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G = Dic66Q8order 192 = 26·3

4th semidirect product of Dic6 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic66Q8, C42.85D6, C12.21SD16, C4⋊C4.88D6, C4⋊Q8.13S3, C4.14(S3×Q8), C35(Q8⋊Q8), C12.41(C2×Q8), (C2×C12).163D4, C4.8(D4.S3), C12⋊C8.23C2, C6.62(C2×SD16), C12.88(C4○D4), (C4×Dic6).19C2, C6.78(C22⋊Q8), (C4×C12).141C22, (C2×C12).412C23, C2.15(D63Q8), C4.37(Q83S3), C6.SD16.16C2, C12.Q8.18C2, C6.100(C8.C22), C4⋊Dic3.351C22, C2.21(Q8.11D6), (C2×Dic6).277C22, (C3×C4⋊Q8).13C2, (C2×C6).543(C2×D4), C2.16(C2×D4.S3), (C2×C3⋊C8).142C22, (C2×C4).192(C3⋊D4), (C3×C4⋊C4).135C22, (C2×C4).509(C22×S3), C22.215(C2×C3⋊D4), SmallGroup(192,653)

Series: Derived Chief Lower central Upper central

C1C2×C12 — Dic66Q8
C1C3C6C12C2×C12C2×Dic6C4×Dic6 — Dic66Q8
C3C6C2×C12 — Dic66Q8
C1C22C42C4⋊Q8

Generators and relations for Dic66Q8
 G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, bd=db, dcd-1=c-1 >

Subgroups: 224 in 96 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×2], C4 [×6], C22, C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×4], Q8 [×5], Dic3 [×3], C12 [×2], C12 [×2], C12 [×3], C2×C6, C42, C42, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×Q8 [×2], C3⋊C8 [×2], Dic6 [×2], Dic6, C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], Q8⋊C4 [×2], C4⋊C8, C4.Q8 [×2], C4×Q8, C4⋊Q8, C2×C3⋊C8 [×2], C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C6×Q8, Q8⋊Q8, C12⋊C8, C12.Q8 [×2], C6.SD16 [×2], C4×Dic6, C3×C4⋊Q8, Dic66Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], SD16 [×2], C2×D4, C2×Q8, C4○D4, C3⋊D4 [×2], C22×S3, C22⋊Q8, C2×SD16, C8.C22, D4.S3 [×2], S3×Q8, Q83S3, C2×C3⋊D4, Q8⋊Q8, C2×D4.S3, Q8.11D6, D63Q8, Dic66Q8

Smallest permutation representation of Dic66Q8
Regular action on 192 points
Generators in S192
(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)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)
(1 142 7 136)(2 141 8 135)(3 140 9 134)(4 139 10 133)(5 138 11 144)(6 137 12 143)(13 105 19 99)(14 104 20 98)(15 103 21 97)(16 102 22 108)(17 101 23 107)(18 100 24 106)(25 91 31 85)(26 90 32 96)(27 89 33 95)(28 88 34 94)(29 87 35 93)(30 86 36 92)(37 174 43 180)(38 173 44 179)(39 172 45 178)(40 171 46 177)(41 170 47 176)(42 169 48 175)(49 66 55 72)(50 65 56 71)(51 64 57 70)(52 63 58 69)(53 62 59 68)(54 61 60 67)(73 184 79 190)(74 183 80 189)(75 182 81 188)(76 181 82 187)(77 192 83 186)(78 191 84 185)(109 132 115 126)(110 131 116 125)(111 130 117 124)(112 129 118 123)(113 128 119 122)(114 127 120 121)(145 161 151 167)(146 160 152 166)(147 159 153 165)(148 158 154 164)(149 157 155 163)(150 168 156 162)
(1 76 169 123)(2 83 170 130)(3 78 171 125)(4 73 172 132)(5 80 173 127)(6 75 174 122)(7 82 175 129)(8 77 176 124)(9 84 177 131)(10 79 178 126)(11 74 179 121)(12 81 180 128)(13 87 161 60)(14 94 162 55)(15 89 163 50)(16 96 164 57)(17 91 165 52)(18 86 166 59)(19 93 167 54)(20 88 168 49)(21 95 157 56)(22 90 158 51)(23 85 159 58)(24 92 160 53)(25 156 69 98)(26 151 70 105)(27 146 71 100)(28 153 72 107)(29 148 61 102)(30 155 62 97)(31 150 63 104)(32 145 64 99)(33 152 65 106)(34 147 66 101)(35 154 67 108)(36 149 68 103)(37 110 143 191)(38 117 144 186)(39 112 133 181)(40 119 134 188)(41 114 135 183)(42 109 136 190)(43 116 137 185)(44 111 138 192)(45 118 139 187)(46 113 140 182)(47 120 141 189)(48 115 142 184)
(1 163 169 15)(2 164 170 16)(3 165 171 17)(4 166 172 18)(5 167 173 19)(6 168 174 20)(7 157 175 21)(8 158 176 22)(9 159 177 23)(10 160 178 24)(11 161 179 13)(12 162 180 14)(25 116 69 185)(26 117 70 186)(27 118 71 187)(28 119 72 188)(29 120 61 189)(30 109 62 190)(31 110 63 191)(32 111 64 192)(33 112 65 181)(34 113 66 182)(35 114 67 183)(36 115 68 184)(37 104 143 150)(38 105 144 151)(39 106 133 152)(40 107 134 153)(41 108 135 154)(42 97 136 155)(43 98 137 156)(44 99 138 145)(45 100 139 146)(46 101 140 147)(47 102 141 148)(48 103 142 149)(49 75 88 122)(50 76 89 123)(51 77 90 124)(52 78 91 125)(53 79 92 126)(54 80 93 127)(55 81 94 128)(56 82 95 129)(57 83 96 130)(58 84 85 131)(59 73 86 132)(60 74 87 121)

