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G = M4(2)×C23order 368 = 24·23

Direct product of C23 and M4(2)

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

Aliases: M4(2)×C23, C4.C92, C83C46, C1847C2, C92.4C4, C22.C92, C92.22C22, (C2×C46).1C4, (C2×C92).8C2, C2.3(C2×C92), (C2×C4).2C46, C4.6(C2×C46), C46.12(C2×C4), SmallGroup(368,23)

Series: Derived Chief Lower central Upper central

C1C2 — M4(2)×C23
C1C2C4C92C184 — M4(2)×C23
C1C2 — M4(2)×C23
C1C92 — M4(2)×C23

Generators and relations for M4(2)×C23
 G = < a,b,c | a23=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

2C2
2C46

Smallest permutation representation of M4(2)×C23
On 184 points
Generators in S184
(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)
(1 99 140 63 124 170 87 37)(2 100 141 64 125 171 88 38)(3 101 142 65 126 172 89 39)(4 102 143 66 127 173 90 40)(5 103 144 67 128 174 91 41)(6 104 145 68 129 175 92 42)(7 105 146 69 130 176 70 43)(8 106 147 47 131 177 71 44)(9 107 148 48 132 178 72 45)(10 108 149 49 133 179 73 46)(11 109 150 50 134 180 74 24)(12 110 151 51 135 181 75 25)(13 111 152 52 136 182 76 26)(14 112 153 53 137 183 77 27)(15 113 154 54 138 184 78 28)(16 114 155 55 116 162 79 29)(17 115 156 56 117 163 80 30)(18 93 157 57 118 164 81 31)(19 94 158 58 119 165 82 32)(20 95 159 59 120 166 83 33)(21 96 160 60 121 167 84 34)(22 97 161 61 122 168 85 35)(23 98 139 62 123 169 86 36)
(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 67)(42 68)(43 69)(44 47)(45 48)(46 49)(93 164)(94 165)(95 166)(96 167)(97 168)(98 169)(99 170)(100 171)(101 172)(102 173)(103 174)(104 175)(105 176)(106 177)(107 178)(108 179)(109 180)(110 181)(111 182)(112 183)(113 184)(114 162)(115 163)

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

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

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

230 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D23A···23V46A···46V46W···46AR92A···92AR92AS···92BN184A···184CJ
order122444888823···2346···4646···4692···9292···92184···184
size11211222221···11···12···21···12···22···2

230 irreducible representations

dim111111111122
type+++
imageC1C2C2C4C4C23C46C46C92C92M4(2)M4(2)×C23
kernelM4(2)×C23C184C2×C92C92C2×C46M4(2)C8C2×C4C4C22C23C1
# reps121222244224444244

Matrix representation of M4(2)×C23 in GL2(𝔽1289) generated by

6690
0669
,
5457
3311284
,
1183
01288
G:=sub<GL(2,GF(1289))| [669,0,0,669],[5,331,457,1284],[1,0,183,1288] >;

M4(2)×C23 in GAP, Magma, Sage, TeX

M_4(2)\times C_{23}
% in TeX

G:=Group("M4(2)xC23");
// GroupNames label

G:=SmallGroup(368,23);
// by ID

G=gap.SmallGroup(368,23);
# by ID

G:=PCGroup([5,-2,-2,-23,-2,-2,460,1861,58]);
// Polycyclic

G:=Group<a,b,c|a^23=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

Export

Subgroup lattice of M4(2)×C23 in TeX

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