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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — M4(2)×C3×C6
 Chief series C1 — C2 — C4 — C12 — C3×C12 — C3×C24 — C32×M4(2) — M4(2)×C3×C6
 Lower central C1 — C2 — M4(2)×C3×C6
 Upper central C1 — C6×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, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C32, C12, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C3×C6, C3×C6, C3×C6, C24, C2×C12, C22×C6, C2×M4(2), C3×C12, C3×C12, C62, C62, C62, C2×C24, C3×M4(2), C22×C12, C3×C24, C6×C12, C6×C12, C2×C62, C6×M4(2), C6×C24, C32×M4(2), C2×C6×C12, M4(2)×C3×C6
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C32, C12, C2×C6, M4(2), C22×C4, C3×C6, C2×C12, C22×C6, C2×M4(2), C3×C12, C62, C3×M4(2), C22×C12, C6×C12, C2×C62, C6×M4(2), C32×M4(2), C2×C6×C12, M4(2)×C3×C6

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

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3H 4A 4B 4C 4D 4E 4F 6A ··· 6X 6Y ··· 6AN 8A ··· 8H 12A ··· 12AF 12AG ··· 12AV 24A ··· 24BL order 1 2 2 2 2 2 3 ··· 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 1 ··· 1 1 1 1 1 2 2 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 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 C3 C4 C4 C6 C6 C6 C12 C12 M4(2) C3×M4(2) kernel M4(2)×C3×C6 C6×C24 C32×M4(2) C2×C6×C12 C6×M4(2) C6×C12 C2×C62 C2×C24 C3×M4(2) C22×C12 C2×C12 C22×C6 C3×C6 C6 # reps 1 2 4 1 8 6 2 16 32 8 48 16 4 32

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

 8 0 0 0 64 0 0 0 64
,
 9 0 0 0 1 0 0 0 1
,
 72 0 0 0 54 71 0 48 19
,
 1 0 0 0 72 0 0 19 1
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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