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G = C62.225C23  order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: C62.225C23, C32⋊19(C4×D4), C3⋊Dic3⋊17D4, C62⋊13(C2×C4), C6.107(S3×D4), C32⋊7D4⋊3C4, (C2×C12).206D6, (C22×C6).87D6, C3⋊4(Dic3⋊4D4), C6.94(D4⋊2S3), C6.11D12⋊20C2, (C6×C12).251C22, C6.Dic6⋊20C2, (C2×C62).64C22, C2.2(C12.D6), C6.67(S3×C2×C4), C2.2(D4×C3⋊S3), (C2×C6)⋊10(C4×S3), C22⋊2(C4×C3⋊S3), (C3×C22⋊C4)⋊9S3, C3⋊Dic3⋊8(C2×C4), C22⋊C4⋊7(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×C32⋊7D4).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

Derived series
C1 — C3×C6 — C62.225C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C32⋊7D4 — C62.225C23
Lower central
C32 — C3×C6 — C62.225C23
Upper central
C1 — C22 — C22⋊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, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4×D4, C3⋊Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C4×C3⋊S3, C2×C3⋊Dic3, C2×C3⋊Dic3, C32⋊7D4, C6×C12, C22×C3⋊S3, C2×C62, Dic3⋊4D4, C4×C3⋊Dic3, C6.Dic6, C6.11D12, C32×C22⋊C4, C2×C4×C3⋊S3, C22×C3⋊Dic3, C2×C32⋊7D4, C62.225C23
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C3⋊S3, C4×S3, C22×S3, C4×D4, C2×C3⋊S3, S3×C2×C4, S3×D4, D4⋊2S3, C4×C3⋊S3, C22×C3⋊S3, Dic3⋊4D4, 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 32 58 18 39 61)(2 33 59 13 40 62)(3 34 60 14 41 63)(4 35 55 15 42 64)(5 36 56 16 37 65)(6 31 57 17 38 66)(7 22 117 139 29 121)(8 23 118 140 30 122)(9 24 119 141 25 123)(10 19 120 142 26 124)(11 20 115 143 27 125)(12 21 116 144 28 126)(43 71 91 53 78 100)(44 72 92 54 73 101)(45 67 93 49 74 102)(46 68 94 50 75 97)(47 69 95 51 76 98)(48 70 96 52 77 99)(79 107 127 89 114 136)(80 108 128 90 109 137)(81 103 129 85 110 138)(82 104 130 86 111 133)(83 105 131 87 112 134)(84 106 132 88 113 135)

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

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

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×D4D4⋊2S3
kernelC62.225C23C4×C3⋊Dic3C6.Dic6C6.11D12C32×C22⋊C4C2×C4×C3⋊S3C22×C3⋊Dic3C2×C32⋊7D4C32⋊7D4C3×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
 

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