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## G = C32×C8.C4order 288 = 25·32

### Direct product of C32 and C8.C4

direct product, metacyclic, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C32×C8.C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C6×C12 — C32×M4(2) — C32×C8.C4
 Lower central C1 — C2 — C4 — C32×C8.C4
 Upper central C1 — C3×C12 — C6×C12 — C32×C8.C4

Generators and relations for C32×C8.C4
G = < a,b,c,d | a3=b3=c8=1, d4=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 108 in 90 conjugacy classes, 72 normal (20 characteristic)
C1, C2, C2, C3 [×4], C4 [×2], C22, C6 [×4], C6 [×4], C8 [×2], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], C2×C8, M4(2) [×2], C3×C6, C3×C6, C24 [×8], C24 [×8], C2×C12 [×4], C8.C4, C3×C12 [×2], C62, C2×C24 [×4], C3×M4(2) [×8], C3×C24 [×2], C3×C24 [×2], C6×C12, C3×C8.C4 [×4], C6×C24, C32×M4(2) [×2], C32×C8.C4
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, D4, Q8, C32, C12 [×8], C2×C6 [×4], C4⋊C4, C3×C6 [×3], C2×C12 [×4], C3×D4 [×4], C3×Q8 [×4], C8.C4, C3×C12 [×2], C62, C3×C4⋊C4 [×4], C6×C12, D4×C32, Q8×C32, C3×C8.C4 [×4], C32×C4⋊C4, C32×C8.C4

Smallest permutation representation of C32×C8.C4
On 144 points
Generators in S144
(1 73 61)(2 74 62)(3 75 63)(4 76 64)(5 77 57)(6 78 58)(7 79 59)(8 80 60)(9 67 44)(10 68 45)(11 69 46)(12 70 47)(13 71 48)(14 72 41)(15 65 42)(16 66 43)(17 29 53)(18 30 54)(19 31 55)(20 32 56)(21 25 49)(22 26 50)(23 27 51)(24 28 52)(33 112 104)(34 105 97)(35 106 98)(36 107 99)(37 108 100)(38 109 101)(39 110 102)(40 111 103)(81 142 89)(82 143 90)(83 144 91)(84 137 92)(85 138 93)(86 139 94)(87 140 95)(88 141 96)(113 130 121)(114 131 122)(115 132 123)(116 133 124)(117 134 125)(118 135 126)(119 136 127)(120 129 128)
(1 31 11)(2 32 12)(3 25 13)(4 26 14)(5 27 15)(6 28 16)(7 29 9)(8 30 10)(17 44 59)(18 45 60)(19 46 61)(20 47 62)(21 48 63)(22 41 64)(23 42 57)(24 43 58)(33 123 140)(34 124 141)(35 125 142)(36 126 143)(37 127 144)(38 128 137)(39 121 138)(40 122 139)(49 71 75)(50 72 76)(51 65 77)(52 66 78)(53 67 79)(54 68 80)(55 69 73)(56 70 74)(81 98 134)(82 99 135)(83 100 136)(84 101 129)(85 102 130)(86 103 131)(87 104 132)(88 97 133)(89 106 117)(90 107 118)(91 108 119)(92 109 120)(93 110 113)(94 111 114)(95 112 115)(96 105 116)
(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)
(1 101 3 99 5 97 7 103)(2 100 4 98 6 104 8 102)(9 86 11 84 13 82 15 88)(10 85 12 83 14 81 16 87)(17 114 19 120 21 118 23 116)(18 113 20 119 22 117 24 115)(25 135 27 133 29 131 31 129)(26 134 28 132 30 130 32 136)(33 80 39 74 37 76 35 78)(34 79 40 73 38 75 36 77)(41 89 43 95 45 93 47 91)(42 96 44 94 46 92 48 90)(49 126 51 124 53 122 55 128)(50 125 52 123 54 121 56 127)(57 105 59 111 61 109 63 107)(58 112 60 110 62 108 64 106)(65 141 67 139 69 137 71 143)(66 140 68 138 70 144 72 142)

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

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

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

126 conjugacy classes

 class 1 2A 2B 3A ··· 3H 4A 4B 4C 6A ··· 6H 6I ··· 6P 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12P 12Q ··· 12X 24A ··· 24AF 24AG ··· 24BL order 1 2 2 3 ··· 3 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 8 8 8 8 12 ··· 12 12 ··· 12 24 ··· 24 24 ··· 24 size 1 1 2 1 ··· 1 1 1 2 1 ··· 1 2 ··· 2 2 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C3 C4 C6 C6 C12 D4 Q8 C3×D4 C3×Q8 C8.C4 C3×C8.C4 kernel C32×C8.C4 C6×C24 C32×M4(2) C3×C8.C4 C3×C24 C2×C24 C3×M4(2) C24 C3×C12 C62 C12 C2×C6 C32 C3 # reps 1 1 2 8 4 8 16 32 1 1 8 8 4 32

Matrix representation of C32×C8.C4 in GL4(𝔽73) generated by

 64 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 8 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 0 72 0 0 0 0 22 0 0 0 20 10
,
 27 0 0 0 0 46 0 0 0 0 52 71 0 0 15 21
G:=sub<GL(4,GF(73))| [64,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,22,20,0,0,0,10],[27,0,0,0,0,46,0,0,0,0,52,15,0,0,71,21] >;

C32×C8.C4 in GAP, Magma, Sage, TeX

C_3^2\times C_8.C_4
% in TeX

G:=Group("C3^2xC8.C4");
// GroupNames label

G:=SmallGroup(288,326);
// by ID

G=gap.SmallGroup(288,326);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,504,533,260,6304,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^8=1,d^4=c^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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