in 228 conjugacy classes, 180 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C23, C32, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C3×C6, C3×C6, C3×C6, C2×C12, C22×C6, C42⋊C2, C3×C12, C3×C12, C62, C62, C62, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×C12, C6×C12, C2×C62, C3×C42⋊C2, C122, C32×C22⋊C4, C32×C4⋊C4, C2×C6×C12, C32×C42⋊C2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C32, C12, C2×C6, C22×C4, C4○D4, C3×C6, C2×C12, C22×C6, C42⋊C2, C3×C12, C62, C22×C12, C3×C4○D4, C6×C12, C2×C62, C3×C42⋊C2, C2×C6×C12, C32×C4○D4, C32×C42⋊C2
►On 144 points
Generators in S144
(1 95 87)(2 96 88)(3 93 85)(4 94 86)(5 34 26)(6 35 27)(7 36 28)(8 33 25)(9 89 17)(10 90 18)(11 91 19)(12 92 20)(13 81 21)(14 82 22)(15 83 23)(16 84 24)(29 77 101)(30 78 102)(31 79 103)(32 80 104)(37 105 97)(38 106 98)(39 107 99)(40 108 100)(41 121 49)(42 122 50)(43 123 51)(44 124 52)(45 113 53)(46 114 54)(47 115 55)(48 116 56)(57 75 65)(58 76 66)(59 73 67)(60 74 68)(61 141 133)(62 142 134)(63 143 135)(64 144 136)(69 137 129)(70 138 130)(71 139 131)(72 140 132)(109 125 117)(110 126 118)(111 127 119)(112 128 120)
(1 83 11)(2 84 12)(3 81 9)(4 82 10)(5 78 38)(6 79 39)(7 80 40)(8 77 37)(13 17 85)(14 18 86)(15 19 87)(16 20 88)(21 89 93)(22 90 94)(23 91 95)(24 92 96)(25 29 97)(26 30 98)(27 31 99)(28 32 100)(33 101 105)(34 102 106)(35 103 107)(36 104 108)(41 109 113)(42 110 114)(43 111 115)(44 112 116)(45 49 117)(46 50 118)(47 51 119)(48 52 120)(53 121 125)(54 122 126)(55 123 127)(56 124 128)(57 61 129)(58 62 130)(59 63 131)(60 64 132)(65 133 137)(66 134 138)(67 135 139)(68 136 140)(69 75 141)(70 76 142)(71 73 143)(72 74 144)
(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 59 111 27)(2 60 112 28)(3 57 109 25)(4 58 110 26)(5 94 76 126)(6 95 73 127)(7 96 74 128)(8 93 75 125)(9 129 41 97)(10 130 42 98)(11 131 43 99)(12 132 44 100)(13 133 45 101)(14 134 46 102)(15 135 47 103)(16 136 48 104)(17 137 49 105)(18 138 50 106)(19 139 51 107)(20 140 52 108)(21 141 53 77)(22 142 54 78)(23 143 55 79)(24 144 56 80)(29 81 61 113)(30 82 62 114)(31 83 63 115)(32 84 64 116)(33 85 65 117)(34 86 66 118)(35 87 67 119)(36 88 68 120)(37 89 69 121)(38 90 70 122)(39 91 71 123)(40 92 72 124)
(2 112)(4 110)(5 76)(7 74)(10 42)(12 44)(14 46)(16 48)(18 50)(20 52)(22 54)(24 56)(26 58)(28 60)(30 62)(32 64)(34 66)(36 68)(38 70)(40 72)(78 142)(80 144)(82 114)(84 116)(86 118)(88 120)(90 122)(92 124)(94 126)(96 128)(98 130)(100 132)(102 134)(104 136)(106 138)(108 140)
G:=sub<Sym(144)| (1,95,87)(2,96,88)(3,93,85)(4,94,86)(5,34,26)(6,35,27)(7,36,28)(8,33,25)(9,89,17)(10,90,18)(11,91,19)(12,92,20)(13,81,21)(14,82,22)(15,83,23)(16,84,24)(29,77,101)(30,78,102)(31,79,103)(32,80,104)(37,105,97)(38,106,98)(39,107,99)(40,108,100)(41,121,49)(42,122,50)(43,123,51)(44,124,52)(45,113,53)(46,114,54)(47,115,55)(48,116,56)(57,75,65)(58,76,66)(59,73,67)(60,74,68)(61,141,133)(62,142,134)(63,143,135)(64,144,136)(69,137,129)(70,138,130)(71,139,131)(72,140,132)(109,125,117)(110,126,118)(111,127,119)(112,128,120), (1,83,11)(2,84,12)(3,81,9)(4,82,10)(5,78,38)(6,79,39)(7,80,40)(8,77,37)(13,17,85)(14,18,86)(15,19,87)(16,20,88)(21,89,93)(22,90,94)(23,91,95)(24,92,96)(25,29,97)(26,30,98)(27,31,99)(28,32,100)(33,101,105)(34,102,106)(35,103,107)(36,104