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 [×6], C4 [×4], C4 [×8], C22, C22 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], C23, C42 [×4], C42 [×4], C2×C8 [×8], C2×C8 [×8], C22×C4, C22×C4 [×6], C2.C42 [×4], C4×C8 [×4], C8⋊C4 [×4], C2×C42, C2×C42 [×2], C22×C8 [×4], C22.7C42 [×4], C42⋊4C4, C2×C4×C8, C2×C8⋊C4, (C4×C8)⋊12C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], C22×C4 [×3], C4○D4 [×4], C2×C42, C42⋊C2 [×6], C8○D4 [×4], C42⋊4C4, C8○2M4(2) [×2], C42.7C22 [×4], (C4×C8)⋊12C4
►Regular action on 128 points
Generators in S128
(1 31 23 87)(2 32 24 88)(3 25 17 81)(4 26 18 82)(5 27 19 83)(6 28 20 84)(7 29 21 85)(8 30 22 86)(9 38 74 94)(10 39 75 95)(11 40 76 96)(12 33 77 89)(13 34 78 90)(14 35 79 91)(15 36 80 92)(16 37 73 93)(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 95)(2 64 104 40)(3 113 97 89)(4 58 98 34)(5 115 99 91)(6 60 100 36)(7 117 101 93)(8 62 102 38)(9 82 72 106)(10 27 65 51)(11 84 66 108)(12 29 67 53)(13 86 68 110)(14 31 69 55)(15 88 70 112)(16 25 71 49)(17 57 41 33)(18 114 42 90)(19 59 43 35)(20 116 44 92)(21 61 45 37)(22 118 46 94)(23 63 47 39)(24 120 48 96)(26 122 50 74)(28 124 52 76)(30 126 54 78)(32 128 56 80)(73 81 121 105)(75 83 123 107)(77 85 125 109)(79 87 127 111)
G:=sub<Sym(128)| (1,31,23,87)(2,32,24,88)(3,25,17,81)(4,26,18,82)(5,27,19,83)(6,28,20,84)(7,29,21,85)(8,30,22,86)(9,38,74,94)(10,39,75,95)(11,40,76,96)(12,33,77,89)(13,34,78,90)(14,35,79,91)(15,36,80,92)(16,37,73,93)(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), (1,119,103,95)(2,64,104,40)(3,113,97,89)(4,58,98,34)(5,115,99,91)(6,60,100,36)(7,117,101,93)(8,62,102,38)(9,82,72,106)(10,27,65,51)(11,84,66,108)(12,29,67,53)(13,86,68,110)(14,31,69,55)(15,88,70,112)(16,25,71,49)(17,57,41,33)(18,114,42,90)(19,59,43,35)(20,116,44,92)(21,61,45,37)(22,118,46,94)(23,63,47,39)(24,120,48,96)(26,122,50,74)(28,124,52,76)(30,126,54,78)(32,128,56,80)(73,81,121,105)(75,83,123,107)(77,85,125,109)(79,87,127,111)>;
G:=Group( (1,31,23,87)(2,32,24,88)(3,25,17,81)(4,26,18,82)(5,27,19,83)(6,28,20,84)(7,29,21,85)(8,30,22,86)(9,38,74,94)(10,39,75,95)(11,40,76,96)(12,33,77,89)(13,34,78,90)(14,35,79,91)(15,36,80,92)(16,37,73,93)(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), (1,119,103,95)(2,64,104,40)(3,113,97,89)(4,58,98,34)(5,115,99,91)(6,60,100,36)(7,117,101,93)(8,62,102,38)(9,82,72,106)(10,27,65,51)(11,84,66,108)(12,29,67,53)(13,86,68,110)(14,31,69,55)(15,88,70,112)(16,25,71,49)(17,57,41,33)(18,114,42,90)(19,59,43,35)(20,116,44,92)(21,61,45,37)(22,118,46,94)(23,63,47,39)(24,120,48,96)(26,122,50,74)(28,124,52,76)(30,126,54,78)(32,128,56,80)(73,81,121,105)(75,83,123,107)(77,85,125,109)(79,87,127,111) );
G=PermutationGroup([(1,31,23,87),(2,32,24,88),(3,25,17,81),(4,26,18,82),(5,27,19,83),(6,28,20,84),(7,29,21,85),(8,30,22,86),(9,38,74,94),(10,39,75,95),(11,40,76,96),(12,33,77,89),(13,34,78,90),(14,35,79,91),(15,36,80,92),(16,37,73,93),(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)], [(1,119,103,95),(2,64,104,40),(3,113,97,89),(4,58,98,34),(5,115,99,91),(6,60,100,36),(7,117,101,93),(8,62,102,38),(9,82,72,106),(10,27,65,51),(11,84,66,108),(12,29,67,53),(13,86,68,110),(14,31,69,55),(15,88,70,112),(16,25,71,49),(17,57,41,33),(18,114,42,90),(19,59,43,35),(20,116,44,92),(21,61,45,37),(22,118,46,94),(23,63,47,39),(24,120,48,96),(26,122,50,74),(28,124,52,76),(30,126,54,78),(32,128,56,80),(73,81,121,105),(75,83,123,107),(77,85,125,109),(79,87,127,111)])
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