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## G = C2×C32⋊4Q8order 144 = 24·32

### Direct product of C2 and C32⋊4Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C32⋊4Q8
 Chief series C1 — C3 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C2×C32⋊4Q8
 Lower central C32 — C3×C6 — C2×C32⋊4Q8
 Upper central C1 — C22 — C2×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)])

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 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 2 2 18 18 18 18 2 ··· 2 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C2 S3 Q8 D6 D6 Dic6 kernel C2×C32⋊4Q8 C32⋊4Q8 C2×C3⋊Dic3 C6×C12 C2×C12 C3×C6 C12 C2×C6 C6 # reps 1 4 2 1 4 2 8 4 16

Matrix representation of C2×C324Q8 in GL6(𝔽13)

 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 1 0 0 0 0 12 0
,
 0 1 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 3 6 0 0 0 0 7 10 0 0 0 0 0 0 3 6 0 0 0 0 7 10 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 4 2 0 0 0 0 11 9 0 0 0 0 0 0 11 2 0 0 0 0 4 2 0 0 0 0 0 0 10 6 0 0 0 0 3 3

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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