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G = M4(2)×C3×C6order 288 = 25·32

Direct product of C3×C6 and M4(2)

direct product, metabelian, nilpotent (class 2), monomial

Aliases: M4(2)×C3×C6, C84C62, (C6×C24)⋊22C2, (C2×C24)⋊14C6, C2414(C2×C6), C4.12(C6×C12), (C6×C12).35C4, (C2×C62).9C4, (C2×C12).25C12, C12.58(C2×C12), (C3×C24)⋊34C22, C23.4(C3×C12), C4.11(C2×C62), (C2×C4).33C62, (C22×C12).37C6, C12.67(C22×C6), C6.41(C22×C12), (C22×C6).13C12, C62.120(C2×C4), C22.11(C6×C12), (C6×C12).382C22, (C3×C12).194C23, (C2×C8)⋊6(C3×C6), C2.6(C2×C6×C12), (C2×C6×C12).26C2, (C2×C4).6(C3×C12), (C2×C6).33(C2×C12), (C22×C4).8(C3×C6), (C3×C12).144(C2×C4), (C2×C12).174(C2×C6), (C3×C6).133(C22×C4), SmallGroup(288,827)

Series: Derived Chief Lower central Upper central

C1C2 — M4(2)×C3×C6
C1C2C4C12C3×C12C3×C24C32×M4(2) — M4(2)×C3×C6
C1C2 — M4(2)×C3×C6
C1C6×C12 — M4(2)×C3×C6

Generators and relations for M4(2)×C3×C6
 G = < a,b,c,d | a3=b6=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 228 in 204 conjugacy classes, 180 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C6 [×12], C6 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C32, C12 [×16], C2×C6 [×12], C2×C6 [×8], C2×C8 [×2], M4(2) [×4], C22×C4, C3×C6, C3×C6 [×2], C3×C6 [×2], C24 [×16], C2×C12 [×24], C22×C6 [×4], C2×M4(2), C3×C12 [×2], C3×C12 [×2], C62, C62 [×2], C62 [×2], C2×C24 [×8], C3×M4(2) [×16], C22×C12 [×4], C3×C24 [×4], C6×C12 [×2], C6×C12 [×4], C2×C62, C6×M4(2) [×4], C6×C24 [×2], C32×M4(2) [×4], C2×C6×C12, M4(2)×C3×C6
Quotients: C1, C2 [×7], C3 [×4], C4 [×4], C22 [×7], C6 [×28], C2×C4 [×6], C23, C32, C12 [×16], C2×C6 [×28], M4(2) [×2], C22×C4, C3×C6 [×7], C2×C12 [×24], C22×C6 [×4], C2×M4(2), C3×C12 [×4], C62 [×7], C3×M4(2) [×8], C22×C12 [×4], C6×C12 [×6], C2×C62, C6×M4(2) [×4], C32×M4(2) [×2], C2×C6×C12, M4(2)×C3×C6

Smallest permutation representation of M4(2)×C3×C6
On 144 points
Generators in S144
(1 97 89)(2 98 90)(3 99 91)(4 100 92)(5 101 93)(6 102 94)(7 103 95)(8 104 96)(9 34 140)(10 35 141)(11 36 142)(12 37 143)(13 38 144)(14 39 137)(15 40 138)(16 33 139)(17 123 115)(18 124 116)(19 125 117)(20 126 118)(21 127 119)(22 128 120)(23 121 113)(24 122 114)(25 51 43)(26 52 44)(27 53 45)(28 54 46)(29 55 47)(30 56 48)(31 49 41)(32 50 42)(57 76 65)(58 77 66)(59 78 67)(60 79 68)(61 80 69)(62 73 70)(63 74 71)(64 75 72)(81 109 135)(82 110 136)(83 111 129)(84 112 130)(85 105 131)(86 106 132)(87 107 133)(88 108 134)
(1 37 17 63 43 135)(2 38 18 64 44 136)(3 39 19 57 45 129)(4 40 20 58 46 130)(5 33 21 59 47 131)(6 34 22 60 48 132)(7 35 23 61 41 133)(8 36 24 62 42 134)(9 120 68 56 106 94)(10 113 69 49 107 95)(11 114 70 50 108 96)(12 115 71 51 109 89)(13 116 72 52 110 90)(14 117 65 53 111 91)(15 118 66 54 112 92)(16 119 67 55 105 93)(25 81 97 143 123 74)(26 82 98 144 124 75)(27 83 99 137 125 76)(28 84 100 138 126 77)(29 85 101 139 127 78)(30 86 102 140 128 79)(31 87 103 141 121 80)(32 88 104 142 122 73)
(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)(2 64)(3 61)(4 58)(5 63)(6 60)(7 57)(8 62)(9 56)(10 53)(11 50)(12 55)(13 52)(14 49)(15 54)(16 51)(17 131)(18 136)(19 133)(20 130)(21 135)(22 132)(23 129)(24 134)(25 139)(26 144)(27 141)(28 138)(29 143)(30 140)(31 137)(32 142)(33 43)(34 48)(35 45)(36 42)(37 47)(38 44)(39 41)(40 46)(65 95)(66 92)(67 89)(68 94)(69 91)(70 96)(71 93)(72 90)(73 104)(74 101)(75 98)(76 103)(77 100)(78 97)(79 102)(80 99)(81 127)(82 124)(83 121)(84 126)(85 123)(86 128)(87 125)(88 122)(105 115)(106 120)(107 117)(108 114)(109 119)(110 116)(111 113)(112 118)

