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G = C4⋊C4×C3⋊S3order 288 = 25·32

Direct product of C4⋊C4 and C3⋊S3

direct product, metabelian, supersoluble, monomial

Aliases: C4⋊C4×C3⋊S3, C62.235C23, C124(C4×S3), C6.45(S3×Q8), C6.113(S3×D4), (C2×C12).211D6, C12⋊Dic320C2, (C6×C12).253C22, C6.Dic622C2, C43(C4×C3⋊S3), C33(S3×C4⋊C4), (C3×C4⋊C4)⋊2S3, (C4×C3⋊S3)⋊3C4, C6.69(S3×C2×C4), C2.3(D4×C3⋊S3), C2.2(Q8×C3⋊S3), C3211(C2×C4⋊C4), (C3×C12)⋊12(C2×C4), (C2×C3⋊S3).71D4, (C2×C3⋊S3).11Q8, (C3×C6).72(C2×Q8), (C32×C4⋊C4)⋊11C2, C3⋊Dic314(C2×C4), (C3×C6).235(C2×D4), (C3×C6).100(C22×C4), (C2×C6).252(C22×S3), C22.16(C22×C3⋊S3), (C22×C3⋊S3).112C22, (C2×C3⋊Dic3).158C22, C2.11(C2×C4×C3⋊S3), (C2×C4×C3⋊S3).22C2, (C2×C4).31(C2×C3⋊S3), (C2×C3⋊S3).51(C2×C4), SmallGroup(288,748)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C4⋊C4×C3⋊S3
C1C3C32C3×C6C62C22×C3⋊S3C2×C4×C3⋊S3 — C4⋊C4×C3⋊S3
C32C3×C6 — C4⋊C4×C3⋊S3
C1C22C4⋊C4

Generators and relations for C4⋊C4×C3⋊S3
 G = < a,b,c,d,e | a4=b4=c3=d3=e2=1, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 948 in 276 conjugacy classes, 93 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×2], C4 [×6], C22, C22 [×6], S3 [×16], C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, C32, Dic3 [×16], C12 [×8], C12 [×8], D6 [×24], C2×C6 [×4], C4⋊C4, C4⋊C4 [×3], C22×C4 [×3], C3⋊S3 [×4], C3×C6 [×3], C4×S3 [×32], C2×Dic3 [×12], C2×C12 [×12], C22×S3 [×4], C2×C4⋊C4, C3⋊Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, Dic3⋊C4 [×8], C4⋊Dic3 [×4], C3×C4⋊C4 [×4], S3×C2×C4 [×12], C4×C3⋊S3 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C2×C3⋊Dic3 [×2], C6×C12, C6×C12 [×2], C22×C3⋊S3, S3×C4⋊C4 [×4], C6.Dic6 [×2], C12⋊Dic3, C32×C4⋊C4, C2×C4×C3⋊S3, C2×C4×C3⋊S3 [×2], C4⋊C4×C3⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×12], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C3⋊S3, C4×S3 [×8], C22×S3 [×4], C2×C4⋊C4, C2×C3⋊S3 [×3], S3×C2×C4 [×4], S3×D4 [×4], S3×Q8 [×4], C4×C3⋊S3 [×2], C22×C3⋊S3, S3×C4⋊C4 [×4], C2×C4×C3⋊S3, D4×C3⋊S3, Q8×C3⋊S3, C4⋊C4×C3⋊S3

