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## G = C6×C32⋊4Q8order 432 = 24·33

### Direct product of C6 and C32⋊4Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C6×C32⋊4Q8
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C3×C3⋊Dic3 — C6×C3⋊Dic3 — C6×C32⋊4Q8
 Lower central C32 — C3×C6 — C6×C32⋊4Q8
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C6×C324Q8
G = < a,b,c,d,e | a6=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: 708 in 292 conjugacy classes, 118 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C32, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C2×Q8, C3×C6, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C62, C62, C62, C2×Dic6, C6×Q8, C32×C6, C32×C6, C3×Dic6, C6×Dic3, C324Q8, C2×C3⋊Dic3, C6×C12, C6×C12, C6×C12, C3×C3⋊Dic3, C32×C12, C3×C62, C6×Dic6, C2×C324Q8, C3×C324Q8, C6×C3⋊Dic3, C3×C6×C12, C6×C324Q8
Quotients:

Smallest permutation representation of C6×C324Q8
On 144 points
Generators in S144
(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 14 49)(2 15 50)(3 16 51)(4 17 52)(5 18 53)(6 13 54)(7 100 131)(8 101 132)(9 102 127)(10 97 128)(11 98 129)(12 99 130)(19 31 72)(20 32 67)(21 33 68)(22 34 69)(23 35 70)(24 36 71)(25 75 56)(26 76 57)(27 77 58)(28 78 59)(29 73 60)(30 74 55)(37 116 44)(38 117 45)(39 118 46)(40 119 47)(41 120 48)(42 115 43)(61 112 81)(62 113 82)(63 114 83)(64 109 84)(65 110 79)(66 111 80)(85 137 126)(86 138 121)(87 133 122)(88 134 123)(89 135 124)(90 136 125)(91 144 103)(92 139 104)(93 140 105)(94 141 106)(95 142 107)(96 143 108)
(1 16 53)(2 17 54)(3 18 49)(4 13 50)(5 14 51)(6 15 52)(7 98 127)(8 99 128)(9 100 129)(10 101 130)(11 102 131)(12 97 132)(19 33 70)(20 34 71)(21 35 72)(22 36 67)(23 31 68)(24 32 69)(25 77 60)(26 78 55)(27 73 56)(28 74 57)(29 75 58)(30 76 59)(37 120 46)(38 115 47)(39 116 48)(40 117 43)(41 118 44)(42 119 45)(61 114 79)(62 109 80)(63 110 81)(64 111 82)(65 112 83)(66 113 84)(85 135 122)(86 136 123)(87 137 124)(88 138 125)(89 133 126)(90 134 121)(91 142 105)(92 143 106)(93 144 107)(94 139 108)(95 140 103)(96 141 104)
(1 64 20 74)(2 65 21 75)(3 66 22 76)(4 61 23 77)(5 62 24 78)(6 63 19 73)(7 96 115 136)(8 91 116 137)(9 92 117 138)(10 93 118 133)(11 94 119 134)(12 95 120 135)(13 114 31 60)(14 109 32 55)(15 110 33 56)(16 111 34 57)(17 112 35 58)(18 113 36 59)(25 50 79 68)(26 51 80 69)(27 52 81 70)(28 53 82 71)(29 54 83 72)(30 49 84 67)(37 85 132 103)(38 86 127 104)(39 87 128 105)(40 88 129 106)(41 89 130 107)(42 90 131 108)(43 125 100 143)(44 126 101 144)(45 121 102 139)(46 122 97 140)(47 123 98 141)(48 124 99 142)
(1 95 20 135)(2 96 21 136)(3 91 22 137)(4 92 23 138)(5 93 24 133)(6 94 19 134)(7 75 115 65)(8 76 116 66)(9 77 117 61)(10 78 118 62)(11 73 119 63)(12 74 120 64)(13 106 31 88)(14 107 32 89)(15 108 33 90)(16 103 34 85)(17 104 35 86)(18 105 36 87)(25 43 79 100)(26 44 80 101)(27 45 81 102)(28 46 82 97)(29 47 83 98)(30 48 84 99)(37 111 132 57)(38 112 127 58)(39 113 128 59)(40 114 129 60)(41 109 130 55)(42 110 131 56)(49 142 67 124)(50 143 68 125)(51 144 69 126)(52 139 70 121)(53 140 71 122)(54 141 72 123)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3N 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6AP 12A ··· 12AZ 12BA ··· 12BH order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 1 1 2 ··· 2 2 2 18 18 18 18 1 ··· 1 2 ··· 2 2 ··· 2 18 ··· 18

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C2 C3 C6 C6 C6 S3 Q8 D6 D6 C3×S3 Dic6 C3×Q8 S3×C6 S3×C6 C3×Dic6 kernel C6×C32⋊4Q8 C3×C32⋊4Q8 C6×C3⋊Dic3 C3×C6×C12 C2×C32⋊4Q8 C32⋊4Q8 C2×C3⋊Dic3 C6×C12 C6×C12 C32×C6 C3×C12 C62 C2×C12 C3×C6 C3×C6 C12 C2×C6 C6 # reps 1 4 2 1 2 8 4 2 4 2 8 4 8 16 4 16 8 32

Matrix representation of C6×C324Q8 in GL4(𝔽13) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 12 1 0 0 12 0
,
 9 0 0 0 7 3 0 0 0 0 12 1 0 0 12 0
,
 8 0 0 0 10 5 0 0 0 0 10 6 0 0 7 3
,
 8 8 0 0 0 5 0 0 0 0 5 8 0 0 0 8
G:=sub<GL(4,GF(13))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,1,0],[9,7,0,0,0,3,0,0,0,0,12,12,0,0,1,0],[8,10,0,0,0,5,0,0,0,0,10,7,0,0,6,3],[8,0,0,0,8,5,0,0,0,0,5,0,0,0,8,8] >;

C6×C324Q8 in GAP, Magma, Sage, TeX

C_6\times C_3^2\rtimes_4Q_8
% in TeX

G:=Group("C6xC3^2:4Q8");
// GroupNames label

G:=SmallGroup(432,710);
// by ID

G=gap.SmallGroup(432,710);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,590,142,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=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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𝔽