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G = C2×C324Q8order 144 = 24·32

Direct product of C2 and C324Q8

direct product, metabelian, supersoluble, monomial

Aliases: C2×C324Q8, C62Dic6, C12.49D6, C62.31C22, (C3×C6)⋊4Q8, C326(C2×Q8), (C2×C12).9S3, (C6×C12).5C2, C33(C2×Dic6), (C2×C6).36D6, C6.30(C22×S3), (C3×C6).29C23, (C3×C12).35C22, C3⋊Dic3.16C22, C4.11(C2×C3⋊S3), (C2×C4).4(C3⋊S3), C22.8(C2×C3⋊S3), C2.3(C22×C3⋊S3), (C2×C3⋊Dic3).9C2, SmallGroup(144,168)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×C324Q8
C1C3C32C3×C6C3⋊Dic3C2×C3⋊Dic3 — C2×C324Q8
C32C3×C6 — C2×C324Q8
C1C22C2×C4

Generators and relations for C2×C324Q8
 G = < a,b,c,d,e | a2=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 290 in 114 conjugacy classes, 59 normal (9 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×4], C22, C6 [×12], C2×C4, C2×C4 [×2], Q8 [×4], C32, Dic3 [×16], C12 [×8], C2×C6 [×4], C2×Q8, C3×C6, C3×C6 [×2], Dic6 [×16], C2×Dic3 [×8], C2×C12 [×4], C3⋊Dic3 [×4], C3×C12 [×2], C62, C2×Dic6 [×4], C324Q8 [×4], C2×C3⋊Dic3 [×2], C6×C12, C2×C324Q8
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], Q8 [×2], C23, D6 [×12], C2×Q8, C3⋊S3, Dic6 [×8], C22×S3 [×4], C2×C3⋊S3 [×3], C2×Dic6 [×4], C324Q8 [×2], C22×C3⋊S3, C2×C324Q8

Smallest permutation representation of C2×C324Q8
Regular action on 144 points
Generators in S144
(1 76)(2 73)(3 74)(4 75)(5 68)(6 65)(7 66)(8 67)(9 121)(10 122)(11 123)(12 124)(13 62)(14 63)(15 64)(16 61)(17 111)(18 112)(19 109)(20 110)(21 47)(22 48)(23 45)(24 46)(25 132)(26 129)(27 130)(28 131)(29 55)(30 56)(31 53)(32 54)(33 50)(34 51)(35 52)(36 49)(37 70)(38 71)(39 72)(40 69)(41 86)(42 87)(43 88)(44 85)(57 93)(58 94)(59 95)(60 96)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(89 125)(90 126)(91 127)(92 128)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)
(1 16 33)(2 13 34)(3 14 35)(4 15 36)(5 44 29)(6 41 30)(7 42 31)(8 43 32)(9 138 118)(10 139 119)(11 140 120)(12 137 117)(17 114 134)(18 115 135)(19 116 136)(20 113 133)(21 58 40)(22 59 37)(23 60 38)(24 57 39)(25 128 143)(26 125 144)(27 126 141)(28 127 142)(45 96 71)(46 93 72)(47 94 69)(48 95 70)(49 75 64)(50 76 61)(51 73 62)(52 74 63)(53 66 87)(54 67 88)(55 68 85)(56 65 86)(77 97 110)(78 98 111)(79 99 112)(80 100 109)(81 124 101)(82 121 102)(83 122 103)(84 123 104)(89 108 129)(90 105 130)(91 106 131)(92 107 132)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 16 58)(6 13 59)(7 14 60)(8 15 57)(9 141 115)(10 142 116)(11 143 113)(12 144 114)(17 117 125)(18 118 126)(19 119 127)(20 120 128)(25 133 140)(26 134 137)(27 135 138)(28 136 139)(33 40 44)(34 37 41)(35 38 42)(36 39 43)(45 53 74)(46 54 75)(47 55 76)(48 56 73)(49 72 88)(50 69 85)(51 70 86)(52 71 87)(61 94 68)(62 95 65)(63 96 66)(64 93 67)(77 123 107)(78 124 108)(79 121 105)(80 122 106)(81 89 111)(82 90 112)(83 91 109)(84 92 110)(97 104 132)(98 101 129)(99 102 130)(100 103 131)
(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)
(1 112 3 110)(2 111 4 109)(5 102 7 104)(6 101 8 103)(9 87 11 85)(10 86 12 88)(13 98 15 100)(14 97 16 99)(17 75 19 73)(18 74 20 76)(21 90 23 92)(22 89 24 91)(25 94 27 96)(26 93 28 95)(29 82 31 84)(30 81 32 83)(33 79 35 77)(34 78 36 80)(37 108 39 106)(38 107 40 105)(41 124 43 122)(42 123 44 121)(45 128 47 126)(46 127 48 125)(49 116 51 114)(50 115 52 113)(53 120 55 118)(54 119 56 117)(57 131 59 129)(58 130 60 132)(61 135 63 133)(62 134 64 136)(65 137 67 139)(66 140 68 138)(69 141 71 143)(70 144 72 142)

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

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

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

C2×C324Q8 is a maximal subgroup of
C12.71D12  C12.73D12  C62.114D4  C6.4Dic12  C12.20D12  C3⋊Dic3.D4  D12.29D6  C62.10C23  Dic36Dic6  D67Dic6  C12.27D12  Dic3⋊Dic6  D63Dic6  C62.83C23  C126Dic6  C1226C2  C626Q8  C62.229C23  C62.231C23  C122Dic6  C62.240C23  C12.31D12  C24.5D6  C6210Q8  C62.254C23  C62.259C23  C62.75D4  C2×S3×Dic6  D12.34D6  C2×Q8×C3⋊S3  C3292- 1+4
C2×C324Q8 is a maximal quotient of
C126Dic6  C12.25Dic6  C626Q8  C122Dic6  C62.234C23  C6210Q8

42 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E4F6A···6L12A···12P
order122233334444446···612···12
size1111222222181818182···22···2

42 irreducible representations

dim111122222
type+++++-++-
imageC1C2C2C2S3Q8D6D6Dic6
kernelC2×C324Q8C324Q8C2×C3⋊Dic3C6×C12C2×C12C3×C6C12C2×C6C6
# reps1421428416

Matrix representation of C2×C324Q8 in GL6(𝔽13)

1200000
0120000
001000
000100
000010
000001
,
100000
010000
001000
000100
0000121
0000120
,
010000
12120000
000100
00121200
000010
000001
,
360000
7100000
003600
0071000
0000120
0000012
,
420000
1190000
0011200
004200
0000106
000033

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,7,0,0,0,0,6,10,0,0,0,0,0,0,3,7,0,0,0,0,6,10,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[4,11,0,0,0,0,2,9,0,0,0,0,0,0,11,4,0,0,0,0,2,2,0,0,0,0,0,0,10,3,0,0,0,0,6,3] >;

C2×C324Q8 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_4Q_8
% in TeX

G:=Group("C2xC3^2:4Q8");
// GroupNames label

G:=SmallGroup(144,168);
// by ID

G=gap.SmallGroup(144,168);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,218,50,964,3461]);
// Polycyclic

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

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×
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