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G = C2×C9⋊C24order 432 = 24·33

Direct product of C2 and C9⋊C24

direct product, metacyclic, supersoluble, monomial

Aliases: C2×C9⋊C24, C18⋊C24, C36.3C12, C62.3Dic3, C9⋊C87C6, C92(C2×C24), (C2×C36).4C6, C4.3(C9⋊C12), C12.94(S3×C6), (C2×C18).1C12, C18.5(C2×C12), (C6×C12).18S3, C36.15(C2×C6), (C3×C12).62D6, C6.14(C6×Dic3), (C3×C12).9Dic3, (C2×3- 1+2)⋊C8, C22.2(C9⋊C12), C12.15(C3×Dic3), 3- 1+22(C2×C8), (C4×3- 1+2).3C4, (C22×3- 1+2).1C4, (C4×3- 1+2).14C22, (C2×C9⋊C8)⋊C3, C3.3(C6×C3⋊C8), C6.6(C3×C3⋊C8), C32.(C2×C3⋊C8), C4.14(C2×C9⋊C6), C2.1(C2×C9⋊C12), (C3×C6).5(C3⋊C8), (C2×C4).5(C9⋊C6), (C2×C12).33(C3×S3), (C2×C6).16(C3×Dic3), (C3×C6).10(C2×Dic3), (C2×C4×3- 1+2).4C2, (C2×3- 1+2).5(C2×C4), SmallGroup(432,142)

Series: Derived Chief Lower central Upper central

C1C9 — C2×C9⋊C24
C1C3C9C18C36C4×3- 1+2C9⋊C24 — C2×C9⋊C24
C9 — C2×C9⋊C24
C1C2×C4

Generators and relations for C2×C9⋊C24
 G = < a,b,c | a2=b9=c24=1, ab=ba, ac=ca, cbc-1=b2 >

Subgroups: 158 in 74 conjugacy classes, 46 normal (32 characteristic)
C1, C2, C2 [×2], C3, C3, C4 [×2], C22, C6, C6 [×2], C6 [×3], C8 [×2], C2×C4, C9, C9, C32, C12 [×2], C12 [×2], C2×C6, C2×C6, C2×C8, C18, C18 [×2], C18 [×3], C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×2], C2×C12, C2×C12, 3- 1+2, C36 [×2], C36 [×2], C2×C18, C2×C18, C3×C12 [×2], C62, C2×C3⋊C8, C2×C24, C2×3- 1+2, C2×3- 1+2 [×2], C9⋊C8 [×2], C2×C36, C2×C36, C3×C3⋊C8 [×2], C6×C12, C4×3- 1+2 [×2], C22×3- 1+2, C2×C9⋊C8, C6×C3⋊C8, C9⋊C24 [×2], C2×C4×3- 1+2, C2×C9⋊C24
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, Dic3 [×2], C12 [×2], D6, C2×C6, C2×C8, C3×S3, C3⋊C8 [×2], C24 [×2], C2×Dic3, C2×C12, C3×Dic3 [×2], S3×C6, C2×C3⋊C8, C2×C24, C9⋊C6, C3×C3⋊C8 [×2], C6×Dic3, C9⋊C12 [×2], C2×C9⋊C6, C6×C3⋊C8, C9⋊C24 [×2], C2×C9⋊C12, C2×C9⋊C24

Smallest permutation representation of C2×C9⋊C24
On 144 points
Generators in S144
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 116)(62 117)(63 118)(64 119)(65 120)(66 97)(67 98)(68 99)(69 100)(70 101)(71 102)(72 103)(73 140)(74 141)(75 142)(76 143)(77 144)(78 121)(79 122)(80 123)(81 124)(82 125)(83 126)(84 127)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)(93 136)(94 137)(95 138)(96 139)
(1 72 88 42 64 96 12 56 80)(2 89 65 13 81 49 43 73 57)(3 66 82 44 58 90 14 50 74)(4 83 59 15 75 67 45 91 51)(5 60 76 46 52 84 16 68 92)(6 77 53 9 93 61 47 85 69)(7 54 94 48 70 78 10 62 86)(8 95 71 11 87 55 41 79 63)(17 103 131 35 119 139 28 111 123)(18 132 120 29 124 104 36 140 112)(19 97 125 37 113 133 30 105 141)(20 126 114 31 142 98 38 134 106)(21 115 143 39 107 127 32 99 135)(22 144 108 25 136 116 40 128 100)(23 109 137 33 101 121 26 117 129)(24 138 102 27 130 110 34 122 118)
(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)

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

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

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

80 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D6A6B6C6D···6I8A···8H9A9B9C12A12B12C12D12E···12L18A···18I24A···24P36A···36L
order122233344446666···68···89991212121212···1218···1824···2436···36
size111123311112223···39···966622223···36···69···96···6

80 irreducible representations

dim111111111111222222222266666
type++++-+-+-+-
imageC1C2C2C3C4C4C6C6C8C12C12C24S3Dic3D6Dic3C3×S3C3⋊C8C3×Dic3S3×C6C3×Dic3C3×C3⋊C8C9⋊C6C9⋊C12C2×C9⋊C6C9⋊C12C9⋊C24
kernelC2×C9⋊C24C9⋊C24C2×C4×3- 1+2C2×C9⋊C8C4×3- 1+2C22×3- 1+2C9⋊C8C2×C36C2×3- 1+2C36C2×C18C18C6×C12C3×C12C3×C12C62C2×C12C3×C6C12C12C2×C6C6C2×C4C4C4C22C2
# reps1212224284416111124222811114

Matrix representation of C2×C9⋊C24 in GL10(𝔽73)

1000000000
0100000000
00720000000
00072000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
0100000000
727200000000
00721000000
00720000000
000011727100
00000017200
00000007201
0000110727272
00000007200
00001007200
,
223800000000
165100000000
006420000000
00119000000
000012510000
000063610000
0000120005161
00006351001022
00006351102200
0000061126300

G:=sub<GL(10,GF(73))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,71,72,72,72,72,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0],[22,16,0,0,0,0,0,0,0,0,38,51,0,0,0,0,0,0,0,0,0,0,64,11,0,0,0,0,0,0,0,0,20,9,0,0,0,0,0,0,0,0,0,0,12,63,12,63,63,0,0,0,0,0,51,61,0,51,51,61,0,0,0,0,0,0,0,0,10,12,0,0,0,0,0,0,0,0,22,63,0,0,0,0,0,0,51,10,0,0,0,0,0,0,0,0,61,22,0,0] >;

C2×C9⋊C24 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes C_{24}
% in TeX

G:=Group("C2xC9:C24");
// GroupNames label

G:=SmallGroup(432,142);
// by ID

G=gap.SmallGroup(432,142);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,80,10085,2035,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^2=b^9=c^24=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations

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