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## G = C38.A4order 456 = 23·3·19

### The non-split extension by C38 of A4 acting via A4/C22=C3

Aliases: C38.A4, C19⋊SL2(𝔽3), Q8⋊(C19⋊C3), C2.(C19⋊A4), (Q8×C19)⋊2C3, SmallGroup(456,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C19 — C38.A4
 Chief series C1 — C2 — C38 — Q8×C19 — C38.A4
 Lower central Q8×C19 — C38.A4
 Upper central C1 — C2

Generators and relations for C38.A4
G = < a,b,c,d | a38=d3=1, b2=c2=a19, ab=ba, ac=ca, dad-1=a11, cbc-1=a19b, dbd-1=a19bc, dcd-1=b >

Smallest permutation representation of C38.A4
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)
(1 113 20 94)(2 114 21 95)(3 77 22 96)(4 78 23 97)(5 79 24 98)(6 80 25 99)(7 81 26 100)(8 82 27 101)(9 83 28 102)(10 84 29 103)(11 85 30 104)(12 86 31 105)(13 87 32 106)(14 88 33 107)(15 89 34 108)(16 90 35 109)(17 91 36 110)(18 92 37 111)(19 93 38 112)(39 117 58 136)(40 118 59 137)(41 119 60 138)(42 120 61 139)(43 121 62 140)(44 122 63 141)(45 123 64 142)(46 124 65 143)(47 125 66 144)(48 126 67 145)(49 127 68 146)(50 128 69 147)(51 129 70 148)(52 130 71 149)(53 131 72 150)(54 132 73 151)(55 133 74 152)(56 134 75 115)(57 135 76 116)
(1 55 20 74)(2 56 21 75)(3 57 22 76)(4 58 23 39)(5 59 24 40)(6 60 25 41)(7 61 26 42)(8 62 27 43)(9 63 28 44)(10 64 29 45)(11 65 30 46)(12 66 31 47)(13 67 32 48)(14 68 33 49)(15 69 34 50)(16 70 35 51)(17 71 36 52)(18 72 37 53)(19 73 38 54)(77 116 96 135)(78 117 97 136)(79 118 98 137)(80 119 99 138)(81 120 100 139)(82 121 101 140)(83 122 102 141)(84 123 103 142)(85 124 104 143)(86 125 105 144)(87 126 106 145)(88 127 107 146)(89 128 108 147)(90 129 109 148)(91 130 110 149)(92 131 111 150)(93 132 112 151)(94 133 113 152)(95 134 114 115)
(2 8 12)(3 15 23)(4 22 34)(5 29 7)(6 36 18)(9 19 13)(10 26 24)(11 33 35)(14 16 30)(17 37 25)(21 27 31)(28 38 32)(39 135 89)(40 142 100)(41 149 111)(42 118 84)(43 125 95)(44 132 106)(45 139 79)(46 146 90)(47 115 101)(48 122 112)(49 129 85)(50 136 96)(51 143 107)(52 150 80)(53 119 91)(54 126 102)(55 133 113)(56 140 86)(57 147 97)(58 116 108)(59 123 81)(60 130 92)(61 137 103)(62 144 114)(63 151 87)(64 120 98)(65 127 109)(66 134 82)(67 141 93)(68 148 104)(69 117 77)(70 124 88)(71 131 99)(72 138 110)(73 145 83)(74 152 94)(75 121 105)(76 128 78)```

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

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

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

37 conjugacy classes

 class 1 2 3A 3B 4 6A 6B 19A ··· 19F 38A ··· 38F 76A ··· 76R order 1 2 3 3 4 6 6 19 ··· 19 38 ··· 38 76 ··· 76 size 1 1 76 76 6 76 76 3 ··· 3 3 ··· 3 6 ··· 6

37 irreducible representations

 dim 1 1 2 2 3 3 3 6 type + - + image C1 C3 SL2(𝔽3) SL2(𝔽3) A4 C19⋊C3 C19⋊A4 C38.A4 kernel C38.A4 Q8×C19 C19 C19 C38 Q8 C2 C1 # reps 1 2 1 2 1 6 18 6

Matrix representation of C38.A4 in GL5(𝔽229)

 228 0 0 0 0 0 228 0 0 0 0 0 155 156 0 0 0 6 143 0 0 0 18 61 165
,
 22 39 0 0 0 105 207 0 0 0 0 0 186 27 0 0 0 186 43 0 0 0 114 123 228
,
 102 2 0 0 0 179 127 0 0 0 0 0 43 202 0 0 0 43 186 0 0 0 57 168 228
,
 1 0 0 0 0 189 134 0 0 0 0 0 79 39 2 0 0 212 90 122 0 0 148 120 60

`G:=sub<GL(5,GF(229))| [228,0,0,0,0,0,228,0,0,0,0,0,155,6,18,0,0,156,143,61,0,0,0,0,165],[22,105,0,0,0,39,207,0,0,0,0,0,186,186,114,0,0,27,43,123,0,0,0,0,228],[102,179,0,0,0,2,127,0,0,0,0,0,43,43,57,0,0,202,186,168,0,0,0,0,228],[1,189,0,0,0,0,134,0,0,0,0,0,79,212,148,0,0,39,90,120,0,0,2,122,60] >;`

C38.A4 in GAP, Magma, Sage, TeX

`C_{38}.A_4`
`% in TeX`

`G:=Group("C38.A4");`
`// GroupNames label`

`G:=SmallGroup(456,23);`
`// by ID`

`G=gap.SmallGroup(456,23);`
`# by ID`

`G:=PCGroup([5,-3,-2,2,-19,-2,61,766,137,1177,582,1683]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^38=d^3=1,b^2=c^2=a^19,a*b=b*a,a*c=c*a,d*a*d^-1=a^11,c*b*c^-1=a^19*b,d*b*d^-1=a^19*b*c,d*c*d^-1=b>;`
`// generators/relations`

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