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## G = C4○D4×C3×C6order 288 = 25·32

### Direct product of C3×C6 and C4○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C3×C6
 Chief series C1 — C2 — C6 — C3×C6 — C62 — D4×C32 — C32×C4○D4 — C4○D4×C3×C6
 Lower central C1 — C2 — C4○D4×C3×C6
 Upper central C1 — C6×C12 — C4○D4×C3×C6

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

Smallest permutation representation of 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

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