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G = C12.3S4order 288 = 25·32

3rd non-split extension by C12 of S4 acting via S4/A4=C2

non-abelian, soluble

Aliases: C12.3S4, Q8.3D18, C6.22(C2×S4), C4○D4.2D9, Q8.D92C2, C4.2(C3.S4), C3.(C4.S4), (C3×Q8).11D6, Q8⋊C9.3C22, Q8.C18.1C2, C2.8(C2×C3.S4), (C3×C4○D4).2S3, SmallGroup(288,338)

Series: Derived Chief Lower central Upper central

C1C2Q8Q8⋊C9 — C12.3S4
C1C2Q8C3×Q8Q8⋊C9Q8.D9 — C12.3S4
Q8⋊C9 — C12.3S4
C1C2C4

Generators and relations for C12.3S4
 G = < a,b,c,d,e | a12=1, b2=c2=e2=a6, d3=a4, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a6b, dbd-1=a6bc, ebe-1=bc, dcd-1=b, ece-1=a6c, ede-1=a8d2 >

Subgroups: 303 in 61 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3, C4, C4 [×3], C22, C6, C6, C8 [×2], C2×C4 [×2], D4, Q8, Q8 [×3], C9, Dic3 [×2], C12, C12, C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C18, C3⋊C8 [×2], Dic6 [×3], C2×Dic3, C2×C12, C3×D4, C3×Q8, C8.C22, Dic9 [×2], C36, C4.Dic3, D4.S3 [×2], C3⋊Q16 [×2], C2×Dic6, C3×C4○D4, Q8⋊C9, Dic18, Q8.14D6, Q8.D9 [×2], Q8.C18, C12.3S4
Quotients: C1, C2 [×3], C22, S3, D6, D9, S4, D18, C2×S4, C3.S4, C4.S4, C2×C3.S4, C12.3S4

Character table of C12.3S4

 class 12A2B34A4B4C4D6A6B8A8B9A9B9C12A12B12C18A18B18C36A36B36C36D36E36F
 size 116226363621236368882212888888888
ρ1111111111111111111111111111    trivial
ρ2111111-1-111-1-1111111111111111    linear of order 2
ρ311-11-111-11-1-11111-1-11111-1-1-1-1-1-1    linear of order 2
ρ411-11-11-111-11-1111-1-11111-1-1-1-1-1-1    linear of order 2
ρ522-22-22002-200-1-1-1-2-22-1-1-1111111    orthogonal lifted from D6
ρ6222222002200-1-1-1222-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ7222-12200-1-100ζ9792ζ9594ζ989-1-1-1ζ9792ζ9594ζ989ζ9792ζ9792ζ9594ζ9594ζ989ζ989    orthogonal lifted from D9
ρ822-2-1-2200-1100ζ9792ζ9594ζ98911-1ζ9792ζ9594ζ9899792979295949594989989    orthogonal lifted from D18
ρ922-2-1-2200-1100ζ9594ζ989ζ979211-1ζ9594ζ989ζ97929594959498998997929792    orthogonal lifted from D18
ρ1022-2-1-2200-1100ζ989ζ9792ζ959411-1ζ989ζ9792ζ95949899899792979295949594    orthogonal lifted from D18
ρ11222-12200-1-100ζ989ζ9792ζ9594-1-1-1ζ989ζ9792ζ9594ζ989ζ989ζ9792ζ9792ζ9594ζ9594    orthogonal lifted from D9
ρ12222-12200-1-100ζ9594ζ989ζ9792-1-1-1ζ9594ζ989ζ9792ζ9594ζ9594ζ989ζ989ζ9792ζ9792    orthogonal lifted from D9
ρ133313-3-1-1131-11000-3-3-1000000000    orthogonal lifted from C2×S4
ρ1433-133-1-1-13-11100033-1000000000    orthogonal lifted from S4
ρ1533-133-1113-1-1-100033-1000000000    orthogonal lifted from S4
ρ163313-3-11-1311-1000-3-3-1000000000    orthogonal lifted from C2×S4
ρ174-4040000-4000-2-2-2000222000000    symplectic lifted from C4.S4, Schur index 2
ρ184-4040000-4000111000-1-1-1-33-33-33    symplectic lifted from C4.S4, Schur index 2
ρ194-4040000-4000111000-1-1-13-33-33-3    symplectic lifted from C4.S4, Schur index 2
ρ204-40-20000200098997929594-23230ζ989ζ9792ζ959443ζ9843ζ9ζ43ζ9843ζ94ζ974ζ92ζ4ζ974ζ92ζ4ζ954ζ944ζ954ζ94    symplectic faithful, Schur index 2
ρ214-40-2000020009899792959423-230ζ989ζ9792ζ9594ζ43ζ9843ζ943ζ9843ζ9ζ4ζ974ζ924ζ974ζ924ζ954ζ94ζ4ζ954ζ94    symplectic faithful, Schur index 2
ρ224-40-20000200097929594989-23230ζ9792ζ9594ζ9894ζ974ζ92ζ4ζ974ζ92ζ4ζ954ζ944ζ954ζ9443ζ9843ζ9ζ43ζ9843ζ9    symplectic faithful, Schur index 2
ρ234-40-2000020009792959498923-230ζ9792ζ9594ζ989ζ4ζ974ζ924ζ974ζ924ζ954ζ94ζ4ζ954ζ94ζ43ζ9843ζ943ζ9843ζ9    symplectic faithful, Schur index 2
ρ244-40-2000020009594989979223-230ζ9594ζ989ζ97924ζ954ζ94ζ4ζ954ζ94ζ43ζ9843ζ943ζ9843ζ9ζ4ζ974ζ924ζ974ζ92    symplectic faithful, Schur index 2
ρ254-40-20000200095949899792-23230ζ9594ζ989ζ9792ζ4ζ954ζ944ζ954ζ9443ζ9843ζ9ζ43ζ9843ζ94ζ974ζ92ζ4ζ974ζ92    symplectic faithful, Schur index 2
ρ2666-2-36-200-3100000-3-31000000000    orthogonal lifted from C3.S4
ρ27662-3-6-200-3-100000331000000000    orthogonal lifted from C2×C3.S4