G:=sub<Sym(192)| (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)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,142,7,136)(2,141,8,135)(3,140,9,134)(4,139,10,133)(5,138,11,144)(6,137,12,143)(13,105,19,99)(14,104,20,98)(15,103,21,97)(16,102,22,108)(17,101,23,107)(18,100,24,106)(25,91,31,85)(26,90,32,96)(27,89,33,95)(28,88,34,94)(29,87,35,93)(30,86,36,92)(37,174,43,180)(38,173,44,179)(39,172,45,178)(40,171,46,177)(41,170,47,176)(42,169,48,175)(49,66,55,72)(50,65,56,71)(51,64,57,70)(52,63,58,69)(53,62,59,68)(54,61,60,67)(73,184,79,190)(74,183,80,189)(75,182,81,188)(76,181,82,187)(77,192,83,186)(78,191,84,185)(109,132,115,126)(110,131,116,125)(111,130,117,124)(112,129,118,123)(113,128,119,122)(114,127,120,121)(145,161,151,167)(146,160,152,166)(147,159,153,165)(148,158,154,164)(149,157,155,163)(150,168,156,162), (1,76,169,123)(2,83,170,130)(3,78,171,125)(4,73,172,132)(5,80,173,127)(6,75,174,122)(7,82,175,129)(8,77,176,124)(9,84,177,131)(10,79,178,126)(11,74,179,121)(12,81,180,128)(13,87,161,60)(14,94,162,55)(15,89,163,50)(16,96,164,57)(17,91,165,52)(18,86,166,59)(19,93,167,54)(20,88,168,49)(21,95,157,56)(22,90,158,51)(23,85,159,58)(24,92,160,53)(25,156,69,98)(26,151,70,105)(27,146,71,100)(28,153,72,107)(29,148,61,102)(30,155,62,97)(31,150,63,104)(32,145,64,99)(33,152,65,106)(34,147,66,101)(35,154,67,108)(36,149,68,103)(37,110,143,191)(38,117,144,186)(39,112,133,181)(40,119,134,188)(41,114,135,183)(42,109,136,190)(43,116,137,185)(44,111,138,192)(45,118,139,187)(46,113,140,182)(47,120,141,189)(48,115,142,184), (1,163,169,15)(2,164,170,16)(3,165,171,17)(4,166,172,18)(5,167,173,19)(6,168,174,20)(7,157,175,21)(8,158,176,22)(9,159,177,23)(10,160,178,24)(11,161,179,13)(12,162,180,14)(25,116,69,185)(26,117,70,186)(27,118,71,187)(28,119,72,188)(29,120,61,189)(30,109,62,190)(31,110,63,191)(32,111,64,192)(33,112,65,181)(34,113,66,182)(35,114,67,183)(36,115,68,184)(37,104,143,150)(38,105,144,151)(39,106,133,152)(40,107,134,153)(41,108,135,154)(42,97,136,155)(43,98,137,156)(44,99,138,145)(45,100,139,146)(46,101,140,147)(47,102,141,148)(48,103,142,149)(49,75,88,122)(50,76,89,123)(51,77,90,124)(52,78,91,125)(53,79,92,126)(54,80,93,127)(55,81,94,128)(56,82,95,129)(57,83,96,130)(58,84,85,131)(59,73,86,132)(60,74,87,121)>;

