direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C8×C4⋊C4, C4⋊2(C4×C8), C8○5(C4⋊C8), C4⋊C8⋊15C4, (C4×C8)⋊13C4, C2.2(C8×D4), C2.1(C8×Q8), C4.44(C4×Q8), (C2×C8).65Q8, C4.168(C4×D4), (C2×C8).400D4, (C2×C4).48C42, C22.90(C4×D4), C22.22(C4×Q8), C42.313(C2×C4), C8○3(C2.C42), C22.26(C8○D4), C22.22(C22×C8), C22.30(C2×C42), C4.73(C42⋊C2), C2.5(C8○2M4(2)), C2.C42.28C4, (C2×C42).994C22, (C22×C8).590C22, C23.259(C22×C4), C8○2(C22.7C42), (C22×C4).1612C23, C22.7C42.49C2, C8○(C2×C4⋊C8), C2.9(C2×C4×C8), C2.3(C4×C4⋊C4), C4⋊C8○(C22×C8), (C2×C4×C8).16C2, (C2×C8)○2(C4⋊C8), C4.73(C2×C4⋊C4), (C2×C4⋊C8).60C2, (C4×C4⋊C4).80C2, (C2×C4⋊C4).82C4, (C2×C4).39(C2×C8), (C2×C8).204(C2×C4), (C2×C4).332(C2×Q8), (C2×C4).1504(C2×D4), (C2×C4).922(C4○D4), (C2×C4).602(C22×C4), (C22×C4).377(C2×C4), (C2×C8)○2(C2.C42), (C2×C8)○2(C22.7C42), (C2×C8)○(C4×C4⋊C4), (C2×C8)○(C2×C4⋊C8), (C22×C8)○(C4×C4⋊C4), SmallGroup(128,501)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8×C4⋊C4
G = < a,b,c | a8=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 188 in 148 conjugacy classes, 108 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×16], C2×C4 [×10], C23, C42 [×4], C42 [×4], C4⋊C4 [×8], C2×C8 [×12], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C4×C8 [×4], C4×C8 [×4], C4⋊C8 [×4], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C22×C8 [×2], C22.7C42 [×2], C4×C4⋊C4, C2×C4×C8, C2×C4×C8 [×2], C2×C4⋊C8, C8×C4⋊C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], C2×C8 [×12], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C4×C8 [×4], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C22×C8 [×2], C8○D4 [×2], C4×C4⋊C4, C2×C4×C8, C8○2M4(2), C8×D4 [×2], C8×Q8 [×2], C8×C4⋊C4
►Regular action on 128 points
(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)
(1 108 33 120)(2 109 34 113)(3 110 35 114)(4 111 36 115)(5 112 37 116)(6 105 38 117)(7 106 39 118)(8 107 40 119)(9 69 25 63)(10 70 26 64)(11 71 27 57)(12 72 28 58)(13 65 29 59)(14 66 30 60)(15 67 31 61)(16 68 32 62)(17 83 125 79)(18 84 126 80)(19 85 127 73)(20 86 128 74)(21 87 121 75)(22 88 122 76)(23 81 123 77)(24 82 124 78)(41 101 53 95)(42 102 54 96)(43 103 55 89)(44 104 56 90)(45 97 49 91)(46 98 50 92)(47 99 51 93)(48 100 52 94)
(1 81 43 58)(2 82 44 59)(3 83 45 60)(4 84 46 61)(5 85 47 62)(6 86 48 63)(7 87 41 64)(8 88 42 57)(9 117 128 94)(10 118 121 95)(11 119 122 96)(12 120 123 89)(13 113 124 90)(14 114 125 91)(15 115 126 92)(16 116 127 93)(17 97 30 110)(18 98 31 111)(19 99 32 112)(20 100 25 105)(21 101 26 106)(22 102 27 107)(23 103 28 108)(24 104 29 109)(33 77 55 72)(34 78 56 65)(35 79 49 66)(36 80 50 67)(37 73 51 68)(38 74 52 69)(39 75 53 70)(40 76 54 71)
G:=sub<Sym(128)| (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), (1,108,33,120)(2,109,34,113)(3,110,35,114)(4,111,36,115)(5,112,37,116)(6,105,38,117)(7,106,39,118)(8,107,40,119)(9,69,25,63)(10,70,26,64)(11,71,27,57)(12,72,28,58)(13,65,29,59)(14,66,30,60)(15,67,31,61)(16,68,32,62)(17,83,125,79)(18,84,126,80)(19,85,127,73)(20,86,128,74)(21,87,121,75)(22,88,122,76)(23,81,123,77)(24,82,124,78)(41,101,53,95)(42,102,54,96)(43,103,55,89)(44,104,56,90)(45,97,49,91)(46,98,50,92)(47,99,51,93)(48,100,52,94), (1,81,43,58)(2,82,44,59)(3,83,45,60)(4,84,46,61)(5,85,47,62)(6,86,48,63)(7,87,41,64)(8,88,42,57)(9,117,128,94)(10,118,121,95)(11,119,122,96)(12,120,123,89)(13,113,124,90)(14,114,125,91)(15,115,126,92)(16,116,127,93)(17,97,30,110)(18,98,31,111)(19,99,32,112)(20,100,25,105)(21,101,26,106)(22,102,27,107)(23,103,28,108)(24,104,29,109)(33,77,55,72)(34,78,56,65)(35,79,49,66)(36,80,50,67)(37,73,51,68)(38,74,52,69)(39,75,53,70)(40,76,54,71)>;
