p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4⋊C4⋊3C8, C2.3(C8×Q8), C2.12(C8×D4), (C2×C8).38Q8, (C2×C8).325D4, C2.6(C8⋊9D4), C2.3(C8⋊4Q8), C22.25(C4×Q8), C22.100(C4×D4), (C2×C4).36M4(2), C22.31(C8○D4), C4.114(C22⋊Q8), (C22×C8).40C22, C22.41(C22×C8), C4.36(C42.C2), C4.46(C42⋊2C2), C2.C42.21C4, (C2×C42).290C22, C23.270(C22×C4), C22.52(C2×M4(2)), C2.9(C42.12C4), (C22×C4).1631C23, C22.7C42.7C2, C22.58(C42⋊C2), C4.137(C22.D4), C2.4(C42.7C22), C2.3(C23.63C23), (C2×C4×C8).20C2, (C2×C4⋊C8).28C2, (C4×C4⋊C4).12C2, (C2×C4⋊C4).56C4, (C2×C4).20(C2×C8), (C2×C4).343(C2×Q8), (C2×C4).1532(C2×D4), (C2×C4).937(C4○D4), (C22×C4).121(C2×C4), SmallGroup(128,648)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C4⋊3C8
G = < a,b,c | a4=b4=c8=1, bab-1=a-1, cac-1=ab2, bc=cb >
Subgroups: 180 in 120 conjugacy classes, 68 normal (52 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2.C42, C4×C8, C4⋊C8, C2×C42, C2×C4⋊C4, C22×C8, C22.7C42, C4×C4⋊C4, C2×C4×C8, C2×C4⋊C8, C4⋊C4⋊3C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C22×C8, C2×M4(2), C8○D4, C23.63C23, C42.12C4, C42.7C22, C8×D4, C8⋊9D4, C8×Q8, C8⋊4Q8, C4⋊C4⋊3C8
►Regular action on 128 points
Generators in S128
(1 40 55 97)(2 125 56 106)(3 34 49 99)(4 127 50 108)(5 36 51 101)(6 121 52 110)(7 38 53 103)(8 123 54 112)(9 37 28 102)(10 122 29 111)(11 39 30 104)(12 124 31 105)(13 33 32 98)(14 126 25 107)(15 35 26 100)(16 128 27 109)(17 113 76 68)(18 82 77 92)(19 115 78 70)(20 84 79 94)(21 117 80 72)(22 86 73 96)(23 119 74 66)(24 88 75 90)(41 116 63 71)(42 85 64 95)(43 118 57 65)(44 87 58 89)(45 120 59 67)(46 81 60 91)(47 114 61 69)(48 83 62 93)
(1 24 12 45)(2 17 13 46)(3 18 14 47)(4 19 15 48)(5 20 16 41)(6 21 9 42)(7 22 10 43)(8 23 11 44)(25 61 49 77)(26 62 50 78)(27 63 51 79)(28 64 52 80)(29 57 53 73)(30 58 54 74)(31 59 55 75)(32 60 56 76)(33 91 125 68)(34 92 126 69)(35 93 127 70)(36 94 128 71)(37 95 121 72)(38 96 122 65)(39 89 123 66)(40 90 124 67)(81 106 113 98)(82 107 114 99)(83 108 115 100)(84 109 116 101)(85 110 117 102)(86 111 118 103)(87 112 119 104)(88 105 120 97)
(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)
G:=sub<Sym(128)| (1,40,55,97)(2,125,56,106)(3,34,49,99)(4,127,50,108)(5,36,51,101)(6,121,52,110)(7,38,53,103)(8,123,54,112)(9,37,28,102)(10,122,29,111)(11,39,30,104)(12,124,31,105)(13,33,32,98)(14,126,25,107)(15,35,26,100)(16,128,27,109)(17,113,76,68)(18,82,77,92)(19,115,78,70)(20,84,79,94)(21,117,80,72)(22,86,73,96)(23,119,74,66)(24,88,75,90)(41,116,63,71)(42,85,64,95)(43,118,57,65)(44,87,58,89)(45,120,59,67)(46,81,60,91)(47,114,61,69)(48,83,62,93), (1,24,12,45)(2,17,13,46)(3,18,14,47)(4,19,15,48)(5,20,16,41)(6,21,9,42)(7,22,10,43)(8,23,11,44)(25,61,49,77)(26,62,50,78)(27,63,51,79)(28,64,52,80)(29,57,53,73)(30,58,54,74)(31,59,55,75)(32,60,56,76)(33,91,125,68)(34,92,126,69)(35,93,127,70)(36,94,128,71)(37,95,121,72)(38,96,122,65)(39,89,123,66)(40,90,124,67)(81,106,113,98)(82,107,114,99)(83,108,115,100)(84,109,116,101)(85,110,117,102)(86,111,118,103)(87,112,119,104)(88,105,120,97), (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)>;
