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## G = M4(2)×C19order 304 = 24·19

### Direct product of C19 and M4(2)

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: M4(2)×C19, C4.C76, C83C38, C1527C2, C76.4C4, C22.C76, C76.22C22, C2.3(C2×C76), (C2×C76).8C2, (C2×C38).1C4, (C2×C4).2C38, C4.6(C2×C38), C38.12(C2×C4), SmallGroup(304,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — M4(2)×C19
 Chief series C1 — C2 — C4 — C76 — C152 — M4(2)×C19
 Lower central C1 — C2 — M4(2)×C19
 Upper central C1 — C76 — M4(2)×C19

Generators and relations for M4(2)×C19
G = < a,b,c | a19=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Smallest permutation representation of M4(2)×C19
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)
(1 108 135 80 38 51 115 63)(2 109 136 81 20 52 116 64)(3 110 137 82 21 53 117 65)(4 111 138 83 22 54 118 66)(5 112 139 84 23 55 119 67)(6 113 140 85 24 56 120 68)(7 114 141 86 25 57 121 69)(8 96 142 87 26 39 122 70)(9 97 143 88 27 40 123 71)(10 98 144 89 28 41 124 72)(11 99 145 90 29 42 125 73)(12 100 146 91 30 43 126 74)(13 101 147 92 31 44 127 75)(14 102 148 93 32 45 128 76)(15 103 149 94 33 46 129 58)(16 104 150 95 34 47 130 59)(17 105 151 77 35 48 131 60)(18 106 152 78 36 49 132 61)(19 107 134 79 37 50 133 62)
(39 96)(40 97)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 94)(59 95)(60 77)(61 78)(62 79)(63 80)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)

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

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

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

190 conjugacy classes

 class 1 2A 2B 4A 4B 4C 8A 8B 8C 8D 19A ··· 19R 38A ··· 38R 38S ··· 38AJ 76A ··· 76AJ 76AK ··· 76BB 152A ··· 152BT order 1 2 2 4 4 4 8 8 8 8 19 ··· 19 38 ··· 38 38 ··· 38 76 ··· 76 76 ··· 76 152 ··· 152 size 1 1 2 1 1 2 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

190 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + image C1 C2 C2 C4 C4 C19 C38 C38 C76 C76 M4(2) M4(2)×C19 kernel M4(2)×C19 C152 C2×C76 C76 C2×C38 M4(2) C8 C2×C4 C4 C22 C19 C1 # reps 1 2 1 2 2 18 36 18 36 36 2 36

Matrix representation of M4(2)×C19 in GL2(𝔽457) generated by

 256 0 0 256
,
 0 1 348 0
,
 1 0 0 456
G:=sub<GL(2,GF(457))| [256,0,0,256],[0,348,1,0],[1,0,0,456] >;

M4(2)×C19 in GAP, Magma, Sage, TeX

M_4(2)\times C_{19}
% in TeX

G:=Group("M4(2)xC19");
// GroupNames label

G:=SmallGroup(304,23);
// by ID

G=gap.SmallGroup(304,23);
# by ID

G:=PCGroup([5,-2,-2,-19,-2,-2,380,1541,58]);
// Polycyclic

G:=Group<a,b,c|a^19=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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