,108)(41,109,113)(42,110,114)(43,111,115)(44,112,116)(45,49,117)(46,50,118)(47,51,119)(48,52,120)(53,121,125)(54,122,126)(55,123,127)(56,124,128)(57,61,129)(58,62,130)(59,63,131)(60,64,132)(65,133,137)(66,134,138)(67,135,139)(68,136,140)(69,75,141)(70,76,142)(71,73,143)(72,74,144), (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), (1,59,111,27)(2,60,112,28)(3,57,109,25)(4,58,110,26)(5,94,76,126)(6,95,73,127)(7,96,74,128)(8,93,75,125)(9,129,41,97)(10,130,42,98)(11,131,43,99)(12,132,44,100)(13,133,45,101)(14,134,46,102)(15,135,47,103)(16,136,48,104)(17,137,49,105)(18,138,50,106)(19,139,51,107)(20,140,52,108)(21,141,53,77)(22,142,54,78)(23,143,55,79)(24,144,56,80)(29,81,61,113)(30,82,62,114)(31,83,63,115)(32,84,64,116)(33,85,65,117)(34,86,66,118)(35,87,67,119)(36,88,68,120)(37,89,69,121)(38,90,70,122)(39,91,71,123)(40,92,72,124), (2,112)(4,110)(5,76)(7,74)(10,42)(12,44)(14,46)(16,48)(18,50)(20,52)(22,54)(24,56)(26,58)(28,60)(30,62)(32,64)(34,66)(36,68)(38,70)(40,72)(78,142)(80,144)(82,114)(84,116)(86,118)(88,120)(90,122)(92,124)(94,126)(96,128)(98,130)(100,132)(102,134)(104,136)(106,138)(108,140)>;
G:=Group( (1,95,87)(2,96,88)(3,93,85)(4,94,86)(5,34,26)(6,35,27)(7,36,28)(8,33,25)(9,89,17)(10,90,18)(11,91,19)(12,92,20)(13,81,21)(14,82,22)(15,83,23)(16,84,24)(29,77,101)(30,78,102)(31,79,103)(32,80,104)(37,105,97)(38,106,98)(39,107,99)(40,108,100)(41,121,49)(42,122,50)(43,123,51)(44,124,52)(45,113,53)(46,114,54)(47,115,55)(48,116,56)(57,75,65)(58,76,66)(59,73,67)(60,74,68)(61,141,133)(62,142,134)(63,143,135)(64,144,136)(69,137,129)(70,138,130)(71,139,131)(72,140,132)(109,125,117)(110,126,118)(111,127,119)(112,128,120), (1,83,11)(2,84,12)(3,81,9)(4,82,10)(5,78,38)(6,79,39)(7,80,40)(8,77,37)(13,17,85)(14,18,86)(15,19,87)(16,20,88)(21,89,93)(22,90,94)(23,91,95)(24,92,96)(25,29,97)(26,30,98)(27,31,99)(28,32,100)(33,101,105)(34,102,106)(35,103,107)(36,104,108)(41,109,113)(42,110,114)(43,111,115)(44,112,116)(45,49,117)(46,50,118)(47,51,119)(48,52,120)(53,121,125)(54,122,126)(55,123,127)(56,124,128)(57,61,129)(58,62,130)(59,63,131)(60,64,132)(65,133,137)(66,134,138)(67,135,139)(68,136,140)(69,75,141)(70,76,142)(71,73,143)(72,74,144), (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), (1,59,111,27)(2,60,112,28)(3,57,109,25)(4,58,110,26)(5,94,76,126)(6,95,73,127)(7,96,74,128)(8,93,75,125)(9,129,41,97)(10,130,42,98)(11,131,43,99)(12,132,44,100)(13,133,45,101)(14,134,46,102)(15,135,47,103)(16,136,48,104)(17,137,49,105)(18,138,50,106)(19,139,51,107)(20,140,52,108)(21,141,53,77)(22,142,54,78)(23,143,55,79)(24,144,56,80)(29,81,61,113)(30,82,62,114)(31,83,63,115)(32,84,64,116)(33,85,65,117)(34,86,66,118)(35,87,67,119)(36,88,68,120)(37,89,69,121)(38,90,70,122)(39,91,71,123)(40,92,72,124), (2,112)(4,110)(5,76)(7,74)(10,42)(12,44)(14,46)(16,48)(18,50)(20,52)(22,54)(24,56)(26,58)(28,60)(30,62)(32,64)(34,66)(36,68)(38,70)(40,72)(78,142)(80,144)(82,114)(84,116)(86,118)(88,120)(90,122)(92,124)(94,126)(96,128)(98,130)(100,132)(102,134)(104,136)(106,138)(108,140) );