G:=Group( (1,40,55,97)(2,125,56,106)(3,34,49,99)(4,127,50,108)(5,36,51,101)(6,121,52,110)(7,38,53,103)(8,123,54,112)(9,37,28,102)(10,122,29,111)(11,39,30,104)(12,124,31,105)(13,33,32,98)(14,126,25,107)(15,35,26,100)(16,128,27,109)(17,113,76,68)(18,82,77,92)(19,115,78,70)(20,84,79,94)(21,117,80,72)(22,86,73,96)(23,119,74,66)(24,88,75,90)(41,116,63,71)(42,85,64,95)(43,118,57,65)(44,87,58,89)(45,120,59,67)(46,81,60,91)(47,114,61,69)(48,83,62,93), (1,24,12,45)(2,17,13,46)(3,18,14,47)(4,19,15,48)(5,20,16,41)(6,21,9,42)(7,22,10,43)(8,23,11,44)(25,61,49,77)(26,62,50,78)(27,63,51,79)(28,64,52,80)(29,57,53,73)(30,58,54,74)(31,59,55,75)(32,60,56,76)(33,91,125,68)(34,92,126,69)(35,93,127,70)(36,94,128,71)(37,95,121,72)(38,96,122,65)(39,89,123,66)(40,90,124,67)(81,106,113,98)(82,107,114,99)(83,108,115,100)(84,109,116,101)(85,110,117,102)(86,111,118,103)(87,112,119,104)(88,105,120,97), (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) );
G=PermutationGroup([[(1,40,55,97),(2,125,56,106),(3,34,49,99),(4,127,50,108),(5,36,51,101),(6,121,52,110),(7,38,53,103),(8,123,54,112),(9,37,28,102),(10,122,29,111),(11,39,30,104),(12,124,31,105),(13,33,32,98),(14,126,25,107),(15,35,26,100),(16,128,27,109),(17,113,76,68),(18,82,77,92),(19,115,78,70),(20,84,79,94),(21,117,80,72),(22,86,73,96),(23,119,74,66),(24,88,75,90),(41,116,63,71),(42,85,64,95),(43,118,57,65),(44,87,58,89),(45,120,59,67),(46,81,60,91),(47,114,61,69),(48,83,62,93)], [(1,24,12,45),(2,17,13,46),(3,18,14,47),(4,19,15,48),(5,20,16,41),(6,21,9,42),(7,22,10,43),(8,23,11,44),(25,61,49,77),(26,62,50,78),(27,63,51,79),(28,64,52,80),(29,57,53,73),(30,58,54,74),(31,59,55,75),(32,60,56,76),(33,91,125,68),(34,92,126,69),(35,93,127,70),(36,94,128,71),(37,95,121,72),(38,96,122,65),(39,89,123,66),(40,90,124,67),(81,106,113,98),(82,107,114,99),(83,108,115,100),(84,109,116,101),(85,110,117,102),(86,111,118,103),(87,112,119,104),(88,105,120,97)], [(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)]])
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 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | M4(2) | C4○D4 | C8○D4 |
| kernel | C4⋊C4⋊3C8 | C22.7C42 | C4×C4⋊C4 | C2×C4×C8 | C2×C4⋊C8 | C2.C42 | C2×C4⋊C4 | C4⋊C4 | C2×C8 | C2×C8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 16 | 2 | 2 | 4 | 8 | 8 |
Matrix representation of C4⋊C4⋊3C8 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 11 | 9 | 0 | 0 |
| 0 | 15 | 6 | 0 | 0 |
| 0 | 0 | 0 | 11 | 2 |
| 0 | 0 | 0 | 7 | 6 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 13 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 8 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,11,15,0,0,0,9,6,0,0,0,0,0,11,7,0,0,0,2,6],[16,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,2,13],[2,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,8,1] >;
C4⋊C4⋊3C8 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_3C_8
% in TeX
G:=Group("C4:C4:3C8"); // GroupNames label
G:=SmallGroup(128,648);
// by ID
G=gap.SmallGroup(128,648);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,58,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^8=1,b*a*b^-1=a^-1,c*a*c^-1=a*b^2,b*c=c*b>;
// generators/relations