G:=Group( (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)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,142,7,136)(2,141,8,135)(3,140,9,134)(4,139,10,133)(5,138,11,144)(6,137,12,143)(13,105,19,99)(14,104,20,98)(15,103,21,97)(16,102,22,108)(17,101,23,107)(18,100,24,106)(25,91,31,85)(26,90,32,96)(27,89,33,95)(28,88,34,94)(29,87,35,93)(30,86,36,92)(37,174,43,180)(38,173,44,179)(39,172,45,178)(40,171,46,177)(41,170,47,176)(42,169,48,175)(49,66,55,72)(50,65,56,71)(51,64,57,70)(52,63,58,69)(53,62,59,68)(54,61,60,67)(73,184,79,190)(74,183,80,189)(75,182,81,188)(76,181,82,187)(77,192,83,186)(78,191,84,185)(109,132,115,126)(110,131,116,125)(111,130,117,124)(112,129,118,123)(113,128,119,122)(114,127,120,121)(145,161,151,167)(146,160,152,166)(147,159,153,165)(148,158,154,164)(149,157,155,163)(150,168,156,162), (1,76,169,123)(2,83,170,130)(3,78,171,125)(4,73,172,132)(5,80,173,127)(6,75,174,122)(7,82,175,129)(8,77,176,124)(9,84,177,131)(10,79,178,126)(11,74,179,121)(12,81,180,128)(13,87,161,60)(14,94,162,55)(15,89,163,50)(16,96,164,57)(17,91,165,52)(18,86,166,59)(19,93,167,54)(20,88,168,49)(21,95,157,56)(22,90,158,51)(23,85,159,58)(24,92,160,53)(25,156,69,98)(26,151,70,105)(27,146,71,100)(28,153,72,107)(29,148,61,102)(30,155,62,97)(31,150,63,104)(32,145,64,99)(33,152,65,106)(34,147,66,101)(35,154,67,108)(36,149,68,103)(37,110,143,191)(38,117,144,186)(39,112,133,181)(40,119,134,188)(41,114,135,183)(42,109,136,190)(43,116,137,185)(44,111,138,192)(45,118,139,187)(46,113,140,182)(47,120,141,189)(48,115,142,184), (1,163,169,15)(2,164,170,16)(3,165,171,17)(4,166,172,18)(5,167,173,19)(6,168,174,20)(7,157,175,21)(8,158,176,22)(9,159,177,23)(10,160,178,24)(11,161,179,13)(12,162,180,14)(25,116,69,185)(26,117,70,186)(27,118,71,187)(28,119,72,188)(29,120,61,189)(30,109,62,190)(31,110,63,191)(32,111,64,192)(33,112,65,181)(34,113,66,182)(35,114,67,183)(36,115,68,184)(37,104,143,150)(38,105,144,151)(39,106,133,152)(40,107,134,153)(41,108,135,154)(42,97,136,155)(43,98,137,156)(44,99,138,145)(45,100,139,146)(46,101,140,147)(47,102,141,148)(48,103,142,149)(49,75,88,122)(50,76,89,123)(51,77,90,124)(52,78,91,125)(53,79,92,126)(54,80,93,127)(55,81,94,128)(56,82,95,129)(57,83,96,130)(58,84,85,131)(59,73,86,132)(60,74,87,121) );

