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G = C112.C4order 448 = 26·7

1st non-split extension by C112 of C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C112.1C4, C28.36D8, C4.18D56, C56.14Q8, C16.1Dic7, C8.13Dic14, C22.1Dic28, (C2×C16).5D7, C56.71(C2×C4), (C2×C112).7C2, (C2×C4).73D28, C28.25(C4⋊C4), (C2×C14).7Q16, C72(C8.4Q8), (C2×C8).311D14, (C2×C28).393D4, C8.16(C2×Dic7), C2.5(C561C4), C14.9(C2.D8), C56.C4.1C2, C4.10(C4⋊Dic7), (C2×C56).383C22, SmallGroup(448,63)

Series: Derived Chief Lower central Upper central

C1C56 — C112.C4
C1C7C14C28C2×C28C2×C56C56.C4 — C112.C4
C7C14C28C56 — C112.C4
C1C4C2×C4C2×C8C2×C16

Generators and relations for C112.C4
 G = < a,b | a112=1, b4=a56, bab-1=a55 >

2C2
2C14
28C8
28C8
14M4(2)
14M4(2)
4C7⋊C8
4C7⋊C8
7C8.C4
7C8.C4
2C4.Dic7
2C4.Dic7
7C8.4Q8

Smallest permutation representation of C112.C4
On 224 points
Generators in S224
(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)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 175 29 147 57 119 85 203)(2 118 30 202 58 174 86 146)(3 173 31 145 59 117 87 201)(4 116 32 200 60 172 88 144)(5 171 33 143 61 115 89 199)(6 114 34 198 62 170 90 142)(7 169 35 141 63 113 91 197)(8 224 36 196 64 168 92 140)(9 167 37 139 65 223 93 195)(10 222 38 194 66 166 94 138)(11 165 39 137 67 221 95 193)(12 220 40 192 68 164 96 136)(13 163 41 135 69 219 97 191)(14 218 42 190 70 162 98 134)(15 161 43 133 71 217 99 189)(16 216 44 188 72 160 100 132)(17 159 45 131 73 215 101 187)(18 214 46 186 74 158 102 130)(19 157 47 129 75 213 103 185)(20 212 48 184 76 156 104 128)(21 155 49 127 77 211 105 183)(22 210 50 182 78 154 106 126)(23 153 51 125 79 209 107 181)(24 208 52 180 80 152 108 124)(25 151 53 123 81 207 109 179)(26 206 54 178 82 150 110 122)(27 149 55 121 83 205 111 177)(28 204 56 176 84 148 112 120)

G:=sub<Sym(224)| (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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,175,29,147,57,119,85,203)(2,118,30,202,58,174,86,146)(3,173,31,145,59,117,87,201)(4,116,32,200,60,172,88,144)(5,171,33,143,61,115,89,199)(6,114,34,198,62,170,90,142)(7,169,35,141,63,113,91,197)(8,224,36,196,64,168,92,140)(9,167,37,139,65,223,93,195)(10,222,38,194,66,166,94,138)(11,165,39,137,67,221,95,193)(12,220,40,192,68,164,96,136)(13,163,41,135,69,219,97,191)(14,218,42,190,70,162,98,134)(15,161,43,133,71,217,99,189)(16,216,44,188,72,160,100,132)(17,159,45,131,73,215,101,187)(18,214,46,186,74,158,102,130)(19,157,47,129,75,213,103,185)(20,212,48,184,76,156,104,128)(21,155,49,127,77,211,105,183)(22,210,50,182,78,154,106,126)(23,153,51,125,79,209,107,181)(24,208,52,180,80,152,108,124)(25,151,53,123,81,207,109,179)(26,206,54,178,82,150,110,122)(27,149,55,121,83,205,111,177)(28,204,56,176,84,148,112,120)>;

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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,175,29,147,57,119,85,203)(2,118,30,202,58,174,86,146)(3,173,31,145,59,117,87,201)(4,116,32,200,60,172,88,144)(5,171,33,143,61,115,89,199)(6,114,34,198,62,170,90,142)(7,169,35,141,63,113,91,197)(8,224,36,196,64,168,92,140)(9,167,37,139,65,223,93,195)(10,222,38,194,66,166,94,138)(11,165,39,137,67,221,95,193)(12,220,40,192,68,164,96,136)(13,163,41,135,69,219,97,191)(14,218,42,190,70,162,98,134)(15,161,43,133,71,217,99,189)(16,216,44,188,72,160,100,132)(17,159,45,131,73,215,101,187)(18,214,46,186,74,158,102,130)(19,157,47,129,75,213,103,185)(20,212,48,184,76,156,104,128)(21,155,49,127,77,211,105,183)(22,210,50,182,78,154,106,126)(23,153,51,125,79,209,107,181)(24,208,52,180,80,152,108,124)(25,151,53,123,81,207,109,179)(26,206,54,178,82,150,110,122)(27,149,55,121,83,205,111,177)(28,204,56,176,84,148,112,120) );

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),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,175,29,147,57,119,85,203),(2,118,30,202,58,174,86,146),(3,173,31,145,59,117,87,201),(4,116,32,200,60,172,88,144),(5,171,33,143,61,115,89,199),(6,114,34,198,62,170,90,142),(7,169,35,141,63,113,91,197),(8,224,36,196,64,168,92,140),(9,167,37,139,65,223,93,195),(10,222,38,194,66,166,94,138),(11,165,39,137,67,221,95,193),(12,220,40,192,68,164,96,136),(13,163,41,135,69,219,97,191),(14,218,42,190,70,162,98,134),(15,161,43,133,71,217,99,189),(16,216,44,188,72,160,100,132),(17,159,45,131,73,215,101,187),(18,214,46,186,74,158,102,130),(19,157,47,129,75,213,103,185),(20,212,48,184,76,156,104,128),(21,155,49,127,77,211,105,183),(22,210,50,182,78,154,106,126),(23,153,51,125,79,209,107,181),(24,208,52,180,80,152,108,124),(25,151,53,123,81,207,109,179),(26,206,54,178,82,150,110,122),(27,149,55,121,83,205,111,177),(28,204,56,176,84,148,112,120)])

118 conjugacy classes

class 1 2A2B4A4B4C7A7B7C8A8B8C8D8E8F8G8H14A···14I16A···16H28A···28L56A···56X112A···112AV
order1224447778888888814···1416···1628···2856···56112···112
size1121122222222565656562···22···22···22···22···2

118 irreducible representations

dim11112222222222222
type+++-+++--+-++-
imageC1C2C2C4Q8D4D7D8Q16Dic7D14Dic14D28C8.4Q8D56Dic28C112.C4
kernelC112.C4C56.C4C2×C112C112C56C2×C28C2×C16C28C2×C14C16C2×C8C8C2×C4C7C4C22C1
# reps12141132263668121248

Matrix representation of C112.C4 in GL2(𝔽113) generated by

680
46108
,
9111
48104
G:=sub<GL(2,GF(113))| [68,46,0,108],[9,48,111,104] >;

C112.C4 in GAP, Magma, Sage, TeX

C_{112}.C_4
% in TeX

G:=Group("C112.C4");
// GroupNames label

G:=SmallGroup(448,63);
// by ID

G=gap.SmallGroup(448,63);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,176,184,675,192,1684,102,18822]);
// Polycyclic

G:=Group<a,b|a^112=1,b^4=a^56,b*a*b^-1=a^55>;
// generators/relations

Export

Subgroup lattice of C112.C4 in TeX

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