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G = C12.28D14order 336 = 24·3·7

7th non-split extension by C12 of D14 acting via D14/D7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D849C2, Dic65D7, D14.7D6, C28.18D6, C12.28D14, C84.21C22, C42.10C23, Dic7.12D6, Dic3.4D14, D42.2C22, (C4×D7)⋊2S3, D21⋊C42C2, (C12×D7)⋊2C2, C217(C4○D4), C71(C4○D12), C3⋊D282C2, C4.14(S3×D7), (C7×Dic6)⋊3C2, C31(Q82D7), (C6×D7).8C22, C6.10(C22×D7), C14.10(C22×S3), (C7×Dic3).4C22, (C3×Dic7).10C22, C2.14(C2×S3×D7), SmallGroup(336,146)

Series: Derived Chief Lower central Upper central

C1C42 — C12.28D14
C1C7C21C42C6×D7C3⋊D28 — C12.28D14
C21C42 — C12.28D14
C1C2C4

Generators and relations for C12.28D14
 G = < a,b,c | a12=c2=1, b14=a6, bab-1=cac=a-1, cbc=a6b13 >

Subgroups: 500 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], S3 [×2], C6, C6, C7, C2×C4 [×3], D4 [×3], Q8, Dic3 [×2], C12, C12, D6 [×2], C2×C6, D7 [×3], C14, C4○D4, C21, Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, Dic7, C28, C28 [×2], D14, D14 [×2], C3×D7, D21 [×2], C42, C4○D12, C4×D7, C4×D7 [×2], D28 [×3], C7×Q8, C7×Dic3 [×2], C3×Dic7, C84, C6×D7, D42 [×2], Q82D7, D21⋊C4 [×2], C3⋊D28 [×2], C12×D7, C7×Dic6, D84, C12.28D14
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], D7, C4○D4, C22×S3, D14 [×3], C4○D12, C22×D7, S3×D7, Q82D7, C2×S3×D7, C12.28D14

Smallest permutation representation of C12.28D14
On 168 points
Generators in S168
(1 142 132 112 52 60 15 156 118 98 38 74)(2 75 39 99 119 157 16 61 53 85 133 143)(3 144 134 86 54 62 17 158 120 100 40 76)(4 77 41 101 121 159 18 63 55 87 135 145)(5 146 136 88 56 64 19 160 122 102 42 78)(6 79 43 103 123 161 20 65 29 89 137 147)(7 148 138 90 30 66 21 162 124 104 44 80)(8 81 45 105 125 163 22 67 31 91 139 149)(9 150 140 92 32 68 23 164 126 106 46 82)(10 83 47 107 127 165 24 69 33 93 113 151)(11 152 114 94 34 70 25 166 128 108 48 84)(12 57 49 109 129 167 26 71 35 95 115 153)(13 154 116 96 36 72 27 168 130 110 50 58)(14 59 51 111 131 141 28 73 37 97 117 155)
(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)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 28)(23 27)(24 26)(29 133)(30 132)(31 131)(32 130)(33 129)(34 128)(35 127)(36 126)(37 125)(38 124)(39 123)(40 122)(41 121)(42 120)(43 119)(44 118)(45 117)(46 116)(47 115)(48 114)(49 113)(50 140)(51 139)(52 138)(53 137)(54 136)(55 135)(56 134)(57 151)(58 150)(59 149)(60 148)(61 147)(62 146)(63 145)(64 144)(65 143)(66 142)(67 141)(68 168)(69 167)(70 166)(71 165)(72 164)(73 163)(74 162)(75 161)(76 160)(77 159)(78 158)(79 157)(80 156)(81 155)(82 154)(83 153)(84 152)(85 89)(86 88)(90 112)(91 111)(92 110)(93 109)(94 108)(95 107)(96 106)(97 105)(98 104)(99 103)(100 102)

G:=sub<Sym(168)| (1,142,132,112,52,60,15,156,118,98,38,74)(2,75,39,99,119,157,16,61,53,85,133,143)(3,144,134,86,54,62,17,158,120,100,40,76)(4,77,41,101,121,159,18,63,55,87,135,145)(5,146,136,88,56,64,19,160,122,102,42,78)(6,79,43,103,123,161,20,65,29,89,137,147)(7,148,138,90,30,66,21,162,124,104,44,80)(8,81,45,105,125,163,22,67,31,91,139,149)(9,150,140,92,32,68,23,164,126,106,46,82)(10,83,47,107,127,165,24,69,33,93,113,151)(11,152,114,94,34,70,25,166,128,108,48,84)(12,57,49,109,129,167,26,71,35,95,115,153)(13,154,116,96,36,72,27,168,130,110,50,58)(14,59,51,111,131,141,28,73,37,97,117,155), (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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,126)(37,125)(38,124)(39,123)(40,122)(41,121)(42,120)(43,119)(44,118)(45,117)(46,116)(47,115)(48,114)(49,113)(50,140)(51,139)(52,138)(53,137)(54,136)(55,135)(56,134)(57,151)(58,150)(59,149)(60,148)(61,147)(62,146)(63,145)(64,144)(65,143)(66,142)(67,141)(68,168)(69,167)(70,166)(71,165)(72,164)(73,163)(74,162)(75,161)(76,160)(77,159)(78,158)(79,157)(80,156)(81,155)(82,154)(83,153)(84,152)(85,89)(86,88)(90,112)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102)>;

