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## G = C4×C3⋊Dic3order 144 = 24·32

### Direct product of C4 and C3⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C4×C3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C4×C3⋊Dic3
 Lower central C32 — C4×C3⋊Dic3
 Upper central C1 — C2×C4

Generators and relations for C4×C3⋊Dic3
G = < a,b,c,d | a4=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 202 in 90 conjugacy classes, 55 normal (9 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C42, C3×C6, C3×C6, C2×Dic3, C2×C12, C3⋊Dic3, C3×C12, C62, C4×Dic3, C2×C3⋊Dic3, C6×C12, C4×C3⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, C3⋊S3, C4×S3, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C4×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C4×C3⋊Dic3

Smallest permutation representation of C4×C3⋊Dic3
Regular action on 144 points
Generators in S144
(1 69 33 50)(2 70 34 51)(3 71 35 52)(4 72 36 53)(5 67 31 54)(6 68 32 49)(7 131 27 134)(8 132 28 135)(9 127 29 136)(10 128 30 137)(11 129 25 138)(12 130 26 133)(13 95 57 66)(14 96 58 61)(15 91 59 62)(16 92 60 63)(17 93 55 64)(18 94 56 65)(19 110 116 80)(20 111 117 81)(21 112 118 82)(22 113 119 83)(23 114 120 84)(24 109 115 79)(37 97 43 75)(38 98 44 76)(39 99 45 77)(40 100 46 78)(41 101 47 73)(42 102 48 74)(85 144 108 121)(86 139 103 122)(87 140 104 123)(88 141 105 124)(89 142 106 125)(90 143 107 126)
(1 41 60)(2 42 55)(3 37 56)(4 38 57)(5 39 58)(6 40 59)(7 125 21)(8 126 22)(9 121 23)(10 122 24)(11 123 19)(12 124 20)(13 36 44)(14 31 45)(15 32 46)(16 33 47)(17 34 48)(18 35 43)(25 140 116)(26 141 117)(27 142 118)(28 143 119)(29 144 120)(30 139 115)(49 78 91)(50 73 92)(51 74 93)(52 75 94)(53 76 95)(54 77 96)(61 67 99)(62 68 100)(63 69 101)(64 70 102)(65 71 97)(66 72 98)(79 137 103)(80 138 104)(81 133 105)(82 134 106)(83 135 107)(84 136 108)(85 114 127)(86 109 128)(87 110 129)(88 111 130)(89 112 131)(90 113 132)
(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 85 4 88)(2 90 5 87)(3 89 6 86)(7 78 10 75)(8 77 11 74)(9 76 12 73)(13 81 16 84)(14 80 17 83)(15 79 18 82)(19 93 22 96)(20 92 23 95)(21 91 24 94)(25 102 28 99)(26 101 29 98)(27 100 30 97)(31 104 34 107)(32 103 35 106)(33 108 36 105)(37 131 40 128)(38 130 41 127)(39 129 42 132)(43 134 46 137)(44 133 47 136)(45 138 48 135)(49 122 52 125)(50 121 53 124)(51 126 54 123)(55 113 58 110)(56 112 59 109)(57 111 60 114)(61 116 64 119)(62 115 65 118)(63 120 66 117)(67 140 70 143)(68 139 71 142)(69 144 72 141)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E ··· 4L 6A ··· 6L 12A ··· 12P order 1 2 2 2 3 3 3 3 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 1 1 1 1 9 ··· 9 2 ··· 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 S3 Dic3 D6 C4×S3 kernel C4×C3⋊Dic3 C2×C3⋊Dic3 C6×C12 C3⋊Dic3 C3×C12 C2×C12 C12 C2×C6 C6 # reps 1 2 1 8 4 4 8 4 16

Matrix representation of C4×C3⋊Dic3 in GL4(𝔽13) generated by

 5 0 0 0 0 5 0 0 0 0 12 0 0 0 0 12
,
 12 1 0 0 12 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 12 1 0 0 0 0 1 12 0 0 1 0
,
 9 2 0 0 11 4 0 0 0 0 4 11 0 0 2 9
G:=sub<GL(4,GF(13))| [5,0,0,0,0,5,0,0,0,0,12,0,0,0,0,12],[12,12,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,1,0,0,0,0,1,1,0,0,12,0],[9,11,0,0,2,4,0,0,0,0,4,2,0,0,11,9] >;

C4×C3⋊Dic3 in GAP, Magma, Sage, TeX

C_4\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("C4xC3:Dic3");
// GroupNames label

G:=SmallGroup(144,92);
// by ID

G=gap.SmallGroup(144,92);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,55,964,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^3=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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