p-group, metabelian, nilpotent (class 2), monomial
Aliases: (C4×C8)⋊12C4, C8⋊C4⋊13C4, (C2×C4).42C42, C42⋊4C4.1C2, C42.295(C2×C4), C2.5(C42⋊4C4), C22.20(C8○D4), C2.C42.7C4, C22.48(C2×C42), C4.69(C42⋊C2), C23.249(C22×C4), (C2×C42).986C22, (C22×C8).375C22, C2.11(C8○2M4(2)), (C22×C4).1602C23, C22.47(C42⋊C2), C22.7C42.39C2, C2.1(C42.7C22), (C2×C4×C8).12C2, (C2×C8).129(C2×C4), (C2×C8⋊C4).22C2, (C2×C4).912(C4○D4), (C22×C4).105(C2×C4), (C2×C4).592(C22×C4), SmallGroup(128,478)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C4×C8)⋊12C4
G = < a,b,c | a4=b8=c4=1, ab=ba, cac-1=ab4, cbc-1=a2b >
Subgroups: 172 in 120 conjugacy classes, 76 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4×C8, C8⋊C4, C2×C42, C2×C42, C22×C8, C22.7C42, C42⋊4C4, C2×C4×C8, C2×C8⋊C4, (C4×C8)⋊12C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C4○D4, C2×C42, C42⋊C2, C8○D4, C42⋊4C4, C8○2M4(2), C42.7C22, (C4×C8)⋊12C4
►Regular action on 128 points
Generators in S128
(1 95 87 31)(2 96 88 32)(3 89 81 25)(4 90 82 26)(5 91 83 27)(6 92 84 28)(7 93 85 29)(8 94 86 30)(9 33 76 17)(10 34 77 18)(11 35 78 19)(12 36 79 20)(13 37 80 21)(14 38 73 22)(15 39 74 23)(16 40 75 24)(41 105 97 49)(42 106 98 50)(43 107 99 51)(44 108 100 52)(45 109 101 53)(46 110 102 54)(47 111 103 55)(48 112 104 56)(57 125 113 67)(58 126 114 68)(59 127 115 69)(60 128 116 70)(61 121 117 71)(62 122 118 72)(63 123 119 65)(64 124 120 66)
(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)
(1 119 103 39)(2 64 104 24)(3 113 97 33)(4 58 98 18)(5 115 99 35)(6 60 100 20)(7 117 101 37)(8 62 102 22)(9 29 125 109)(10 94 126 54)(11 31 127 111)(12 96 128 56)(13 25 121 105)(14 90 122 50)(15 27 123 107)(16 92 124 52)(17 81 57 41)(19 83 59 43)(21 85 61 45)(23 87 63 47)(26 72 106 73)(28 66 108 75)(30 68 110 77)(32 70 112 79)(34 82 114 42)(36 84 116 44)(38 86 118 46)(40 88 120 48)(49 80 89 71)(51 74 91 65)(53 76 93 67)(55 78 95 69)
G:=sub<Sym(128)| (1,95,87,31)(2,96,88,32)(3,89,81,25)(4,90,82,26)(5,91,83,27)(6,92,84,28)(7,93,85,29)(8,94,86,30)(9,33,76,17)(10,34,77,18)(11,35,78,19)(12,36,79,20)(13,37,80,21)(14,38,73,22)(15,39,74,23)(16,40,75,24)(41,105,97,49)(42,106,98,50)(43,107,99,51)(44,108,100,52)(45,109,101,53)(46,110,102,54)(47,111,103,55)(48,112,104,56)(57,125,113,67)(58,126,114,68)(59,127,115,69)(60,128,116,70)(61,121,117,71)(62,122,118,72)(63,123,119,65)(64,124,120,66), (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), (1,119,103,39)(2,64,104,24)(3,113,97,33)(4,58,98,18)(5,115,99,35)(6,60,100,20)(7,117,101,37)(8,62,102,22)(9,29,125,109)(10,94,126,54)(11,31,127,111)(12,96,128,56)(13,25,121,105)(14,90,122,50)(15,27,123,107)(16,92,124,52)(17,81,57,41)(19,83,59,43)(21,85,61,45)(23,87,63,47)(26,72,106,73)(28,66,108,75)(30,68,110,77)(32,70,112,79)(34,82,114,42)(36,84,116,44)(38,86,118,46)(40,88,120,48)(49,80,89,71)(51,74,91,65)(53,76,93,67)(55,78,95,69)>;
