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## G = C32×C42⋊2C2order 288 = 25·32

### Direct product of C32 and C42⋊2C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×C42⋊2C2
 Chief series C1 — C2 — C22 — C2×C6 — C62 — C2×C62 — C32×C22⋊C4 — C32×C42⋊2C2
 Lower central C1 — C22 — C32×C42⋊2C2
 Upper central C1 — C62 — C32×C42⋊2C2

Generators and relations for C32×C422C2
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, cd=dc, ece=cd2, ede=c2d-1 >

Subgroups: 252 in 180 conjugacy classes, 120 normal (10 characteristic)
C1, C2 [×3], C2, C3 [×4], C4 [×6], C22, C22 [×3], C6 [×12], C6 [×4], C2×C4 [×6], C23, C32, C12 [×24], C2×C6 [×4], C2×C6 [×12], C42, C22⋊C4 [×3], C4⋊C4 [×3], C3×C6 [×3], C3×C6, C2×C12 [×24], C22×C6 [×4], C422C2, C3×C12 [×6], C62, C62 [×3], C4×C12 [×4], C3×C22⋊C4 [×12], C3×C4⋊C4 [×12], C6×C12 [×6], C2×C62, C3×C422C2 [×4], C122, C32×C22⋊C4 [×3], C32×C4⋊C4 [×3], C32×C422C2
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], C23, C32, C2×C6 [×28], C4○D4 [×3], C3×C6 [×7], C22×C6 [×4], C422C2, C62 [×7], C3×C4○D4 [×12], C2×C62, C3×C422C2 [×4], C32×C4○D4 [×3], C32×C422C2

Smallest permutation representation of C32×C422C2
On 144 points
Generators in S144
(1 127 119)(2 128 120)(3 125 117)(4 126 118)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(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 73 65)(58 74 66)(59 75 67)(60 76 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)(93 109 101)(94 110 102)(95 111 103)(96 112 104)
(1 115 43)(2 116 44)(3 113 41)(4 114 42)(5 9 77)(6 10 78)(7 11 79)(8 12 80)(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)(37 105 109)(38 106 110)(39 107 111)(40 108 112)(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 73 141)(70 74 142)(71 75 143)(72 76 144)
(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 93 5 59)(2 94 6 60)(3 95 7 57)(4 96 8 58)(9 63 115 97)(10 64 116 98)(11 61 113 99)(12 62 114 100)(13 67 119 101)(14 68 120 102)(15 65 117 103)(16 66 118 104)(17 71 123 105)(18 72 124 106)(19 69 121 107)(20 70 122 108)(21 75 127 109)(22 76 128 110)(23 73 125 111)(24 74 126 112)(25 77 131 43)(26 78 132 44)(27 79 129 41)(28 80 130 42)(29 81 135 47)(30 82 136 48)(31 83 133 45)(32 84 134 46)(33 85 139 51)(34 86 140 52)(35 87 137 49)(36 88 138 50)(37 89 143 55)(38 90 144 56)(39 91 141 53)(40 92 142 54)
(2 6)(4 8)(10 116)(12 114)(14 120)(16 118)(18 124)(20 122)(22 128)(24 126)(25 129)(26 28)(27 131)(29 133)(30 32)(31 135)(33 137)(34 36)(35 139)(37 141)(38 40)(39 143)(42 80)(44 78)(46 84)(48 82)(50 88)(52 86)(54 92)(56 90)(57 93)(58 60)(59 95)(61 97)(62 64)(63 99)(65 101)(66 68)(67 103)(69 105)(70 72)(71 107)(73 109)(74 76)(75 111)(94 96)(98 100)(102 104)(106 108)(110 112)(130 132)(134 136)(138 140)(142 144)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 3A ··· 3H 4A ··· 4F 4G 4H 4I 6A ··· 6X 6Y ··· 6AF 12A ··· 12AV 12AW ··· 12BT order 1 2 2 2 2 3 ··· 3 4 ··· 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 4 1 ··· 1 2 ··· 2 4 4 4 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C4○D4 C3×C4○D4 kernel C32×C42⋊2C2 C122 C32×C22⋊C4 C32×C4⋊C4 C3×C42⋊2C2 C4×C12 C3×C22⋊C4 C3×C4⋊C4 C3×C6 C6 # reps 1 1 3 3 8 8 24 24 6 48

Matrix representation of C32×C422C2 in GL4(𝔽13) generated by

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

C32×C422C2 in GAP, Magma, Sage, TeX

C_3^2\times C_4^2\rtimes_2C_2
% in TeX

G:=Group("C3^2xC4^2:2C2");
// GroupNames label

G:=SmallGroup(288,823);
// by ID

G=gap.SmallGroup(288,823);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,1520,3110,394]);
// 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,c*d=d*c,e*c*e=c*d^2,e*d*e=c^2*d^-1>;
// generators/relations

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