G:=sub<Sym(144)| (1,97,89)(2,98,90)(3,99,91)(4,100,92)(5,101,93)(6,102,94)(7,103,95)(8,104,96)(9,34,140)(10,35,141)(11,36,142)(12,37,143)(13,38,144)(14,39,137)(15,40,138)(16,33,139)(17,123,115)(18,124,116)(19,125,117)(20,126,118)(21,127,119)(22,128,120)(23,121,113)(24,122,114)(25,51,43)(26,52,44)(27,53,45)(28,54,46)(29,55,47)(30,56,48)(31,49,41)(32,50,42)(57,76,65)(58,77,66)(59,78,67)(60,79,68)(61,80,69)(62,73,70)(63,74,71)(64,75,72)(81,109,135)(82,110,136)(83,111,129)(84,112,130)(85,105,131)(86,106,132)(87,107,133)(88,108,134), (1,37,17,63,43,135)(2,38,18,64,44,136)(3,39,19,57,45,129)(4,40,20,58,46,130)(5,33,21,59,47,131)(6,34,22,60,48,132)(7,35,23,61,41,133)(8,36,24,62,42,134)(9,120,68,56,106,94)(10,113,69,49,107,95)(11,114,70,50,108,96)(12,115,71,51,109,89)(13,116,72,52,110,90)(14,117,65,53,111,91)(15,118,66,54,112,92)(16,119,67,55,105,93)(25,81,97,143,123,74)(26,82,98,144,124,75)(27,83,99,137,125,76)(28,84,100,138,126,77)(29,85,101,139,127,78)(30,86,102,140,128,79)(31,87,103,141,121,80)(32,88,104,142,122,73), (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)(2,64)(3,61)(4,58)(5,63)(6,60)(7,57)(8,62)(9,56)(10,53)(11,50)(12,55)(13,52)(14,49)(15,54)(16,51)(17,131)(18,136)(19,133)(20,130)(21,135)(22,132)(23,129)(24,134)(25,139)(26,144)(27,141)(28,138)(29,143)(30,140)(31,137)(32,142)(33,43)(34,48)(35,45)(36,42)(37,47)(38,44)(39,41)(40,46)(65,95)(66,92)(67,89)(68,94)(69,91)(70,96)(71,93)(72,90)(73,104)(74,101)(75,98)(76,103)(77,100)(78,97)(79,102)(80,99)(81,127)(82,124)(83,121)(84,126)(85,123)(86,128)(87,125)(88,122)(105,115)(106,120)(107,117)(108,114)(109,119)(110,116)(111,113)(112,118)>;

G:=Group( (1,97,89)(2,98,90)(3,99,91)(4,100,92)(5,101,93)(6,102,94)(7,103,95)(8,104,96)(9,34,140)(10,35,141)(11,36,142)(12,37,143)(13,38,144)(14,39,137)(15,40,138)(16,33,139)(17,123,115)(18,124,116)(19,125,117)(20,126,118)(21,127,119)(22,128,120)(23,121,113)(24,122,114)(25,51,43)(26,52,44)(27,53,45)(28,54,46)(29,55,47)(30,56,48)(31,49,41)(32,50,42)(57,76,65)(58,77,66)(59,78,67)(60,79,68)(61,80,69)(62,73,70)(63,74,71)(64,75,72)(81,109,135)(82,110,136)(83,111,129)(84,112,130)(85,105,131)(86,106,132)(87,107,133)(88,108,134), (1,37,17,63,43,135)(2,38,18,64,44,136)(3,39,19,57,45,129)(4,40,20,58,46,130)(5,33,21,59,47,131)(6,34,22,60,48,132)(7,35,23,61,41,133)(8,36,24,62,42,134)(9,120,68,56,106,94)(10,113,69,49,107,95)(11,114,70,50,108,96)(12,115,71,51,109,89)(13,116,72,52,110,90)(14,117,65,53,111,91)(15,118,66,54,112,92)(16,119,67,55,105,93)(25,81,97,143,123,74)(26,82,98,144,124,75)(27,83,99,137,125,76)(28,84,100,138,126,77)(29,85,101,139,127,78)(30,86,102,140,128,79)(31,87,103,141,121,80)(32,88,104,142,122,73), (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)(2,64)(3,61)(4,58)(5,63)(6,60)(7,57)(8,62)(9,56)(10,53)(11,50)(12,55)(13,52)(14,49)(15,54)(16,51)(17,131)(18,136)(19,133)(20,130)(21,135)(22,132)(23,129)(24,134)(25,139)(26,144)(27,141)(28,138)(29,143)(30,140)(31,137)(32,142)(33,43)(34,48)(35,45)(36,42)(37,47)(38,44)(39,41)(40,46)(65,95)(66,92)(67,89)(68,94)(69,91)(70,96)(71,93)(72,90)(73,104)(74,101)(75,98)(76,103)(77,100)(78,97)(79,102)(80,99)(81,127)(82,124)(83,121)(84,126)(85,123)(86,128)(87,125)(88,122)(105,115)(106,120)(107,117)(108,114)(109,119)(110,116)(111,113)(112,118) );