Smallest permutation representation of C4⋊C4×C3⋊S3
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 59 77 90)(2 58 78 89)(3 57 79 92)(4 60 80 91)(5 76 51 117)(6 75 52 120)(7 74 49 119)(8 73 50 118)(9 44 109 36)(10 43 110 35)(11 42 111 34)(12 41 112 33)(13 95 45 113)(14 94 46 116)(15 93 47 115)(16 96 48 114)(17 81 54 122)(18 84 55 121)(19 83 56 124)(20 82 53 123)(21 137 62 126)(22 140 63 125)(23 139 64 128)(24 138 61 127)(25 141 38 108)(26 144 39 107)(27 143 40 106)(28 142 37 105)(29 133 70 101)(30 136 71 104)(31 135 72 103)(32 134 69 102)(65 85 100 129)(66 88 97 132)(67 87 98 131)(68 86 99 130)
(1 96 121)(2 93 122)(3 94 123)(4 95 124)(5 141 9)(6 142 10)(7 143 11)(8 144 12)(13 56 91)(14 53 92)(15 54 89)(16 55 90)(17 58 47)(18 59 48)(19 60 45)(20 57 46)(21 32 130)(22 29 131)(23 30 132)(24 31 129)(25 36 117)(26 33 118)(27 34 119)(28 35 120)(37 43 75)(38 44 76)(39 41 73)(40 42 74)(49 106 111)(50 107 112)(51 108 109)(52 105 110)(61 72 85)(62 69 86)(63 70 87)(64 71 88)(65 138 135)(66 139 136)(67 140 133)(68 137 134)(77 114 84)(78 115 81)(79 116 82)(80 113 83)(97 128 104)(98 125 101)(99 126 102)(100 127 103)
(1 76 32)(2 73 29)(3 74 30)(4 75 31)(5 102 90)(6 103 91)(7 104 92)(8 101 89)(9 126 55)(10 127 56)(11 128 53)(12 125 54)(13 142 100)(14 143 97)(15 144 98)(16 141 99)(17 112 140)(18 109 137)(19 110 138)(20 111 139)(21 121 44)(22 122 41)(23 123 42)(24 124 43)(25 86 114)(26 87 115)(27 88 116)(28 85 113)(33 63 81)(34 64 82)(35 61 83)(36 62 84)(37 129 95)(38 130 96)(39 131 93)(40 132 94)(45 105 65)(46 106 66)(47 107 67)(48 108 68)(49 136 57)(50 133 58)(51 134 59)(52 135 60)(69 77 117)(70 78 118)(71 79 119)(72 80 120)
(1 77)(2 78)(3 79)(4 80)(5 134)(6 135)(7 136)(8 133)(9 68)(10 65)(11 66)(12 67)(13 19)(14 20)(15 17)(16 18)(21 25)(22 26)(23 27)(24 28)(29 118)(30 119)(31 120)(32 117)(33 131)(34 132)(35 129)(36 130)(37 61)(38 62)(39 63)(40 64)(41 87)(42 88)(43 85)(44 86)(45 56)(46 53)(47 54)(48 55)(49 104)(50 101)(51 102)(52 103)(57 92)(58 89)(59 90)(60 91)(69 76)(70 73)(71 74)(72 75)(81 93)(82 94)(83 95)(84 96)(97 111)(98 112)(99 109)(100 110)(105 127)(106 128)(107 125)(108 126)(113 124)(114 121)(115 122)(116 123)(137 141)(138 142)(139 143)(140 144)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D4A···4F4G···4L6A···6L12A···12X
order1222222233334···44···46···612···12
size1111999922222···218···182···24···4

60 irreducible representations

dim1111112222244
type+++++++-++-
imageC1C2C2C2C2C4S3D4Q8D6C4×S3S3×D4S3×Q8
kernelC4⋊C4×C3⋊S3C6.Dic6C12⋊Dic3C32×C4⋊C4C2×C4×C3⋊S3C4×C3⋊S3C3×C4⋊C4C2×C3⋊S3C2×C3⋊S3C2×C12C12C6C6
# reps121138422121644

Matrix representation of C4⋊C4×C3⋊S3 in GL6(𝔽13)

1200000
0120000
001000
000100
000053
000008
,
800000
080000
0012000
0001200
000050
000058
,
100000
010000
000100
00121200
000010
000001
,
12120000
100000
001000
000100
000010
000001
,
1200000
110000
0001200
0012000
0000120
0000012

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,3,8],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,5,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

C4⋊C4×C3⋊S3 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\times C_3\rtimes S_3
% in TeX

G:=Group("C4:C4xC3:S3");
// GroupNames label

G:=SmallGroup(288,748);
// by ID

G=gap.SmallGroup(288,748);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,58,2693,9414]);
// Polycyclic

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

׿
×
𝔽