G:=Group( (1,142,132,112,52,60,15,156,118,98,38,74)(2,75,39,99,119,157,16,61,53,85,133,143)(3,144,134,86,54,62,17,158,120,100,40,76)(4,77,41,101,121,159,18,63,55,87,135,145)(5,146,136,88,56,64,19,160,122,102,42,78)(6,79,43,103,123,161,20,65,29,89,137,147)(7,148,138,90,30,66,21,162,124,104,44,80)(8,81,45,105,125,163,22,67,31,91,139,149)(9,150,140,92,32,68,23,164,126,106,46,82)(10,83,47,107,127,165,24,69,33,93,113,151)(11,152,114,94,34,70,25,166,128,108,48,84)(12,57,49,109,129,167,26,71,35,95,115,153)(13,154,116,96,36,72,27,168,130,110,50,58)(14,59,51,111,131,141,28,73,37,97,117,155), (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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,126)(37,125)(38,124)(39,123)(40,122)(41,121)(42,120)(43,119)(44,118)(45,117)(46,116)(47,115)(48,114)(49,113)(50,140)(51,139)(52,138)(53,137)(54,136)(55,135)(56,134)(57,151)(58,150)(59,149)(60,148)(61,147)(62,146)(63,145)(64,144)(65,143)(66,142)(67,141)(68,168)(69,167)(70,166)(71,165)(72,164)(73,163)(74,162)(75,161)(76,160)(77,159)(78,158)(79,157)(80,156)(81,155)(82,154)(83,153)(84,152)(85,89)(86,88)(90,112)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102) );

G=PermutationGroup([(1,142,132,112,52,60,15,156,118,98,38,74),(2,75,39,99,119,157,16,61,53,85,133,143),(3,144,134,86,54,62,17,158,120,100,40,76),(4,77,41,101,121,159,18,63,55,87,135,145),(5,146,136,88,56,64,19,160,122,102,42,78),(6,79,43,103,123,161,20,65,29,89,137,147),(7,148,138,90,30,66,21,162,124,104,44,80),(8,81,45,105,125,163,22,67,31,91,139,149),(9,150,140,92,32,68,23,164,126,106,46,82),(10,83,47,107,127,165,24,69,33,93,113,151),(11,152,114,94,34,70,25,166,128,108,48,84),(12,57,49,109,129,167,26,71,35,95,115,153),(13,154,116,96,36,72,27,168,130,110,50,58),(14,59,51,111,131,141,28,73,37,97,117,155)], [(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),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,28),(23,27),(24,26),(29,133),(30,132),(31,131),(32,130),(33,129),(34,128),(35,127),(36,126),(37,125),(38,124),(39,123),(40,122),(41,121),(42,120),(43,119),(44,118),(45,117),(46,116),(47,115),(48,114),(49,113),(50,140),(51,139),(52,138),(53,137),(54,136),(55,135),(56,134),(57,151),(58,150),(59,149),(60,148),(61,147),(62,146),(63,145),(64,144),(65,143),(66,142),(67,141),(68,168),(69,167),(70,166),(71,165),(72,164),(73,163),(74,162),(75,161),(76,160),(77,159),(78,158),(79,157),(80,156),(81,155),(82,154),(83,153),(84,152),(85,89),(86,88),(90,112),(91,111),(92,110),(93,109),(94,108),(95,107),(96,106),(97,105),(98,104),(99,103),(100,102)])

45 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E6A6B6C7A7B7C12A12B12C12D14A14B14C21A21B21C28A28B28C28D···28I42A42B42C84A···84F
order122223444446667771212121214141421212128282828···2842424284···84
size111442422266772141422222141422244444412···124444···4

45 irreducible representations

dim1111112222222224444
type+++++++++++++++++
imageC1C2C2C2C2C2S3D6D6D6D7C4○D4D14D14C4○D12S3×D7Q82D7C2×S3×D7C12.28D14
kernelC12.28D14D21⋊C4C3⋊D28C12×D7C7×Dic6D84C4×D7Dic7C28D14Dic6C21Dic3C12C7C4C3C2C1
# reps1221111111326343336

Matrix representation of C12.28D14 in GL4(𝔽337) generated by

29231500
46000
0010
0001
,
25710900
2698000
000109
0034303
,
1000
28933600
0034109
00178303
G:=sub<GL(4,GF(337))| [292,46,0,0,315,0,0,0,0,0,1,0,0,0,0,1],[257,269,0,0,109,80,0,0,0,0,0,34,0,0,109,303],[1,289,0,0,0,336,0,0,0,0,34,178,0,0,109,303] >;

C12.28D14 in GAP, Magma, Sage, TeX

C_{12}._{28}D_{14}
% in TeX

G:=Group("C12.28D14");
// GroupNames label

G:=SmallGroup(336,146);
// by ID

G=gap.SmallGroup(336,146);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,121,55,116,50,490,10373]);
// Polycyclic

G:=Group<a,b,c|a^12=c^2=1,b^14=a^6,b*a*b^-1=c*a*c=a^-1,c*b*c=a^6*b^13>;
// generators/relations

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