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G = C62.228C23order 288 = 25·32

73rd non-split extension by C62 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial

Aliases: C62.228C23, C3⋊Dic315D4, C6.110(S3×D4), C33(Dic3⋊D4), (C2×C12).208D6, (C22×C6).90D6, C6.99(C4○D12), C3217(C4⋊D4), (C6×C12).13C22, C6.Dic65C2, C6.11D1221C2, (C2×C62).67C22, C2.11(C12.59D6), C2.9(D4×C3⋊S3), (C2×C3⋊S3)⋊11D4, (C3×C22⋊C4)⋊6S3, (C2×C12⋊S3)⋊5C2, C22⋊C44(C3⋊S3), (C3×C6).232(C2×D4), C23.10(C2×C3⋊S3), (C2×C327D4)⋊8C2, (C32×C22⋊C4)⋊7C2, (C3×C6).115(C4○D4), (C2×C6).245(C22×S3), C22.43(C22×C3⋊S3), (C22×C3⋊S3).40C22, (C2×C3⋊Dic3).80C22, (C2×C4×C3⋊S3)⋊20C2, (C2×C4).6(C2×C3⋊S3), SmallGroup(288,741)

Series: Derived Chief Lower central Upper central

C1C62 — C62.228C23
C1C3C32C3×C6C62C22×C3⋊S3C2×C4×C3⋊S3 — C62.228C23
C32C62 — C62.228C23
C1C22C22⋊C4

Generators and relations for C62.228C23
 G = < a,b,c,d,e | a6=b6=c2=e2=1, d2=a3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=a3b3c, ede=b3d >

Subgroups: 1260 in 282 conjugacy classes, 69 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×5], C22, C22 [×10], S3 [×12], C6 [×12], C6 [×4], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C32, Dic3 [×12], C12 [×8], D6 [×28], C2×C6 [×4], C2×C6 [×12], C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4 [×3], C3⋊S3 [×3], C3×C6 [×3], C3×C6, C4×S3 [×8], D12 [×8], C2×Dic3 [×8], C3⋊D4 [×16], C2×C12 [×8], C22×S3 [×8], C22×C6 [×4], C4⋊D4, C3⋊Dic3 [×2], C3⋊Dic3, C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×5], C62, C62 [×3], Dic3⋊C4 [×4], D6⋊C4 [×4], C3×C22⋊C4 [×4], S3×C2×C4 [×4], C2×D12 [×4], C2×C3⋊D4 [×8], C4×C3⋊S3 [×2], C12⋊S3 [×2], C2×C3⋊Dic3 [×2], C327D4 [×4], C6×C12 [×2], C22×C3⋊S3 [×2], C2×C62, Dic3⋊D4 [×4], C6.Dic6, C6.11D12, C32×C22⋊C4, C2×C4×C3⋊S3, C2×C12⋊S3, C2×C327D4 [×2], C62.228C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×4], C23, D6 [×12], C2×D4 [×2], C4○D4, C3⋊S3, C22×S3 [×4], C4⋊D4, C2×C3⋊S3 [×3], C4○D12 [×4], S3×D4 [×8], C22×C3⋊S3, Dic3⋊D4 [×4], C12.59D6, D4×C3⋊S3 [×2], C62.228C23

Smallest permutation representation of C62.228C23
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 18 58 31 39 61)(2 13 59 32 40 62)(3 14 60 33 41 63)(4 15 55 34 42 64)(5 16 56 35 37 65)(6 17 57 36 38 66)(7 126 27 24 140 118)(8 121 28 19 141 119)(9 122 29 20 142 120)(10 123 30 21 143 115)(11 124 25 22 144 116)(12 125 26 23 139 117)(43 92 51 73 101 71)(44 93 52 74 102 72)(45 94 53 75 97 67)(46 95 54 76 98 68)(47 96 49 77 99 69)(48 91 50 78 100 70)(79 128 87 109 137 107)(80 129 88 110 138 108)(81 130 89 111 133 103)(82 131 90 112 134 104)(83 132 85 113 135 105)(84 127 86 114 136 106)
(1 139)(2 144)(3 143)(4 142)(5 141)(6 140)(7 38)(8 37)(9 42)(10 41)(11 40)(12 39)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)(25 59)(26 58)(27 57)(28 56)(29 55)(30 60)(31 125)(32 124)(33 123)(34 122)(35 121)(36 126)(43 138)(44 137)(45 136)(46 135)(47 134)(48 133)(49 90)(50 89)(51 88)(52 87)(53 86)(54 85)(61 117)(62 116)(63 115)(64 120)(65 119)(66 118)(67 106)(68 105)(69 104)(70 103)(71 108)(72 107)(73 129)(74 128)(75 127)(76 132)(77 131)(78 130)(79 102)(80 101)(81 100)(82 99)(83 98)(84 97)(91 111)(92 110)(93 109)(94 114)(95 113)(96 112)
(1 106 4 103)(2 107 5 104)(3 108 6 105)(7 95 10 92)(8 96 11 93)(9 91 12 94)(13 79 16 82)(14 80 17 83)(15 81 18 84)(19 99 22 102)(20 100 23 97)(21 101 24 98)(25 74 28 77)(26 75 29 78)(27 76 30 73)(31 86 34 89)(32 87 35 90)(33 88 36 85)(37 112 40 109)(38 113 41 110)(39 114 42 111)(43 118 46 115)(44 119 47 116)(45 120 48 117)(49 124 52 121)(50 125 53 122)(51 126 54 123)(55 130 58 127)(56 131 59 128)(57 132 60 129)(61 136 64 133)(62 137 65 134)(63 138 66 135)(67 142 70 139)(68 143 71 140)(69 144 72 141)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 134)(8 135)(9 136)(10 137)(11 138)(12 133)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 132)(20 127)(21 128)(22 129)(23 130)(24 131)(25 80)(26 81)(27 82)(28 83)(29 84)(30 79)(31 53)(32 54)(33 49)(34 50)(35 51)(36 52)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(85 141)(86 142)(87 143)(88 144)(89 139)(90 140)(103 125)(104 126)(105 121)(106 122)(107 123)(108 124)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D4A4B4C4D4E4F6A···6L6M···6T12A···12P
order1222222233334444446···66···612···12
size1111418183622222241818362···24···44···4

54 irreducible representations

dim111111122222224
type+++++++++++++
imageC1C2C2C2C2C2C2S3D4D4D6D6C4○D4C4○D12S3×D4
kernelC62.228C23C6.Dic6C6.11D12C32×C22⋊C4C2×C4×C3⋊S3C2×C12⋊S3C2×C327D4C3×C22⋊C4C3⋊Dic3C2×C3⋊S3C2×C12C22×C6C3×C6C6C6
# reps1111112422842168

Matrix representation of C62.228C23 in GL6(𝔽13)

1200000
0120000
001000
000100
000011
0000120
,
1200000
0120000
0001200
0011200
00001212
000010
,
1200000
0120000
000100
001000
0000103
000063
,
010000
1200000
0012000
0001200
000050
000005
,
010000
100000
0012000
0001200
0000119
000042

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,6,0,0,0,0,3,3],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,11,4,0,0,0,0,9,2] >;

C62.228C23 in GAP, Magma, Sage, TeX

C_6^2._{228}C_2^3
% in TeX

G:=Group("C6^2.228C2^3");
// GroupNames label

G:=SmallGroup(288,741);
// by ID

G=gap.SmallGroup(288,741);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,219,2693,9414]);
// Polycyclic

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

׿
×
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