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## G = M4(2)×C22order 352 = 25·11

### Direct product of C22 and M4(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — M4(2)×C22
 Chief series C1 — C2 — C4 — C44 — C88 — C11×M4(2) — M4(2)×C22
 Lower central C1 — C2 — M4(2)×C22
 Upper central C1 — C2×C44 — M4(2)×C22

Generators and relations for M4(2)×C22
G = < a,b,c | a22=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Subgroups: 76 in 68 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C11, C2×C8, M4(2), C22×C4, C22, C22, C22, C2×M4(2), C44, C44, C2×C22, C2×C22, C2×C22, C88, C2×C44, C2×C44, C22×C22, C2×C88, C11×M4(2), C22×C44, M4(2)×C22
Quotients: C1, C2, C4, C22, C2×C4, C23, C11, M4(2), C22×C4, C22, C2×M4(2), C44, C2×C22, C2×C44, C22×C22, C11×M4(2), C22×C44, M4(2)×C22

Smallest permutation representation of M4(2)×C22
On 176 points
Generators in S176
(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)
(1 114 46 171 109 72 37 145)(2 115 47 172 110 73 38 146)(3 116 48 173 89 74 39 147)(4 117 49 174 90 75 40 148)(5 118 50 175 91 76 41 149)(6 119 51 176 92 77 42 150)(7 120 52 155 93 78 43 151)(8 121 53 156 94 79 44 152)(9 122 54 157 95 80 23 153)(10 123 55 158 96 81 24 154)(11 124 56 159 97 82 25 133)(12 125 57 160 98 83 26 134)(13 126 58 161 99 84 27 135)(14 127 59 162 100 85 28 136)(15 128 60 163 101 86 29 137)(16 129 61 164 102 87 30 138)(17 130 62 165 103 88 31 139)(18 131 63 166 104 67 32 140)(19 132 64 167 105 68 33 141)(20 111 65 168 106 69 34 142)(21 112 66 169 107 70 35 143)(22 113 45 170 108 71 36 144)
(67 131)(68 132)(69 111)(70 112)(71 113)(72 114)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 121)(80 122)(81 123)(82 124)(83 125)(84 126)(85 127)(86 128)(87 129)(88 130)(133 159)(134 160)(135 161)(136 162)(137 163)(138 164)(139 165)(140 166)(141 167)(142 168)(143 169)(144 170)(145 171)(146 172)(147 173)(148 174)(149 175)(150 176)(151 155)(152 156)(153 157)(154 158)

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

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

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

220 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A ··· 8H 11A ··· 11J 22A ··· 22AD 22AE ··· 22AX 44A ··· 44AN 44AO ··· 44BH 88A ··· 88CB order 1 2 2 2 2 2 4 4 4 4 4 4 8 ··· 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 88 ··· 88 size 1 1 1 1 2 2 1 1 1 1 2 2 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

220 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C4 C4 C11 C22 C22 C22 C44 C44 M4(2) C11×M4(2) kernel M4(2)×C22 C2×C88 C11×M4(2) C22×C44 C2×C44 C22×C22 C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C23 C22 C2 # reps 1 2 4 1 6 2 10 20 40 10 60 20 4 40

Matrix representation of M4(2)×C22 in GL3(𝔽89) generated by

 88 0 0 0 64 0 0 0 64
,
 55 0 0 0 13 2 0 66 76
,
 88 0 0 0 1 0 0 76 88
G:=sub<GL(3,GF(89))| [88,0,0,0,64,0,0,0,64],[55,0,0,0,13,66,0,2,76],[88,0,0,0,1,76,0,0,88] >;

M4(2)×C22 in GAP, Magma, Sage, TeX

M_4(2)\times C_{22}
% in TeX

G:=Group("M4(2)xC22");
// GroupNames label

G:=SmallGroup(352,165);
// by ID

G=gap.SmallGroup(352,165);
# by ID

G:=PCGroup([6,-2,-2,-2,-11,-2,-2,528,2137,88]);
// Polycyclic

G:=Group<a,b,c|a^22=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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