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## G = C32×C4.4D4order 288 = 25·32

### Direct product of C32 and C4.4D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×C4.4D4
 Chief series C1 — C2 — C22 — C2×C6 — C62 — C2×C62 — C32×C22⋊C4 — C32×C4.4D4
 Lower central C1 — C22 — C32×C4.4D4
 Upper central C1 — C62 — C32×C4.4D4

Generators and relations for C32×C4.4D4
G = < a,b,c,d,e | a3=b3=c4=d4=1, e2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=c2d-1 >

Subgroups: 348 in 228 conjugacy classes, 132 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×2], C4 [×4], C22, C22 [×6], C6 [×12], C6 [×8], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C32, C12 [×8], C12 [×16], C2×C6 [×4], C2×C6 [×24], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×C12 [×20], C3×D4 [×8], C3×Q8 [×8], C22×C6 [×8], C4.4D4, C3×C12 [×2], C3×C12 [×4], C62, C62 [×6], C4×C12 [×4], C3×C22⋊C4 [×16], C6×D4 [×4], C6×Q8 [×4], C6×C12, C6×C12 [×4], D4×C32 [×2], Q8×C32 [×2], C2×C62 [×2], C3×C4.4D4 [×4], C122, C32×C22⋊C4 [×4], D4×C3×C6, Q8×C3×C6, C32×C4.4D4
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], D4 [×2], C23, C32, C2×C6 [×28], C2×D4, C4○D4 [×2], C3×C6 [×7], C3×D4 [×8], C22×C6 [×4], C4.4D4, C62 [×7], C6×D4 [×4], C3×C4○D4 [×8], D4×C32 [×2], C2×C62, C3×C4.4D4 [×4], D4×C3×C6, C32×C4○D4 [×2], C32×C4.4D4

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

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3H 4A ··· 4F 4G 4H 6A ··· 6X 6Y ··· 6AN 12A ··· 12AV 12AW ··· 12BL order 1 2 2 2 2 2 3 ··· 3 4 ··· 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 4 4 1 ··· 1 2 ··· 2 4 4 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 C4○D4 C3×D4 C3×C4○D4 kernel C32×C4.4D4 C122 C32×C22⋊C4 D4×C3×C6 Q8×C3×C6 C3×C4.4D4 C4×C12 C3×C22⋊C4 C6×D4 C6×Q8 C3×C12 C3×C6 C12 C6 # reps 1 1 4 1 1 8 8 32 8 8 2 4 16 32

Matrix representation of C32×C4.4D4 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 6 10 0 0 8 7 0 0 0 0 12 0 0 0 0 12
,
 4 11 0 0 1 9 0 0 0 0 8 2 0 0 0 5
,
 4 11 0 0 2 9 0 0 0 0 8 2 0 0 1 5
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[6,8,0,0,10,7,0,0,0,0,12,0,0,0,0,12],[4,1,0,0,11,9,0,0,0,0,8,0,0,0,2,5],[4,2,0,0,11,9,0,0,0,0,8,1,0,0,2,5] >;

C32×C4.4D4 in GAP, Magma, Sage, TeX

C_3^2\times C_4._4D_4
% in TeX

G:=Group("C3^2xC4.4D4");
// GroupNames label

G:=SmallGroup(288,821);
// by ID

G=gap.SmallGroup(288,821);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,1016,3110,394]);
// Polycyclic

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

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