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## G = C2×C32⋊4C8order 144 = 24·32

### Direct product of C2 and C32⋊4C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C32⋊4C8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C2×C32⋊4C8
 Lower central C32 — C2×C32⋊4C8
 Upper central C1 — C2×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, C3, C4, C22, C6, C8, C2×C4, C32, C12, C2×C6, C2×C8, C3×C6, C3×C6, C3⋊C8, C2×C12, C3×C12, C62, C2×C3⋊C8, C324C8, C6×C12, C2×C324C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, C3⋊S3, C3⋊C8, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C2×C3⋊C8, C324C8, 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 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 81)(24 82)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)(33 112)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(49 127)(50 128)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(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 62 136)(18 129 63)(19 64 130)(20 131 57)(21 58 132)(22 133 59)(23 60 134)(24 135 61)(25 110 50)(26 51 111)(27 112 52)(28 53 105)(29 106 54)(30 55 107)(31 108 56)(32 49 109)(33 122 47)(34 48 123)(35 124 41)(36 42 125)(37 126 43)(38 44 127)(39 128 45)(40 46 121)(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)(9 122 73)(10 74 123)(11 124 75)(12 76 125)(13 126 77)(14 78 127)(15 128 79)(16 80 121)(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 72 88)(42 81 65)(43 66 82)(44 83 67)(45 68 84)(46 85 69)(47 70 86)(48 87 71)(49 62 120)(50 113 63)(51 64 114)(52 115 57)(53 58 116)(54 117 59)(55 60 118)(56 119 61)(89 133 106)(90 107 134)(91 135 108)(92 109 136)(93 129 110)(94 111 130)(95 131 112)(96 105 132)
(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)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,81)(24,82)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(33,112)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(49,127)(50,128)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(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)(17,62,136)(18,129,63)(19,64,130)(20,131,57)(21,58,132)(22,133,59)(23,60,134)(24,135,61)(25,110,50)(26,51,111)(27,112,52)(28,53,105)(29,106,54)(30,55,107)(31,108,56)(32,49,109)(33,122,47)(34,48,123)(35,124,41)(36,42,125)(37,126,43)(38,44,127)(39,128,45)(40,46,121)(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)(9,122,73)(10,74,123)(11,124,75)(12,76,125)(13,126,77)(14,78,127)(15,128,79)(16,80,121)(33,137,98)(34,99,138)(35,139,100)(36,101,140)(37,141,102)(38,103,142)(39,143,104)(40,97,144)(41,72,88)(42,81,65)(43,66,82)(44,83,67)(45,68,84)(46,85,69)(47,70,86)(48,87,71)(49,62,120)(50,113,63)(51,64,114)(52,115,57)(53,58,116)(54,117,59)(55,60,118)(56,119,61)(89,133,106)(90,107,134)(91,135,108)(92,109,136)(93,129,110)(94,111,130)(95,131,112)(96,105,132), (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)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,81)(24,82)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(33,112)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(49,127)(50,128)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(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)(17,62,136)(18,129,63)(19,64,130)(20,131,57)(21,58,132)(22,133,59)(23,60,134)(24,135,61)(25,110,50)(26,51,111)(27,112,52)(28,53,105)(29,106,54)(30,55,107)(31,108,56)(32,49,109)(33,122,47)(34,48,123)(35,124,41)(36,42,125)(37,126,43)(38,44,127)(39,128,45)(40,46,121)(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)(9,122,73)(10,74,123)(11,124,75)(12,76,125)(13,126,77)(14,78,127)(15,128,79)(16,80,121)(33,137,98)(34,99,138)(35,139,100)(36,101,140)(37,141,102)(38,103,142)(39,143,104)(40,97,144)(41,72,88)(42,81,65)(43,66,82)(44,83,67)(45,68,84)(46,85,69)(47,70,86)(48,87,71)(49,62,120)(50,113,63)(51,64,114)(52,115,57)(53,58,116)(54,117,59)(55,60,118)(56,119,61)(89,133,106)(90,107,134)(91,135,108)(92,109,136)(93,129,110)(94,111,130)(95,131,112)(96,105,132), (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),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,83),(18,84),(19,85),(20,86),(21,87),(22,88),(23,81),(24,82),(25,45),(26,46),(27,47),(28,48),(29,41),(30,42),(31,43),(32,44),(33,112),(34,105),(35,106),(36,107),(37,108),(38,109),(39,110),(40,111),(49,127),(50,128),(51,121),(52,122),(53,123),(54,124),(55,125),(56,126),(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),(17,62,136),(18,129,63),(19,64,130),(20,131,57),(21,58,132),(22,133,59),(23,60,134),(24,135,61),(25,110,50),(26,51,111),(27,112,52),(28,53,105),(29,106,54),(30,55,107),(31,108,56),(32,49,109),(33,122,47),(34,48,123),(35,124,41),(36,42,125),(37,126,43),(38,44,127),(39,128,45),(40,46,121),(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),(9,122,73),(10,74,123),(11,124,75),(12,76,125),(13,126,77),(14,78,127),(15,128,79),(16,80,121),(33,137,98),(34,99,138),(35,139,100),(36,101,140),(37,141,102),(38,103,142),(39,143,104),(40,97,144),(41,72,88),(42,81,65),(43,66,82),(44,83,67),(45,68,84),(46,85,69),(47,70,86),(48,87,71),(49,62,120),(50,113,63),(51,64,114),(52,115,57),(53,58,116),(54,117,59),(55,60,118),(56,119,61),(89,133,106),(90,107,134),(91,135,108),(92,109,136),(93,129,110),(94,111,130),(95,131,112),(96,105,132)], [(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)]])

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6L 8A ··· 8H 12A ··· 12P order 1 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 2 2 2 2 1 1 1 1 2 ··· 2 9 ··· 9 2 ··· 2

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 S3 Dic3 D6 Dic3 C3⋊C8 kernel C2×C32⋊4C8 C32⋊4C8 C6×C12 C3×C12 C62 C3×C6 C2×C12 C12 C12 C2×C6 C6 # reps 1 2 1 2 2 8 4 4 4 4 16

Matrix representation of C2×C324C8 in GL5(𝔽73)

 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 72 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 72 0
,
 1 0 0 0 0 0 0 1 0 0 0 72 72 0 0 0 0 0 1 0 0 0 0 0 1
,
 51 0 0 0 0 0 39 47 0 0 0 8 34 0 0 0 0 0 46 27 0 0 0 0 27

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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