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## G = C32×C4⋊D4order 288 = 25·32

### Direct product of C32 and C4⋊D4

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×C4⋊D4
 Chief series C1 — C2 — C22 — C2×C6 — C62 — C2×C62 — D4×C3×C6 — C32×C4⋊D4
 Lower central C1 — C22 — C32×C4⋊D4
 Upper central C1 — C62 — C32×C4⋊D4

Generators and relations for C32×C4⋊D4
G = < a,b,c,d,e | a3=b3=c4=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 444 in 282 conjugacy classes, 144 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C6 [×12], C6 [×16], C2×C4 [×2], C2×C4 [×2], C2×C4 [×2], D4 [×6], C23, C23 [×2], C32, C12 [×8], C12 [×12], C2×C6 [×12], C2×C6 [×32], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], C3×C6 [×3], C3×C6 [×4], C2×C12 [×16], C2×C12 [×8], C3×D4 [×24], C22×C6 [×12], C4⋊D4, C3×C12 [×2], C3×C12 [×3], C62, C62 [×2], C62 [×8], C3×C22⋊C4 [×8], C3×C4⋊C4 [×4], C22×C12 [×4], C6×D4 [×12], C6×C12 [×2], C6×C12 [×2], C6×C12 [×2], D4×C32 [×6], C2×C62, C2×C62 [×2], C3×C4⋊D4 [×4], C32×C22⋊C4 [×2], C32×C4⋊C4, C2×C6×C12, D4×C3×C6, D4×C3×C6 [×2], C32×C4⋊D4
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], D4 [×4], C23, C32, C2×C6 [×28], C2×D4 [×2], C4○D4, C3×C6 [×7], C3×D4 [×16], C22×C6 [×4], C4⋊D4, C62 [×7], C6×D4 [×8], C3×C4○D4 [×4], D4×C32 [×4], C2×C62, C3×C4⋊D4 [×4], D4×C3×C6 [×2], C32×C4○D4, C32×C4⋊D4

Smallest permutation representation of C32×C4⋊D4
On 144 points
Generators in S144
(1 23 15)(2 24 16)(3 21 13)(4 22 14)(5 101 93)(6 102 94)(7 103 95)(8 104 96)(9 89 81)(10 90 82)(11 91 83)(12 92 84)(17 85 77)(18 86 78)(19 87 79)(20 88 80)(25 105 33)(26 106 34)(27 107 35)(28 108 36)(29 97 37)(30 98 38)(31 99 39)(32 100 40)(41 121 49)(42 122 50)(43 123 51)(44 124 52)(45 113 53)(46 114 54)(47 115 55)(48 116 56)(57 76 65)(58 73 66)(59 74 67)(60 75 68)(61 141 133)(62 142 134)(63 143 135)(64 144 136)(69 137 129)(70 138 130)(71 139 131)(72 140 132)(109 125 117)(110 126 118)(111 127 119)(112 128 120)
(1 11 79)(2 12 80)(3 9 77)(4 10 78)(5 37 105)(6 38 106)(7 39 107)(8 40 108)(13 81 85)(14 82 86)(15 83 87)(16 84 88)(17 21 89)(18 22 90)(19 23 91)(20 24 92)(25 93 97)(26 94 98)(27 95 99)(28 96 100)(29 33 101)(30 34 102)(31 35 103)(32 36 104)(41 109 113)(42 110 114)(43 111 115)(44 112 116)(45 49 117)(46 50 118)(47 51 119)(48 52 120)(53 121 125)(54 122 126)(55 123 127)(56 124 128)(57 61 129)(58 62 130)(59 63 131)(60 64 132)(65 133 137)(66 134 138)(67 135 139)(68 136 140)(69 76 141)(70 73 142)(71 74 143)(72 75 144)
(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 95 111 59)(2 94 112 58)(3 93 109 57)(4 96 110 60)(5 125 76 21)(6 128 73 24)(7 127 74 23)(8 126 75 22)(9 97 113 61)(10 100 114 64)(11 99 115 63)(12 98 116 62)(13 101 117 65)(14 104 118 68)(15 103 119 67)(16 102 120 66)(17 105 121 69)(18 108 122 72)(19 107 123 71)(20 106 124 70)(25 41 129 77)(26 44 130 80)(27 43 131 79)(28 42 132 78)(29 45 133 81)(30 48 134 84)(31 47 135 83)(32 46 136 82)(33 49 137 85)(34 52 138 88)(35 51 139 87)(36 50 140 86)(37 53 141 89)(38 56 142 92)(39 55 143 91)(40 54 144 90)
(2 4)(5 76)(6 75)(7 74)(8 73)(10 12)(14 16)(18 20)(22 24)(25 129)(26 132)(27 131)(28 130)(29 133)(30 136)(31 135)(32 134)(33 137)(34 140)(35 139)(36 138)(37 141)(38 144)(39 143)(40 142)(42 44)(46 48)(50 52)(54 56)(57 93)(58 96)(59 95)(60 94)(61 97)(62 100)(63 99)(64 98)(65 101)(66 104)(67 103)(68 102)(69 105)(70 108)(71 107)(72 106)(78 80)(82 84)(86 88)(90 92)(110 112)(114 116)(118 120)(122 124)(126 128)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3H 4A 4B 4C 4D 4E 4F 6A ··· 6X 6Y ··· 6AN 6AO ··· 6BD 12A ··· 12AF 12AG ··· 12AV order 1 2 2 2 2 2 2 2 3 ··· 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 4 4 1 ··· 1 2 2 2 2 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D4 C4○D4 C3×D4 C3×D4 C3×C4○D4 kernel C32×C4⋊D4 C32×C22⋊C4 C32×C4⋊C4 C2×C6×C12 D4×C3×C6 C3×C4⋊D4 C3×C22⋊C4 C3×C4⋊C4 C22×C12 C6×D4 C3×C12 C62 C3×C6 C12 C2×C6 C6 # reps 1 2 1 1 3 8 16 8 8 24 2 2 2 16 16 16

Matrix representation of C32×C4⋊D4 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 9 0 0 0 0 9
,
 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 5 10 0 0 0 8
,
 0 12 0 0 1 0 0 0 0 0 1 2 0 0 12 12
,
 1 0 0 0 0 12 0 0 0 0 1 0 0 0 12 12
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,9,0,0,0,0,9],[3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,5,0,0,0,10,8],[0,1,0,0,12,0,0,0,0,0,1,12,0,0,2,12],[1,0,0,0,0,12,0,0,0,0,1,12,0,0,0,12] >;

C32×C4⋊D4 in GAP, Magma, Sage, TeX

C_3^2\times C_4\rtimes D_4
% in TeX

G:=Group("C3^2xC4:D4");
// GroupNames label

G:=SmallGroup(288,818);
// by ID

G=gap.SmallGroup(288,818);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,512,3110]);
// Polycyclic

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

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