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G = C4×C3⋊Dic3order 144 = 24·32

Direct product of C4 and C3⋊Dic3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C4×C3⋊Dic3, C122Dic3, C324C42, C62.21C22, (C3×C12)⋊6C4, C6.13(C4×S3), C32(C4×Dic3), (C2×C6).30D6, (C6×C12).13C2, (C2×C12).17S3, C6.14(C2×Dic3), C2.2(C4×C3⋊S3), (C2×C4).6(C3⋊S3), (C3×C6).24(C2×C4), C22.3(C2×C3⋊S3), C2.2(C2×C3⋊Dic3), (C2×C3⋊Dic3).10C2, SmallGroup(144,92)

Series: Derived Chief Lower central Upper central

C1C32 — C4×C3⋊Dic3
C1C3C32C3×C6C62C2×C3⋊Dic3 — C4×C3⋊Dic3
C32 — C4×C3⋊Dic3
C1C2×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)]])

C4×C3⋊Dic3 is a maximal subgroup of
C6.(S3×C8)  C2.Dic32  D124Dic3  C12.15Dic6  C12.30Dic6  C24⋊Dic3  C62.37D4  C62.39D4  (C3×C12)⋊4C8  C322C8⋊C4  C325(C4⋊C8)  C62.8C23  C62.13C23  C62.25C23  C62.32C23  C62.33C23  C62.40C23  C62.42C23  C62.43C23  C4×S3×Dic3  C62.48C23  D12⋊Dic3  C62.72C23  C62.84C23  C62.85C23  C12⋊Dic6  C42×C3⋊S3  C12216C2  C62.221C23  C62.223C23  C62.225C23  C62.229C23  C62.231C23  C122Dic6  C62.233C23  C62.234C23  C62.236C23  C62.237C23  C62.242C23  C62.247C23  C62.254C23  C62.258C23  C62.259C23  C62.262C23
C4×C3⋊Dic3 is a maximal quotient of
C122.C2  C24⋊Dic3  C62.15Q8

48 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E···4L6A···6L12A···12P
order1222333344444···46···612···12
size1111222211119···92···22···2

48 irreducible representations

dim111112222
type++++-+
imageC1C2C2C4C4S3Dic3D6C4×S3
kernelC4×C3⋊Dic3C2×C3⋊Dic3C6×C12C3⋊Dic3C3×C12C2×C12C12C2×C6C6
# reps1218448416

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

5000
0500
00120
00012
,
12100
12000
0010
0001
,
0100
12100
00112
0010
,
9200
11400
00411
0029
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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