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, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4×C8, C4×C8, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C22×C8, C22×C8, C22.7C42, C4×C4⋊C4, C2×C4×C8, C2×C4×C8, C2×C4⋊C8, C8×C4⋊C4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C42, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, C4○D4, C4×C8, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22×C8, C8○D4, C4×C4⋊C4, C2×C4×C8, C8○2M4(2), C8×D4, C8×Q8, 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 65 25 58)(10 66 26 59)(11 67 27 60)(12 68 28 61)(13 69 29 62)(14 70 30 63)(15 71 31 64)(16 72 32 57)(17 87 124 75)(18 88 125 76)(19 81 126 77)(20 82 127 78)(21 83 128 79)(22 84 121 80)(23 85 122 73)(24 86 123 74)(41 97 53 90)(42 98 54 91)(43 99 55 92)(44 100 56 93)(45 101 49 94)(46 102 50 95)(47 103 51 96)(48 104 52 89)
(1 81 43 61)(2 82 44 62)(3 83 45 63)(4 84 46 64)(5 85 47 57)(6 86 48 58)(7 87 41 59)(8 88 42 60)(9 117 123 89)(10 118 124 90)(11 119 125 91)(12 120 126 92)(13 113 127 93)(14 114 128 94)(15 115 121 95)(16 116 122 96)(17 97 26 106)(18 98 27 107)(19 99 28 108)(20 100 29 109)(21 101 30 110)(22 102 31 111)(23 103 32 112)(24 104 25 105)(33 77 55 68)(34 78 56 69)(35 79 49 70)(36 80 50 71)(37 73 51 72)(38 74 52 65)(39 75 53 66)(40 76 54 67)
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)(9,65,25,58)(10,66,26,59)(11,67,27,60)(12,68,28,61)(13,69,29,62)(14,70,30,63)(15,71,31,64)(16,72,32,57)(17,87,124,75)(18,88,125,76)(19,81,126,77)(20,82,127,78)(21,83,128,79)(22,84,121,80)(23,85,122,73)(24,86,123,74)(41,97,53,90)(42,98,54,91)(43,99,55,92)(44,100,56,93)(45,101,49,94)(46,102,50,95)(47,103,51,96)(48,104,52,89), (1,81,43,61)(2,82,44,62)(3,83,45,63)(4,84,46,64)(5,85,47,57)(6,86,48,58)(7,87,41,59)(8,88,42,60)(9,117,123,89)(10,118,124,90)(11,119,125,91)(12,120,126,92)(13,113,127,93)(14,114,128,94)(15,115,121,95)(16,116,122,96)(17,97,26,106)(18,98,27,107)(19,99,28,108)(20,100,29,109)(21,101,30,110)(22,102,31,111)(23,103,32,112)(24,104,25,105)(33,77,55,68)(34,78,56,69)(35,79,49,70)(36,80,50,71)(37,73,51,72)(38,74,52,65)(39,75,53,66)(40,76,54,67)>;
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)(9,65,25,58)(10,66,26,59)(11,67,27,60)(12,68,28,61)(13,69,29,62)(14,70,30,63)(15,71,31,64)(16,72,32,57)(17,87,124,75)(18,88,125,76)(19,81,126,77)(20,82,127,78)(21,83,128,79)(22,84,121,80)(23,85,122,73)(24,86,123,74)(41,97,53,90)(42,98,54,91)(43,99,55,92)(44,100,56,93)(45,101,49,94)(46,102,50,95)(47,103,51,96)(48,104,52,89), (1,81,43,61)(2,82,44,62)(3,83,45,63)(4,84,46,64)(5,85,47,57)(6,86,48,58)(7,87,41,59)(8,88,42,60)(9,117,123,89)(10,118,124,90)(11,119,125,91)(12,120,126,92)(13,113,127,93)(14,114,128,94)(15,115,121,95)(16,116,122,96)(17,97,26,106)(18,98,27,107)(19,99,28,108)(20,100,29,109)(21,101,30,110)(22,102,31,111)(23,103,32,112)(24,104,25,105)(33,77,55,68)(34,78,56,69)(35,79,49,70)(36,80,50,71)(37,73,51,72)(38,74,52,65)(39,75,53,66)(40,76,54,67) );
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),(9,65,25,58),(10,66,26,59),(11,67,27,60),(12,68,28,61),(13,69,29,62),(14,70,30,63),(15,71,31,64),(16,72,32,57),(17,87,124,75),(18,88,125,76),(19,81,126,77),(20,82,127,78),(21,83,128,79),(22,84,121,80),(23,85,122,73),(24,86,123,74),(41,97,53,90),(42,98,54,91),(43,99,55,92),(44,100,56,93),(45,101,49,94),(46,102,50,95),(47,103,51,96),(48,104,52,89)], [(1,81,43,61),(2,82,44,62),(3,83,45,63),(4,84,46,64),(5,85,47,57),(6,86,48,58),(7,87,41,59),(8,88,42,60),(9,117,123,89),(10,118,124,90),(11,119,125,91),(12,120,126,92),(13,113,127,93),(14,114,128,94),(15,115,121,95),(16,116,122,96),(17,97,26,106),(18,98,27,107),(19,99,28,108),(20,100,29,109),(21,101,30,110),(22,102,31,111),(23,103,32,112),(24,104,25,105),(33,77,55,68),(34,78,56,69),(35,79,49,70),(36,80,50,71),(37,73,51,72),(38,74,52,65),(39,75,53,66),(40,76,54,67)]])
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