G=PermutationGroup([(1,97,89),(2,98,90),(3,99,91),(4,100,92),(5,101,93),(6,102,94),(7,103,95),(8,104,96),(9,34,140),(10,35,141),(11,36,142),(12,37,143),(13,38,144),(14,39,137),(15,40,138),(16,33,139),(17,123,115),(18,124,116),(19,125,117),(20,126,118),(21,127,119),(22,128,120),(23,121,113),(24,122,114),(25,51,43),(26,52,44),(27,53,45),(28,54,46),(29,55,47),(30,56,48),(31,49,41),(32,50,42),(57,76,65),(58,77,66),(59,78,67),(60,79,68),(61,80,69),(62,73,70),(63,74,71),(64,75,72),(81,109,135),(82,110,136),(83,111,129),(84,112,130),(85,105,131),(86,106,132),(87,107,133),(88,108,134)], [(1,37,17,63,43,135),(2,38,18,64,44,136),(3,39,19,57,45,129),(4,40,20,58,46,130),(5,33,21,59,47,131),(6,34,22,60,48,132),(7,35,23,61,41,133),(8,36,24,62,42,134),(9,120,68,56,106,94),(10,113,69,49,107,95),(11,114,70,50,108,96),(12,115,71,51,109,89),(13,116,72,52,110,90),(14,117,65,53,111,91),(15,118,66,54,112,92),(16,119,67,55,105,93),(25,81,97,143,123,74),(26,82,98,144,124,75),(27,83,99,137,125,76),(28,84,100,138,126,77),(29,85,101,139,127,78),(30,86,102,140,128,79),(31,87,103,141,121,80),(32,88,104,142,122,73)], [(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),(2,64),(3,61),(4,58),(5,63),(6,60),(7,57),(8,62),(9,56),(10,53),(11,50),(12,55),(13,52),(14,49),(15,54),(16,51),(17,131),(18,136),(19,133),(20,130),(21,135),(22,132),(23,129),(24,134),(25,139),(26,144),(27,141),(28,138),(29,143),(30,140),(31,137),(32,142),(33,43),(34,48),(35,45),(36,42),(37,47),(38,44),(39,41),(40,46),(65,95),(66,92),(67,89),(68,94),(69,91),(70,96),(71,93),(72,90),(73,104),(74,101),(75,98),(76,103),(77,100),(78,97),(79,102),(80,99),(81,127),(82,124),(83,121),(84,126),(85,123),(86,128),(87,125),(88,122),(105,115),(106,120),(107,117),(108,114),(109,119),(110,116),(111,113),(112,118)])

180 conjugacy classes

class 1 2A2B2C2D2E3A···3H4A4B4C4D4E4F6A···6X6Y···6AN8A···8H12A···12AF12AG···12AV24A···24BL
order1222223···34444446···66···68···812···1212···1224···24
size1111221···11111221···12···22···21···12···22···2

180 irreducible representations

dim11111111111122
type++++
imageC1C2C2C2C3C4C4C6C6C6C12C12M4(2)C3×M4(2)
kernelM4(2)×C3×C6C6×C24C32×M4(2)C2×C6×C12C6×M4(2)C6×C12C2×C62C2×C24C3×M4(2)C22×C12C2×C12C22×C6C3×C6C6
# reps1241862163284816432

Matrix representation of M4(2)×C3×C6 in GL3(𝔽73) generated by

800
0640
0064
,
900
010
001
,
7200
05471
04819
,
100
0720
0191
G:=sub<GL(3,GF(73))| [8,0,0,0,64,0,0,0,64],[9,0,0,0,1,0,0,0,1],[72,0,0,0,54,48,0,71,19],[1,0,0,0,72,19,0,0,1] >;

M4(2)×C3×C6 in GAP, Magma, Sage, TeX

M_4(2)\times C_3\times C_6
% in TeX

G:=Group("M4(2)xC3xC6");
// GroupNames label

G:=SmallGroup(288,827);
// by ID

G=gap.SmallGroup(288,827);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,504,2045,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

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