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G = C62.225C23order 288 = 25·32

70th non-split extension by C62 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial

Aliases: C62.225C23, C3219(C4×D4), C3⋊Dic317D4, C6213(C2×C4), C6.107(S3×D4), C327D43C4, (C2×C12).206D6, (C22×C6).87D6, C34(Dic34D4), C6.94(D42S3), C6.11D1220C2, (C6×C12).251C22, C6.Dic620C2, (C2×C62).64C22, C2.2(C12.D6), C6.67(S3×C2×C4), C2.2(D4×C3⋊S3), (C2×C6)⋊10(C4×S3), C222(C4×C3⋊S3), (C3×C22⋊C4)⋊9S3, C3⋊Dic38(C2×C4), C22⋊C47(C3⋊S3), (C4×C3⋊Dic3)⋊22C2, (C3×C6).230(C2×D4), C23.19(C2×C3⋊S3), (C3×C6).98(C22×C4), (C22×C3⋊Dic3)⋊5C2, (C2×C327D4).9C2, (C3×C6).143(C4○D4), (C32×C22⋊C4)⋊17C2, (C2×C6).242(C22×S3), C22.14(C22×C3⋊S3), (C22×C3⋊S3).82C22, (C2×C3⋊Dic3).155C22, C2.9(C2×C4×C3⋊S3), (C2×C4×C3⋊S3)⋊18C2, (C2×C3⋊S3)⋊10(C2×C4), (C2×C4).27(C2×C3⋊S3), SmallGroup(288,738)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C62.225C23
C1C3C32C3×C6C62C22×C3⋊S3C2×C327D4 — C62.225C23
C32C3×C6 — C62.225C23
C1C22C22⋊C4

Generators and relations for C62.225C23
 G = < a,b,c,d,e | a6=b6=c2=d2=1, e2=a3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd=b3c, ce=ec, ede-1=b3d >

Subgroups: 996 in 282 conjugacy classes, 87 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×7], C22, C22 [×2], C22 [×6], S3 [×8], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, C32, Dic3 [×20], C12 [×8], D6 [×16], C2×C6 [×12], C2×C6 [×8], C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4 [×2], C2×D4, C3⋊S3 [×2], C3×C6 [×3], C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×20], C3⋊D4 [×16], C2×C12 [×8], C22×S3 [×4], C22×C6 [×4], C4×D4, C3⋊Dic3 [×4], C3⋊Dic3, C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×2], C62 [×2], C4×Dic3 [×4], Dic3⋊C4 [×4], D6⋊C4 [×4], C3×C22⋊C4 [×4], S3×C2×C4 [×4], C22×Dic3 [×4], C2×C3⋊D4 [×4], C4×C3⋊S3 [×2], C2×C3⋊Dic3 [×3], C2×C3⋊Dic3 [×2], C327D4 [×4], C6×C12 [×2], C22×C3⋊S3, C2×C62, Dic34D4 [×4], C4×C3⋊Dic3, C6.Dic6, C6.11D12, C32×C22⋊C4, C2×C4×C3⋊S3, C22×C3⋊Dic3, C2×C327D4, C62.225C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], D4 [×2], C23, D6 [×12], C22×C4, C2×D4, C4○D4, C3⋊S3, C4×S3 [×8], C22×S3 [×4], C4×D4, C2×C3⋊S3 [×3], S3×C2×C4 [×4], S3×D4 [×4], D42S3 [×4], C4×C3⋊S3 [×2], C22×C3⋊S3, Dic34D4 [×4], C2×C4×C3⋊S3, D4×C3⋊S3, C12.D6, C62.225C23

Smallest permutation representation of C62.225C23
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 15 58 33 39 61)(2 16 59 34 40 62)(3 17 60 35 41 63)(4 18 55 36 42 64)(5 13 56 31 37 65)(6 14 57 32 38 66)(7 22 123 116 29 140)(8 23 124 117 30 141)(9 24 125 118 25 142)(10 19 126 119 26 143)(11 20 121 120 27 144)(12 21 122 115 28 139)(43 92 67 73 101 50)(44 93 68 74 102 51)(45 94 69 75 97 52)(46 95 70 76 98 53)(47 96 71 77 99 54)(48 91 72 78 100 49)(79 128 103 109 137 86)(80 129 104 110 138 87)(81 130 105 111 133 88)(82 131 106 112 134 89)(83 132 107 113 135 90)(84 127 108 114 136 85)
(2 6)(3 5)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 63)(14 62)(15 61)(16 66)(17 65)(18 64)(25 116)(26 115)(27 120)(28 119)(29 118)(30 117)(31 35)(32 34)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 99)(44 98)(45 97)(46 102)(47 101)(48 100)(50 54)(51 53)(67 71)(68 70)(73 96)(74 95)(75 94)(76 93)(77 92)(78 91)(79 132)(80 131)(81 130)(82 129)(83 128)(84 127)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(109 135)(110 134)(111 133)(112 138)(113 137)(114 136)(121 144)(122 143)(123 142)(124 141)(125 140)(126 139)
(1 88)(2 89)(3 90)(4 85)(5 86)(6 87)(7 74)(8 75)(9 76)(10 77)(11 78)(12 73)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 99)(20 100)(21 101)(22 102)(23 97)(24 98)(25 95)(26 96)(27 91)(28 92)(29 93)(30 94)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(1 49 4 52)(2 50 5 53)(3 51 6 54)(7 80 10 83)(8 81 11 84)(9 82 12 79)(13 46 16 43)(14 47 17 44)(15 48 18 45)(19 132 22 129)(20 127 23 130)(21 128 24 131)(25 134 28 137)(26 135 29 138)(27 136 30 133)(31 70 34 67)(32 71 35 68)(33 72 36 69)(37 76 40 73)(38 77 41 74)(39 78 42 75)(55 94 58 91)(56 95 59 92)(57 96 60 93)(61 100 64 97)(62 101 65 98)(63 102 66 99)(85 141 88 144)(86 142 89 139)(87 143 90 140)(103 125 106 122)(104 126 107 123)(105 121 108 124)(109 118 112 115)(110 119 113 116)(111 120 114 117)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D4A4B4C4D4E4F4G4H4I4J4K4L6A···6L6M···6T12A···12P
order1222222233334444444444446···66···612···12
size1111221818222222229999181818182···24···44···4

60 irreducible representations

dim11111111122222244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C4S3D4D6D6C4○D4C4×S3S3×D4D42S3
kernelC62.225C23C4×C3⋊Dic3C6.Dic6C6.11D12C32×C22⋊C4C2×C4×C3⋊S3C22×C3⋊Dic3C2×C327D4C327D4C3×C22⋊C4C3⋊Dic3C2×C12C22×C6C3×C6C2×C6C6C6
# reps111111118428421644

Matrix representation of C62.225C23 in GL6(𝔽13)

1210000
1200000
001100
0012000
000010
000001
,
0120000
1120000
001000
000100
0000120
0000012
,
100000
1120000
001000
00121200
000010
0000112
,
100000
010000
001000
000100
0000111
0000012
,
1200000
0120000
005000
000500
000010
0000112

G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,11,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,1,0,0,0,0,0,12] >;

C62.225C23 in GAP, Magma, Sage, TeX

C_6^2._{225}C_2^3
% in TeX

G:=Group("C6^2.225C2^3");
// GroupNames label

G:=SmallGroup(288,738);
// by ID

G=gap.SmallGroup(288,738);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,219,58,2693,9414]);
// Polycyclic

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

׿
×
𝔽