Smallest permutation representation of C12.3S4
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 10 7 4)(2 11 8 5)(3 12 9 6)(13 122 19 128)(14 123 20 129)(15 124 21 130)(16 125 22 131)(17 126 23 132)(18 127 24 121)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 46 43 40)(38 47 44 41)(39 48 45 42)(49 143 55 137)(50 144 56 138)(51 133 57 139)(52 134 58 140)(53 135 59 141)(54 136 60 142)(61 83 67 77)(62 84 68 78)(63 73 69 79)(64 74 70 80)(65 75 71 81)(66 76 72 82)(85 119 91 113)(86 120 92 114)(87 109 93 115)(88 110 94 116)(89 111 95 117)(90 112 96 118)(97 100 103 106)(98 101 104 107)(99 102 105 108)
(1 106 7 100)(2 107 8 101)(3 108 9 102)(4 97 10 103)(5 98 11 104)(6 99 12 105)(13 131 19 125)(14 132 20 126)(15 121 21 127)(16 122 22 128)(17 123 23 129)(18 124 24 130)(25 47 31 41)(26 48 32 42)(27 37 33 43)(28 38 34 44)(29 39 35 45)(30 40 36 46)(49 140 55 134)(50 141 56 135)(51 142 57 136)(52 143 58 137)(53 144 59 138)(54 133 60 139)(61 64 67 70)(62 65 68 71)(63 66 69 72)(73 82 79 76)(74 83 80 77)(75 84 81 78)(85 88 91 94)(86 89 92 95)(87 90 93 96)(109 118 115 112)(110 119 116 113)(111 120 117 114)
(1 73 134 5 77 138 9 81 142)(2 74 135 6 78 139 10 82 143)(3 75 136 7 79 140 11 83 144)(4 76 137 8 80 141 12 84 133)(13 29 94 17 33 86 21 25 90)(14 30 95 18 34 87 22 26 91)(15 31 96 19 35 88 23 27 92)(16 32 85 20 36 89 24 28 93)(37 117 124 41 109 128 45 113 132)(38 118 125 42 110 129 46 114 121)(39 119 126 43 111 130 47 115 122)(40 120 127 44 112 131 48 116 123)(49 101 61 53 105 65 57 97 69)(50 102 62 54 106 66 58 98 70)(51 103 63 55 107 67 59 99 71)(52 104 64 56 108 68 60 100 72)
(1 130 7 124)(2 129 8 123)(3 128 9 122)(4 127 10 121)(5 126 11 132)(6 125 12 131)(13 99 19 105)(14 98 20 104)(15 97 21 103)(16 108 22 102)(17 107 23 101)(18 106 24 100)(25 51 31 57)(26 50 32 56)(27 49 33 55)(28 60 34 54)(29 59 35 53)(30 58 36 52)(37 134 43 140)(38 133 44 139)(39 144 45 138)(40 143 46 137)(41 142 47 136)(42 141 48 135)(61 94 67 88)(62 93 68 87)(63 92 69 86)(64 91 70 85)(65 90 71 96)(66 89 72 95)(73 111 79 117)(74 110 80 116)(75 109 81 115)(76 120 82 114)(77 119 83 113)(78 118 84 112)

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

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

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

Matrix representation of C12.3S4 in GL4(𝔽73) generated by

70000
07000
1471240
4943024
,
27000
274600
52501
6762720
,
271900
04600
1056046
1070460
,
195200
475200
48534343
3368766
,
2638710
426721
13392819
24136266
G:=sub<GL(4,GF(73))| [70,0,14,49,0,70,71,43,0,0,24,0,0,0,0,24],[27,27,52,67,0,46,5,62,0,0,0,72,0,0,1,0],[27,0,10,10,19,46,56,70,0,0,0,46,0,0,46,0],[19,47,48,33,52,52,53,68,0,0,43,7,0,0,43,66],[26,4,13,24,38,26,39,13,71,72,28,62,0,1,19,66] >;

C12.3S4 in GAP, Magma, Sage, TeX

C_{12}._3S_4
% in TeX

G:=Group("C12.3S4");
// GroupNames label

G:=SmallGroup(288,338);
// by ID

G=gap.SmallGroup(288,338);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,2045,1016,422,142,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^12=1,b^2=c^2=e^2=a^6,d^3=a^4,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c^-1=a^6*b,d*b*d^-1=a^6*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^6*c,e*d*e^-1=a^8*d^2>;
// generators/relations

Export

Character table of C12.3S4 in TeX

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