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), (1,108,33,120)(2,109,34,113)(3,110,35,114)(4,111,36,115)(5,112,37,116)(6,105,38,117)(7,106,39,118)(8,107,40,119)(9,69,25,63)(10,70,26,64)(11,71,27,57)(12,72,28,58)(13,65,29,59)(14,66,30,60)(15,67,31,61)(16,68,32,62)(17,83,125,79)(18,84,126,80)(19,85,127,73)(20,86,128,74)(21,87,121,75)(22,88,122,76)(23,81,123,77)(24,82,124,78)(41,101,53,95)(42,102,54,96)(43,103,55,89)(44,104,56,90)(45,97,49,91)(46,98,50,92)(47,99,51,93)(48,100,52,94), (1,81,43,58)(2,82,44,59)(3,83,45,60)(4,84,46,61)(5,85,47,62)(6,86,48,63)(7,87,41,64)(8,88,42,57)(9,117,128,94)(10,118,121,95)(11,119,122,96)(12,120,123,89)(13,113,124,90)(14,114,125,91)(15,115,126,92)(16,116,127,93)(17,97,30,110)(18,98,31,111)(19,99,32,112)(20,100,25,105)(21,101,26,106)(22,102,27,107)(23,103,28,108)(24,104,29,109)(33,77,55,72)(34,78,56,65)(35,79,49,66)(36,80,50,67)(37,73,51,68)(38,74,52,69)(39,75,53,70)(40,76,54,71) );
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)], [(1,108,33,120),(2,109,34,113),(3,110,35,114),(4,111,36,115),(5,112,37,116),(6,105,38,117),(7,106,39,118),(8,107,40,119),(9,69,25,63),(10,70,26,64),(11,71,27,57),(12,72,28,58),(13,65,29,59),(14,66,30,60),(15,67,31,61),(16,68,32,62),(17,83,125,79),(18,84,126,80),(19,85,127,73),(20,86,128,74),(21,87,121,75),(22,88,122,76),(23,81,123,77),(24,82,124,78),(41,101,53,95),(42,102,54,96),(43,103,55,89),(44,104,56,90),(45,97,49,91),(46,98,50,92),(47,99,51,93),(48,100,52,94)], [(1,81,43,58),(2,82,44,59),(3,83,45,60),(4,84,46,61),(5,85,47,62),(6,86,48,63),(7,87,41,64),(8,88,42,57),(9,117,128,94),(10,118,121,95),(11,119,122,96),(12,120,123,89),(13,113,124,90),(14,114,125,91),(15,115,126,92),(16,116,127,93),(17,97,30,110),(18,98,31,111),(19,99,32,112),(20,100,25,105),(21,101,26,106),(22,102,27,107),(23,103,28,108),(24,104,29,109),(33,77,55,72),(34,78,56,65),(35,79,49,66),(36,80,50,67),(37,73,51,68),(38,74,52,69),(39,75,53,70),(40,76,54,71)])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AF | 8A | ··· | 8P | 8Q | ··· | 8AN |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | D4 | Q8 | C4○D4 | C8○D4 |
| kernel | C8×C4⋊C4 | C22.7C42 | C4×C4⋊C4 | C2×C4×C8 | C2×C4⋊C8 | C2.C42 | C4×C8 | C4⋊C8 | C2×C4⋊C4 | C4⋊C4 | C2×C8 | C2×C8 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 3 | 1 | 4 | 8 | 8 | 4 | 32 | 2 | 2 | 4 | 8 |
Matrix representation of C8×C4⋊C4 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,2,0,0,0,0,13,0,0,0,0,13],[16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0] >;
C8×C4⋊C4 in GAP, Magma, Sage, TeX
C_8\times C_4\rtimes C_4
% in TeX
G:=Group("C8xC4:C4"); // GroupNames label
G:=SmallGroup(128,501);
// by ID
G=gap.SmallGroup(128,501);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,142,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations