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G = Dic38⋊C3order 456 = 23·3·19

The semidirect product of Dic38 and C3 acting faithfully

metacyclic, supersoluble, monomial

Aliases: Dic38⋊C3, C76.1C6, Dic19.C6, C19⋊C3⋊Q8, C19⋊(C3×Q8), C19⋊C12.C2, C4.(C19⋊C6), C38.1(C2×C6), C2.3(C2×C19⋊C6), (C4×C19⋊C3).1C2, (C2×C19⋊C3).1C22, SmallGroup(456,7)

Series: Derived Chief Lower central Upper central

C1C38 — Dic38⋊C3
C1C19C38C2×C19⋊C3C19⋊C12 — Dic38⋊C3
C19C38 — Dic38⋊C3
C1C2C4

Generators and relations for Dic38⋊C3
 G = < a,b,c | a76=c3=1, b2=a38, bab-1=a-1, cac-1=a49, bc=cb >

19C3
19C4
19C4
19C6
19Q8
19C12
19C12
19C12
19C3×Q8

Character table of Dic38⋊C3

 class 123A3B4A4B4C6A6B12A12B12C12D12E12F19A19B19C38A38B38C76A76B76C76D76E76F
 size 111919238381919383838383838666666666666
ρ1111111111111111111111111111    trivial
ρ211111-1-111-1-1-11-11111111111111    linear of order 2
ρ31111-11-111-111-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ41111-1-11111-1-1-11-1111111-1-1-1-1-1-1    linear of order 2
ρ511ζ3ζ321-1-1ζ3ζ32ζ65ζ6ζ65ζ3ζ6ζ32111111111111    linear of order 6
ρ611ζ32ζ31-1-1ζ32ζ3ζ6ζ65ζ6ζ32ζ65ζ3111111111111    linear of order 6
ρ711ζ3ζ32-1-11ζ3ζ32ζ3ζ6ζ65ζ65ζ32ζ6111111-1-1-1-1-1-1    linear of order 6
ρ811ζ32ζ3111ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ3111111111111    linear of order 3
ρ911ζ32ζ3-1-11ζ32ζ3ζ32ζ65ζ6ζ6ζ3ζ65111111-1-1-1-1-1-1    linear of order 6
ρ1011ζ32ζ3-11-1ζ32ζ3ζ6ζ3ζ32ζ6ζ65ζ65111111-1-1-1-1-1-1    linear of order 6
ρ1111ζ3ζ32111ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ32111111111111    linear of order 3
ρ1211ζ3ζ32-11-1ζ3ζ32ζ65ζ32ζ3ζ65ζ6ζ6111111-1-1-1-1-1-1    linear of order 6
ρ132-222000-2-2000000222-2-2-2000000    symplectic lifted from Q8, Schur index 2
ρ142-2-1+-3-1--30001--31+-3000000222-2-2-2000000    complex lifted from C3×Q8
ρ152-2-1--3-1+-30001+-31--3000000222-2-2-2000000    complex lifted from C3×Q8
ρ16660060000000000ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ19181912191119819719ζ191519131910199196194ζ191719161914195193192ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194    orthogonal lifted from C19⋊C6
ρ176600-60000000000ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ191519131910199196194ζ19181912191119819719ζ1917191619141951931921918191219111981971919151913191019919619419171916191419519319219181912191119819719191719161914195193192191519131910199196194    orthogonal lifted from C2×C19⋊C6
ρ18660060000000000ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ191519131910199196194ζ191719161914195193192ζ19181912191119819719ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192    orthogonal lifted from C19⋊C6
ρ19660060000000000ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ191719161914195193192ζ19181912191119819719ζ191519131910199196194ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719    orthogonal lifted from C19⋊C6
ρ206600-60000000000ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ191719161914195193192ζ191519131910199196194ζ191819121911198197191915191319101991961941917191619141951931921918191219111981971919151913191019919619419181912191119819719191719161914195193192    orthogonal lifted from C2×C19⋊C6
ρ216600-60000000000ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ19181912191119819719ζ191719161914195193192ζ1915191319101991961941917191619141951931921918191219111981971919151913191019919619419171916191419519319219151913191019919619419181912191119819719    orthogonal lifted from C2×C19⋊C6
ρ226-60000000000000ζ191719161914195193192ζ191519131910199196194ζ1918191219111981971919151913191019919619419181912191119819719191719161914195193192ζ4ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ4ζ19154ζ19134ζ19104ζ1994ζ1964ζ194ζ4ζ19174ζ19164ζ19144ζ1954ζ1934ζ1924ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ43ζ191743ζ191643ζ191443ζ19543ζ19343ζ192ζ43ζ191543ζ191343ζ191043ζ19943ζ19643ζ194    symplectic faithful, Schur index 2
ρ236-60000000000000ζ19181912191119819719ζ191719161914195193192ζ19151913191019919619419171916191419519319219151913191019919619419181912191119819719ζ4ζ19154ζ19134ζ19104ζ1994ζ1964ζ194ζ43ζ191743ζ191643ζ191443ζ19543ζ19343ζ1924ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ43ζ191543ζ191343ζ191043ζ19943ζ19643ζ194ζ4ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ4ζ19174ζ19164ζ19144ζ1954ζ1934ζ192    symplectic faithful, Schur index 2
ρ246-60000000000000ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719191519131910199196194191819121911198197191917191619141951931924ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ43ζ191543ζ191343ζ191043ζ19943ζ19643ζ194ζ43ζ191743ζ191643ζ191443ζ19543ζ19343ζ192ζ4ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ4ζ19174ζ19164ζ19144ζ1954ζ1934ζ192ζ4ζ19154ζ19134ζ19104ζ1994ζ1964ζ194    symplectic faithful, Schur index 2
ρ256-60000000000000ζ191519131910199196194ζ19181912191119819719ζ19171916191419519319219181912191119819719191719161914195193192191519131910199196194ζ4ζ19174ζ19164ζ19144ζ1954ζ1934ζ1924ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ4ζ19154ζ19134ζ19104ζ1994ζ1964ζ194ζ43ζ191743ζ191643ζ191443ζ19543ζ19343ζ192ζ43ζ191543ζ191343ζ191043ζ19943ζ19643ζ194ζ4ζ19184ζ19124ζ19114ζ1984ζ1974ζ19    symplectic faithful, Schur index 2
ρ266-60000000000000ζ19181912191119819719ζ191719161914195193192ζ19151913191019919619419171916191419519319219151913191019919619419181912191119819719ζ43ζ191543ζ191343ζ191043ζ19943ζ19643ζ194ζ4ζ19174ζ19164ζ19144ζ1954ζ1934ζ192ζ4ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ4ζ19154ζ19134ζ19104ζ1994ζ1964ζ1944ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ43ζ191743ζ191643ζ191443ζ19543ζ19343ζ192    symplectic faithful, Schur index 2
ρ276-60000000000000ζ191519131910199196194ζ19181912191119819719ζ19171916191419519319219181912191119819719191719161914195193192191519131910199196194ζ43ζ191743ζ191643ζ191443ζ19543ζ19343ζ192ζ4ζ19184ζ19124ζ19114ζ1984ζ1974ζ19ζ43ζ191543ζ191343ζ191043ζ19943ζ19643ζ194ζ4ζ19174ζ19164ζ19144ζ1954ζ1934ζ192ζ4ζ19154ζ19134ζ19104ζ1994ζ1964ζ1944ζ19184ζ19124ζ19114ζ1984ζ1974ζ19    symplectic faithful, Schur index 2

Smallest permutation representation of Dic38⋊C3
On 152 points
Generators in S152
(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)
(1 96 39 134)(2 95 40 133)(3 94 41 132)(4 93 42 131)(5 92 43 130)(6 91 44 129)(7 90 45 128)(8 89 46 127)(9 88 47 126)(10 87 48 125)(11 86 49 124)(12 85 50 123)(13 84 51 122)(14 83 52 121)(15 82 53 120)(16 81 54 119)(17 80 55 118)(18 79 56 117)(19 78 57 116)(20 77 58 115)(21 152 59 114)(22 151 60 113)(23 150 61 112)(24 149 62 111)(25 148 63 110)(26 147 64 109)(27 146 65 108)(28 145 66 107)(29 144 67 106)(30 143 68 105)(31 142 69 104)(32 141 70 103)(33 140 71 102)(34 139 72 101)(35 138 73 100)(36 137 74 99)(37 136 75 98)(38 135 76 97)
(2 46 50)(3 15 23)(4 60 72)(5 29 45)(6 74 18)(7 43 67)(8 12 40)(9 57 13)(10 26 62)(11 71 35)(14 54 30)(16 68 52)(17 37 25)(19 51 47)(21 65 69)(22 34 42)(24 48 64)(27 31 59)(28 76 32)(33 73 49)(36 56 44)(38 70 66)(41 53 61)(55 75 63)(78 122 126)(79 91 99)(80 136 148)(81 105 121)(82 150 94)(83 119 143)(84 88 116)(85 133 89)(86 102 138)(87 147 111)(90 130 106)(92 144 128)(93 113 101)(95 127 123)(97 141 145)(98 110 118)(100 124 140)(103 107 135)(104 152 108)(109 149 125)(112 132 120)(114 146 142)(117 129 137)(131 151 139)

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

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

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

Matrix representation of Dic38⋊C3 in GL8(𝔽229)

18131000000
6648000000
0012110000
0022701000
00100100
0010900010
0021100001
001121228126125127
,
20980000000
7820000000
0010414114318421579
00971958212829226
001219145124542
0019220842925125
002920214501348
00148108122038221
,
940000000
094000000
002119215598144142
007220740943159
001749045211121154
00167225103102476
0021418211583142166
00132124179104207170

G:=sub<GL(8,GF(229))| [181,66,0,0,0,0,0,0,31,48,0,0,0,0,0,0,0,0,121,227,1,109,211,1,0,0,1,0,0,0,0,121,0,0,0,1,0,0,0,228,0,0,0,0,1,0,0,126,0,0,0,0,0,1,0,125,0,0,0,0,0,0,1,127],[209,78,0,0,0,0,0,0,80,20,0,0,0,0,0,0,0,0,104,97,12,192,29,148,0,0,141,195,19,208,202,108,0,0,143,82,145,42,145,12,0,0,184,128,12,9,0,20,0,0,215,29,45,25,13,38,0,0,79,226,42,125,48,221],[94,0,0,0,0,0,0,0,0,94,0,0,0,0,0,0,0,0,21,72,174,167,214,132,0,0,192,207,90,225,182,124,0,0,155,40,45,103,115,179,0,0,98,9,211,102,83,104,0,0,144,43,121,47,142,207,0,0,142,159,154,6,166,170] >;

Dic38⋊C3 in GAP, Magma, Sage, TeX

{\rm Dic}_{38}\rtimes C_3
% in TeX

G:=Group("Dic38:C3");
// GroupNames label

G:=SmallGroup(456,7);
// by ID

G=gap.SmallGroup(456,7);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-19,60,141,66,10804,1064]);
// Polycyclic

G:=Group<a,b,c|a^76=c^3=1,b^2=a^38,b*a*b^-1=a^-1,c*a*c^-1=a^49,b*c=c*b>;
// generators/relations

Export

Subgroup lattice of Dic38⋊C3 in TeX
Character table of Dic38⋊C3 in TeX

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