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## G = C2×Q8×3- 1+2order 432 = 24·33

### Direct product of C2, Q8 and 3- 1+2

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×Q8×3- 1+2
 Chief series C1 — C3 — C6 — C3×C6 — C2×3- 1+2 — C4×3- 1+2 — Q8×3- 1+2 — C2×Q8×3- 1+2
 Lower central C1 — C6 — C2×Q8×3- 1+2
 Upper central C1 — C2×C6 — C2×Q8×3- 1+2

Generators and relations for C2×Q8×3- 1+2
G = < a,b,c,d,e | a2=b4=d9=e3=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 190 in 152 conjugacy classes, 133 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, Q8, C9, C32, C12, C12, C2×C6, C2×C6, C2×Q8, C18, C3×C6, C3×C6, C2×C12, C2×C12, C3×Q8, C3×Q8, 3- 1+2, C36, C2×C18, C3×C12, C62, C6×Q8, C6×Q8, C2×3- 1+2, C2×3- 1+2, C2×C36, Q8×C9, C6×C12, Q8×C32, C4×3- 1+2, C22×3- 1+2, Q8×C18, Q8×C3×C6, C2×C4×3- 1+2, Q8×3- 1+2, C2×Q8×3- 1+2
Quotients: C1, C2, C3, C22, C6, Q8, C23, C32, C2×C6, C2×Q8, C3×C6, C3×Q8, C22×C6, 3- 1+2, C62, C6×Q8, C2×3- 1+2, Q8×C32, C2×C62, C22×3- 1+2, Q8×C3×C6, Q8×3- 1+2, C23×3- 1+2, C2×Q8×3- 1+2

Smallest permutation representation of C2×Q8×3- 1+2
On 144 points
Generators in S144
(1 45)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 125)(11 126)(12 118)(13 119)(14 120)(15 121)(16 122)(17 123)(18 124)(19 35)(20 36)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 81)(53 73)(54 74)(55 71)(56 72)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(82 111)(83 112)(84 113)(85 114)(86 115)(87 116)(88 117)(89 109)(90 110)(91 107)(92 108)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(127 143)(128 144)(129 136)(130 137)(131 138)(132 139)(133 140)(134 141)(135 142)
(1 81 27 70)(2 73 19 71)(3 74 20 72)(4 75 21 64)(5 76 22 65)(6 77 23 66)(7 78 24 67)(8 79 25 68)(9 80 26 69)(10 89 143 91)(11 90 144 92)(12 82 136 93)(13 83 137 94)(14 84 138 95)(15 85 139 96)(16 86 140 97)(17 87 141 98)(18 88 142 99)(28 57 39 46)(29 58 40 47)(30 59 41 48)(31 60 42 49)(32 61 43 50)(33 62 44 51)(34 63 45 52)(35 55 37 53)(36 56 38 54)(100 118 111 129)(101 119 112 130)(102 120 113 131)(103 121 114 132)(104 122 115 133)(105 123 116 134)(106 124 117 135)(107 125 109 127)(108 126 110 128)
(1 117 27 106)(2 109 19 107)(3 110 20 108)(4 111 21 100)(5 112 22 101)(6 113 23 102)(7 114 24 103)(8 115 25 104)(9 116 26 105)(10 55 143 53)(11 56 144 54)(12 57 136 46)(13 58 137 47)(14 59 138 48)(15 60 139 49)(16 61 140 50)(17 62 141 51)(18 63 142 52)(28 93 39 82)(29 94 40 83)(30 95 41 84)(31 96 42 85)(32 97 43 86)(33 98 44 87)(34 99 45 88)(35 91 37 89)(36 92 38 90)(64 129 75 118)(65 130 76 119)(66 131 77 120)(67 132 78 121)(68 133 79 122)(69 134 80 123)(70 135 81 124)(71 127 73 125)(72 128 74 126)
(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)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(37 43 40)(38 41 44)(47 53 50)(48 51 54)(55 61 58)(56 59 62)(65 71 68)(66 69 72)(73 79 76)(74 77 80)(83 89 86)(84 87 90)(91 97 94)(92 95 98)(101 107 104)(102 105 108)(109 115 112)(110 113 116)(119 125 122)(120 123 126)(127 133 130)(128 131 134)(137 143 140)(138 141 144)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A ··· 4F 6A ··· 6F 6G ··· 6L 9A ··· 9F 12A ··· 12L 12M ··· 12X 18A ··· 18R 36A ··· 36AJ order 1 2 2 2 3 3 3 3 4 ··· 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 1 1 3 3 2 ··· 2 1 ··· 1 3 ··· 3 3 ··· 3 2 ··· 2 6 ··· 6 3 ··· 3 6 ··· 6

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 6 type + + + - image C1 C2 C2 C3 C3 C6 C6 C6 C6 Q8 C3×Q8 C3×Q8 3- 1+2 C2×3- 1+2 C2×3- 1+2 Q8×3- 1+2 kernel C2×Q8×3- 1+2 C2×C4×3- 1+2 Q8×3- 1+2 Q8×C18 Q8×C3×C6 C2×C36 Q8×C9 C6×C12 Q8×C32 C2×3- 1+2 C18 C3×C6 C2×Q8 C2×C4 Q8 C2 # reps 1 3 4 6 2 18 24 6 8 2 12 4 2 6 8 4

Matrix representation of C2×Q8×3- 1+2 in GL7(𝔽37)

 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 0 36 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36
,
 5 14 0 0 0 0 0 14 32 0 0 0 0 0 0 0 5 14 0 0 0 0 0 14 32 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36
,
 10 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 26 0 0 0 0 1 0 0
,
 10 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 10

G:=sub<GL(7,GF(37))| [36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[5,14,0,0,0,0,0,14,32,0,0,0,0,0,0,0,5,14,0,0,0,0,0,14,32,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[10,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,26,0],[10,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,10] >;

C2×Q8×3- 1+2 in GAP, Magma, Sage, TeX

C_2\times Q_8\times 3_-^{1+2}
% in TeX

G:=Group("C2xQ8xES-(3,1)");
// GroupNames label

G:=SmallGroup(432,408);
// by ID

G=gap.SmallGroup(432,408);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,504,1037,512,528,760]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^9=e^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^4>;
// generators/relations

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