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## G = Q8×C22order 176 = 24·11

### Direct product of C22 and Q8

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

Aliases: Q8×C22, C44.20C22, C22.12C23, (C2×C44).9C2, (C2×C4).3C22, C4.4(C2×C22), C22.4(C2×C22), C2.2(C22×C22), (C2×C22).15C22, SmallGroup(176,39)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C22
 Chief series C1 — C2 — C22 — C44 — Q8×C11 — Q8×C22
 Lower central C1 — C2 — Q8×C22
 Upper central C1 — C2×C22 — Q8×C22

Generators and relations for Q8×C22
G = < a,b,c | a22=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of Q8×C22
Regular action 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 55 135 42)(2 56 136 43)(3 57 137 44)(4 58 138 23)(5 59 139 24)(6 60 140 25)(7 61 141 26)(8 62 142 27)(9 63 143 28)(10 64 144 29)(11 65 145 30)(12 66 146 31)(13 45 147 32)(14 46 148 33)(15 47 149 34)(16 48 150 35)(17 49 151 36)(18 50 152 37)(19 51 153 38)(20 52 154 39)(21 53 133 40)(22 54 134 41)(67 92 170 130)(68 93 171 131)(69 94 172 132)(70 95 173 111)(71 96 174 112)(72 97 175 113)(73 98 176 114)(74 99 155 115)(75 100 156 116)(76 101 157 117)(77 102 158 118)(78 103 159 119)(79 104 160 120)(80 105 161 121)(81 106 162 122)(82 107 163 123)(83 108 164 124)(84 109 165 125)(85 110 166 126)(86 89 167 127)(87 90 168 128)(88 91 169 129)
(1 160 135 79)(2 161 136 80)(3 162 137 81)(4 163 138 82)(5 164 139 83)(6 165 140 84)(7 166 141 85)(8 167 142 86)(9 168 143 87)(10 169 144 88)(11 170 145 67)(12 171 146 68)(13 172 147 69)(14 173 148 70)(15 174 149 71)(16 175 150 72)(17 176 151 73)(18 155 152 74)(19 156 153 75)(20 157 154 76)(21 158 133 77)(22 159 134 78)(23 123 58 107)(24 124 59 108)(25 125 60 109)(26 126 61 110)(27 127 62 89)(28 128 63 90)(29 129 64 91)(30 130 65 92)(31 131 66 93)(32 132 45 94)(33 111 46 95)(34 112 47 96)(35 113 48 97)(36 114 49 98)(37 115 50 99)(38 116 51 100)(39 117 52 101)(40 118 53 102)(41 119 54 103)(42 120 55 104)(43 121 56 105)(44 122 57 106)

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

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

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

Q8×C22 is a maximal subgroup of   Q8⋊Dic11  C44.10D4  C44.C23  Dic11⋊Q8  D223Q8  C44.23D4  Q8.10D22

110 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 11A ··· 11J 22A ··· 22AD 44A ··· 44BH order 1 2 2 2 4 ··· 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + - image C1 C2 C2 C11 C22 C22 Q8 Q8×C11 kernel Q8×C22 C2×C44 Q8×C11 C2×Q8 C2×C4 Q8 C22 C2 # reps 1 3 4 10 30 40 2 20

Matrix representation of Q8×C22 in GL3(𝔽89) generated by

 88 0 0 0 73 0 0 0 73
,
 1 0 0 0 0 1 0 88 0
,
 88 0 0 0 2 23 0 23 87
G:=sub<GL(3,GF(89))| [88,0,0,0,73,0,0,0,73],[1,0,0,0,0,88,0,1,0],[88,0,0,0,2,23,0,23,87] >;

Q8×C22 in GAP, Magma, Sage, TeX

Q_8\times C_{22}
% in TeX

G:=Group("Q8xC22");
// GroupNames label

G:=SmallGroup(176,39);
// by ID

G=gap.SmallGroup(176,39);
# by ID

G:=PCGroup([5,-2,-2,-2,-11,-2,440,901,446]);
// Polycyclic

G:=Group<a,b,c|a^22=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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