G=PermutationGroup([(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),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,142,7,136),(2,141,8,135),(3,140,9,134),(4,139,10,133),(5,138,11,144),(6,137,12,143),(13,105,19,99),(14,104,20,98),(15,103,21,97),(16,102,22,108),(17,101,23,107),(18,100,24,106),(25,91,31,85),(26,90,32,96),(27,89,33,95),(28,88,34,94),(29,87,35,93),(30,86,36,92),(37,174,43,180),(38,173,44,179),(39,172,45,178),(40,171,46,177),(41,170,47,176),(42,169,48,175),(49,66,55,72),(50,65,56,71),(51,64,57,70),(52,63,58,69),(53,62,59,68),(54,61,60,67),(73,184,79,190),(74,183,80,189),(75,182,81,188),(76,181,82,187),(77,192,83,186),(78,191,84,185),(109,132,115,126),(110,131,116,125),(111,130,117,124),(112,129,118,123),(113,128,119,122),(114,127,120,121),(145,161,151,167),(146,160,152,166),(147,159,153,165),(148,158,154,164),(149,157,155,163),(150,168,156,162)], [(1,76,169,123),(2,83,170,130),(3,78,171,125),(4,73,172,132),(5,80,173,127),(6,75,174,122),(7,82,175,129),(8,77,176,124),(9,84,177,131),(10,79,178,126),(11,74,179,121),(12,81,180,128),(13,87,161,60),(14,94,162,55),(15,89,163,50),(16,96,164,57),(17,91,165,52),(18,86,166,59),(19,93,167,54),(20,88,168,49),(21,95,157,56),(22,90,158,51),(23,85,159,58),(24,92,160,53),(25,156,69,98),(26,151,70,105),(27,146,71,100),(28,153,72,107),(29,148,61,102),(30,155,62,97),(31,150,63,104),(32,145,64,99),(33,152,65,106),(34,147,66,101),(35,154,67,108),(36,149,68,103),(37,110,143,191),(38,117,144,186),(39,112,133,181),(40,119,134,188),(41,114,135,183),(42,109,136,190),(43,116,137,185),(44,111,138,192),(45,118,139,187),(46,113,140,182),(47,120,141,189),(48,115,142,184)], [(1,163,169,15),(2,164,170,16),(3,165,171,17),(4,166,172,18),(5,167,173,19),(6,168,174,20),(7,157,175,21),(8,158,176,22),(9,159,177,23),(10,160,178,24),(11,161,179,13),(12,162,180,14),(25,116,69,185),(26,117,70,186),(27,118,71,187),(28,119,72,188),(29,120,61,189),(30,109,62,190),(31,110,63,191),(32,111,64,192),(33,112,65,181),(34,113,66,182),(35,114,67,183),(36,115,68,184),(37,104,143,150),(38,105,144,151),(39,106,133,152),(40,107,134,153),(41,108,135,154),(42,97,136,155),(43,98,137,156),(44,99,138,145),(45,100,139,146),(46,101,140,147),(47,102,141,148),(48,103,142,149),(49,75,88,122),(50,76,89,123),(51,77,90,124),(52,78,91,125),(53,79,92,126),(54,80,93,127),(55,81,94,128),(56,82,95,129),(57,83,96,130),(58,84,85,131),(59,73,86,132),(60,74,87,121)])

33 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J4K6A6B6C8A8B8C8D12A···12F12G12H12I12J
order1222344444444444666888812···1212121212
size11112222248812121212222121212124···48888

33 irreducible representations

dim1111112222222244444
type+++++++-+++---+
imageC1C2C2C2C2C2S3Q8D4D6D6SD16C4○D4C3⋊D4C8.C22D4.S3S3×Q8Q83S3Q8.11D6
kernelDic66Q8C12⋊C8C12.Q8C6.SD16C4×Dic6C3×C4⋊Q8C4⋊Q8Dic6C2×C12C42C4⋊C4C12C12C2×C4C6C4C4C4C2
# reps1122111221242412112

Matrix representation of Dic66Q8 in GL6(𝔽73)

100000
010000
0072100
0072000
000001
0000720
,
100000
010000
0034200
00457000
00002850
00005045
,
0720000
100000
0072000
0007200
00005943
00004314
,
43620000
62300000
0072000
0007200
000010
000001

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,45,0,0,0,0,42,70,0,0,0,0,0,0,28,50,0,0,0,0,50,45],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,59,43,0,0,0,0,43,14],[43,62,0,0,0,0,62,30,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic66Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_6Q_8
% in TeX

G:=Group("Dic6:6Q8");
// GroupNames label

G:=SmallGroup(192,653);
// by ID

G=gap.SmallGroup(192,653);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,100,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=c^2,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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