G:=Group( (1,95,87,31)(2,96,88,32)(3,89,81,25)(4,90,82,26)(5,91,83,27)(6,92,84,28)(7,93,85,29)(8,94,86,30)(9,33,76,17)(10,34,77,18)(11,35,78,19)(12,36,79,20)(13,37,80,21)(14,38,73,22)(15,39,74,23)(16,40,75,24)(41,105,97,49)(42,106,98,50)(43,107,99,51)(44,108,100,52)(45,109,101,53)(46,110,102,54)(47,111,103,55)(48,112,104,56)(57,125,113,67)(58,126,114,68)(59,127,115,69)(60,128,116,70)(61,121,117,71)(62,122,118,72)(63,123,119,65)(64,124,120,66), (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), (1,119,103,39)(2,64,104,24)(3,113,97,33)(4,58,98,18)(5,115,99,35)(6,60,100,20)(7,117,101,37)(8,62,102,22)(9,29,125,109)(10,94,126,54)(11,31,127,111)(12,96,128,56)(13,25,121,105)(14,90,122,50)(15,27,123,107)(16,92,124,52)(17,81,57,41)(19,83,59,43)(21,85,61,45)(23,87,63,47)(26,72,106,73)(28,66,108,75)(30,68,110,77)(32,70,112,79)(34,82,114,42)(36,84,116,44)(38,86,118,46)(40,88,120,48)(49,80,89,71)(51,74,91,65)(53,76,93,67)(55,78,95,69) );
G=PermutationGroup([[(1,95,87,31),(2,96,88,32),(3,89,81,25),(4,90,82,26),(5,91,83,27),(6,92,84,28),(7,93,85,29),(8,94,86,30),(9,33,76,17),(10,34,77,18),(11,35,78,19),(12,36,79,20),(13,37,80,21),(14,38,73,22),(15,39,74,23),(16,40,75,24),(41,105,97,49),(42,106,98,50),(43,107,99,51),(44,108,100,52),(45,109,101,53),(46,110,102,54),(47,111,103,55),(48,112,104,56),(57,125,113,67),(58,126,114,68),(59,127,115,69),(60,128,116,70),(61,121,117,71),(62,122,118,72),(63,123,119,65),(64,124,120,66)], [(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)], [(1,119,103,39),(2,64,104,24),(3,113,97,33),(4,58,98,18),(5,115,99,35),(6,60,100,20),(7,117,101,37),(8,62,102,22),(9,29,125,109),(10,94,126,54),(11,31,127,111),(12,96,128,56),(13,25,121,105),(14,90,122,50),(15,27,123,107),(16,92,124,52),(17,81,57,41),(19,83,59,43),(21,85,61,45),(23,87,63,47),(26,72,106,73),(28,66,108,75),(30,68,110,77),(32,70,112,79),(34,82,114,42),(36,84,116,44),(38,86,118,46),(40,88,120,48),(49,80,89,71),(51,74,91,65),(53,76,93,67),(55,78,95,69)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4X | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4○D4 | C8○D4 |
| kernel | (C4×C8)⋊12C4 | C22.7C42 | C42⋊4C4 | C2×C4×C8 | C2×C8⋊C4 | C2.C42 | C4×C8 | C8⋊C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 16 |
Matrix representation of (C4×C8)⋊12C4 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 8 | 0 | 0 |
| 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 15 | 0 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 2 | 15 | 0 | 0 |
| 0 | 10 | 15 | 0 | 0 |
| 0 | 0 | 0 | 7 | 16 |
| 0 | 0 | 0 | 16 | 10 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,16,4,0,0,0,8,1,0,0,0,0,0,0,15,0,0,0,2,0],[13,0,0,0,0,0,2,10,0,0,0,15,15,0,0,0,0,0,7,16,0,0,0,16,10] >;
(C4×C8)⋊12C4 in GAP, Magma, Sage, TeX
(C_4\times C_8)\rtimes_{12}C_4 % in TeX
G:=Group("(C4xC8):12C4"); // GroupNames label
G:=SmallGroup(128,478);
// by ID
G=gap.SmallGroup(128,478);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,58,172]);
// Polycyclic
G:=Group<a,b,c|a^4=b^8=c^4=1,a*b=b*a,c*a*c^-1=a*b^4,c*b*c^-1=a^2*b>;
// generators/relations