G=PermutationGroup([[(1,95,87),(2,96,88),(3,93,85),(4,94,86),(5,34,26),(6,35,27),(7,36,28),(8,33,25),(9,89,17),(10,90,18),(11,91,19),(12,92,20),(13,81,21),(14,82,22),(15,83,23),(16,84,24),(29,77,101),(30,78,102),(31,79,103),(32,80,104),(37,105,97),(38,106,98),(39,107,99),(40,108,100),(41,121,49),(42,122,50),(43,123,51),(44,124,52),(45,113,53),(46,114,54),(47,115,55),(48,116,56),(57,75,65),(58,76,66),(59,73,67),(60,74,68),(61,141,133),(62,142,134),(63,143,135),(64,144,136),(69,137,129),(70,138,130),(71,139,131),(72,140,132),(109,125,117),(110,126,118),(111,127,119),(112,128,120)], [(1,83,11),(2,84,12),(3,81,9),(4,82,10),(5,78,38),(6,79,39),(7,80,40),(8,77,37),(13,17,85),(14,18,86),(15,19,87),(16,20,88),(21,89,93),(22,90,94),(23,91,95),(24,92,96),(25,29,97),(26,30,98),(27,31,99),(28,32,100),(33,101,105),(34,102,106),(35,103,107),(36,104,108),(41,109,113),(42,110,114),(43,111,115),(44,112,116),(45,49,117),(46,50,118),(47,51,119),(48,52,120),(53,121,125),(54,122,126),(55,123,127),(56,124,128),(57,61,129),(58,62,130),(59,63,131),(60,64,132),(65,133,137),(66,134,138),(67,135,139),(68,136,140),(69,75,141),(70,76,142),(71,73,143),(72,74,144)], [(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)], [(1,59,111,27),(2,60,112,28),(3,57,109,25),(4,58,110,26),(5,94,76,126),(6,95,73,127),(7,96,74,128),(8,93,75,125),(9,129,41,97),(10,130,42,98),(11,131,43,99),(12,132,44,100),(13,133,45,101),(14,134,46,102),(15,135,47,103),(16,136,48,104),(17,137,49,105),(18,138,50,106),(19,139,51,107),(20,140,52,108),(21,141,53,77),(22,142,54,78),(23,143,55,79),(24,144,56,80),(29,81,61,113),(30,82,62,114),(31,83,63,115),(32,84,64,116),(33,85,65,117),(34,86,66,118),(35,87,67,119),(36,88,68,120),(37,89,69,121),(38,90,70,122),(39,91,71,123),(40,92,72,124)], [(2,112),(4,110),(5,76),(7,74),(10,42),(12,44),(14,46),(16,48),(18,50),(20,52),(22,54),(24,56),(26,58),(28,60),(30,62),(32,64),(34,66),(36,68),(38,70),(40,72),(78,142),(80,144),(82,114),(84,116),(86,118),(88,120),(90,122),(92,124),(94,126),(96,128),(98,130),(100,132),(102,134),(104,136),(106,138),(108,140)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 6A | ··· | 6X | 6Y | ··· | 6AN | 12A | ··· | 12AF | 12AG | ··· | 12DH |
| 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 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | C4○D4 | C3×C4○D4 |
| kernel | C32×C42⋊C2 | C122 | C32×C22⋊C4 | C32×C4⋊C4 | C2×C6×C12 | C3×C42⋊C2 | C6×C12 | C4×C12 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C2×C12 | C3×C6 | C6 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 8 | 16 | 16 | 16 | 8 | 64 | 4 | 32 |
Matrix representation of C32×C42⋊C2 ►in GL3(𝔽13) generated by
| 3 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 9 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 5 | 0 | 0 |
| 0 | 8 | 11 |
| 0 | 0 | 5 |
| 1 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 12 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 8 | 12 |
G:=sub<GL(3,GF(13))| [3,0,0,0,1,0,0,0,1],[9,0,0,0,3,0,0,0,3],[5,0,0,0,8,0,0,11,5],[1,0,0,0,5,0,0,0,5],[12,0,0,0,1,8,0,0,12] >;
C32×C42⋊C2 in GAP, Magma, Sage, TeX
C_3^2\times C_4^2\rtimes C_2
% in TeX
G:=Group("C3^2xC4^2:C2"); // GroupNames label
G:=SmallGroup(288,814);
// by ID
G=gap.SmallGroup(288,814);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1008,1037,394]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=d^4=e^2=1,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,e*c*e=c*d^2,d*e=e*d>;
// generators/relations