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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 [×2], C3 [×4], C4 [×2], C4 [×4], C22, C6 [×12], C2×C4, C2×C4 [×2], C32, Dic3 [×16], C12 [×8], C2×C6 [×4], C42, C3×C6, C3×C6 [×2], C2×Dic3 [×8], C2×C12 [×4], C3⋊Dic3 [×4], C3×C12 [×2], C62, C4×Dic3 [×4], C2×C3⋊Dic3 [×2], C6×C12, C4×C3⋊Dic3
Quotients: C1, C2 [×3], C4 [×6], C22, S3 [×4], C2×C4 [×3], Dic3 [×8], D6 [×4], C42, C3⋊S3, C4×S3 [×8], C2×Dic3 [×4], C3⋊Dic3 [×2], C2×C3⋊S3, C4×Dic3 [×4], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C4×C3⋊Dic3

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

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

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

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

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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