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G = C2×C324C8order 144 = 24·32

Direct product of C2 and C324C8

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

Aliases: C2×C324C8, C12.57D6, C62.6C4, C12.8Dic3, C6⋊(C3⋊C8), (C3×C6)⋊4C8, C329(C2×C8), (C3×C12).9C4, (C6×C12).12C2, (C2×C12).16S3, (C2×C6).7Dic3, C4.3(C3⋊Dic3), C6.12(C2×Dic3), (C3×C12).48C22, C22.2(C3⋊Dic3), C32(C2×C3⋊C8), C4.14(C2×C3⋊S3), (C2×C4).5(C3⋊S3), (C3×C6).31(C2×C4), C2.1(C2×C3⋊Dic3), SmallGroup(144,90)

Series: Derived Chief Lower central Upper central

C1C32 — C2×C324C8
C1C3C32C3×C6C3×C12C324C8 — C2×C324C8
C32 — C2×C324C8
C1C2×C4

Generators and relations for C2×C324C8
 G = < a,b,c,d | a2=b3=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 114 in 66 conjugacy classes, 51 normal (13 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], C2×C8, C3×C6, C3×C6 [×2], C3⋊C8 [×8], C2×C12 [×4], C3×C12 [×2], C62, C2×C3⋊C8 [×4], C324C8 [×2], C6×C12, C2×C324C8
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C8 [×2], C2×C4, Dic3 [×8], D6 [×4], C2×C8, C3⋊S3, C3⋊C8 [×8], C2×Dic3 [×4], C3⋊Dic3 [×2], C2×C3⋊S3, C2×C3⋊C8 [×4], C324C8 [×2], C2×C3⋊Dic3, C2×C324C8

Smallest permutation representation of C2×C324C8
Regular action on 144 points
Generators in S144
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 65)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 81)(24 82)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 121)(49 109)(50 110)(51 111)(52 112)(53 105)(54 106)(55 107)(56 108)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 113)(80 114)(89 139)(90 140)(91 141)(92 142)(93 143)(94 144)(95 137)(96 138)(97 130)(98 131)(99 132)(100 133)(101 134)(102 135)(103 136)(104 129)
(1 91 119)(2 120 92)(3 93 113)(4 114 94)(5 95 115)(6 116 96)(7 89 117)(8 118 90)(9 86 98)(10 99 87)(11 88 100)(12 101 81)(13 82 102)(14 103 83)(15 84 104)(16 97 85)(17 44 136)(18 129 45)(19 46 130)(20 131 47)(21 48 132)(22 133 41)(23 42 134)(24 135 43)(25 128 50)(26 51 121)(27 122 52)(28 53 123)(29 124 54)(30 55 125)(31 126 56)(32 49 127)(33 112 57)(34 58 105)(35 106 59)(36 60 107)(37 108 61)(38 62 109)(39 110 63)(40 64 111)(65 76 140)(66 141 77)(67 78 142)(68 143 79)(69 80 144)(70 137 73)(71 74 138)(72 139 75)
(1 24 31)(2 32 17)(3 18 25)(4 26 19)(5 20 27)(6 28 21)(7 22 29)(8 30 23)(9 112 73)(10 74 105)(11 106 75)(12 76 107)(13 108 77)(14 78 109)(15 110 79)(16 80 111)(33 137 98)(34 99 138)(35 139 100)(36 101 140)(37 141 102)(38 103 142)(39 143 104)(40 97 144)(41 54 117)(42 118 55)(43 56 119)(44 120 49)(45 50 113)(46 114 51)(47 52 115)(48 116 53)(57 70 86)(58 87 71)(59 72 88)(60 81 65)(61 66 82)(62 83 67)(63 68 84)(64 85 69)(89 133 124)(90 125 134)(91 135 126)(92 127 136)(93 129 128)(94 121 130)(95 131 122)(96 123 132)
(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)

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

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

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

C2×C324C8 is a maximal subgroup of
Dic3×C3⋊C8  C3⋊C8⋊Dic3  C12.77D12  D123Dic3  Dic6⋊Dic3  C12.81D12  C12.6Dic6  C12.8Dic6  C62.5Q8  C122.C2  C12.57D12  C12.9Dic6  C12.10Dic6  C62.113D4  C62.114D4  C8×C3⋊Dic3  C12.30Dic6  C24⋊Dic3  C12.60D12  C62.8Q8  C627C8  C62.116D4  C62.117D4  C62.4C8  C2×S3×C3⋊C8  D12.Dic3  D12.30D6  C2×C8×C3⋊S3  C24.47D6  D4.(C3⋊Dic3)  C62.74D4
C2×C324C8 is a maximal quotient of
C12.57D12  C24.94D6  C627C8

48 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D6A···6L8A···8H12A···12P
order1222333344446···68···812···12
size1111222211112···29···92···2

48 irreducible representations

dim11111122222
type++++-+-
imageC1C2C2C4C4C8S3Dic3D6Dic3C3⋊C8
kernelC2×C324C8C324C8C6×C12C3×C12C62C3×C6C2×C12C12C12C2×C6C6
# reps121228444416

Matrix representation of C2×C324C8 in GL5(𝔽73)

10000
01000
00100
000720
000072
,
10000
0727200
01000
000721
000720
,
10000
00100
0727200
00010
00001
,
510000
0394700
083400
0004627
000027

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,72,1,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,1,0],[1,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,0,1,0,0,0,0,0,1],[51,0,0,0,0,0,39,8,0,0,0,47,34,0,0,0,0,0,46,0,0,0,0,27,27] >;

C2×C324C8 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_4C_8
% in TeX

G:=Group("C2xC3^2:4C8");
// GroupNames label

G:=SmallGroup(144,90);
// by ID

G=gap.SmallGroup(144,90);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,50,964,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^3=d^8=1,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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