p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊8C8, C43.8C2, C42.54Q8, C42.321D4, C4.12(C4⋊C8), (C2×C42).44C4, (C2×C4).77M4(2), C2.3(C42⋊8C4), C4.80(C4.4D4), (C22×C8).19C22, C22.36(C22×C8), C4.34(C42.C2), C2.4(C4⋊M4(2)), C2.6(C42.6C4), C23.265(C22×C4), C22.47(C2×M4(2)), C2.7(C42.12C4), (C22×C4).1621C23, (C2×C42).1054C22, C22.7C42.3C2, C22.56(C42⋊C2), C2.9(C2×C4⋊C8), (C2×C4⋊C8).24C2, (C2×C4).61(C2×C8), C22.64(C2×C4⋊C4), (C2×C4).337(C2×Q8), (C2×C4).129(C4⋊C4), (C2×C4).1519(C2×D4), (C2×C4).929(C4○D4), (C22×C4).443(C2×C4), SmallGroup(128,563)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊8C8
G = < a,b,c | a4=b4=c8=1, ab=ba, cac-1=ab2, cbc-1=a2b >
Subgroups: 188 in 130 conjugacy classes, 80 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, C22×C4, C22×C4, C4⋊C8, C2×C42, C2×C42, C22×C8, C22.7C42, C43, C2×C4⋊C8, C42⋊8C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4⋊C8, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, C22×C8, C2×M4(2), C42⋊8C4, C2×C4⋊C8, C4⋊M4(2), C42.12C4, C42.6C4, C42⋊8C8
►Regular action on 128 points
Generators in S128
(1 115 103 91)(2 60 104 36)(3 117 97 93)(4 62 98 38)(5 119 99 95)(6 64 100 40)(7 113 101 89)(8 58 102 34)(9 86 70 110)(10 31 71 55)(11 88 72 112)(12 25 65 49)(13 82 66 106)(14 27 67 51)(15 84 68 108)(16 29 69 53)(17 61 41 37)(18 118 42 94)(19 63 43 39)(20 120 44 96)(21 57 45 33)(22 114 46 90)(23 59 47 35)(24 116 48 92)(26 126 50 78)(28 128 52 80)(30 122 54 74)(32 124 56 76)(73 85 121 109)(75 87 123 111)(77 81 125 105)(79 83 127 107)
(1 27 23 83)(2 52 24 108)(3 29 17 85)(4 54 18 110)(5 31 19 87)(6 56 20 112)(7 25 21 81)(8 50 22 106)(9 62 74 118)(10 39 75 95)(11 64 76 120)(12 33 77 89)(13 58 78 114)(14 35 79 91)(15 60 80 116)(16 37 73 93)(26 46 82 102)(28 48 84 104)(30 42 86 98)(32 44 88 100)(34 126 90 66)(36 128 92 68)(38 122 94 70)(40 124 96 72)(41 109 97 53)(43 111 99 55)(45 105 101 49)(47 107 103 51)(57 125 113 65)(59 127 115 67)(61 121 117 69)(63 123 119 71)
(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)| (1,115,103,91)(2,60,104,36)(3,117,97,93)(4,62,98,38)(5,119,99,95)(6,64,100,40)(7,113,101,89)(8,58,102,34)(9,86,70,110)(10,31,71,55)(11,88,72,112)(12,25,65,49)(13,82,66,106)(14,27,67,51)(15,84,68,108)(16,29,69,53)(17,61,41,37)(18,118,42,94)(19,63,43,39)(20,120,44,96)(21,57,45,33)(22,114,46,90)(23,59,47,35)(24,116,48,92)(26,126,50,78)(28,128,52,80)(30,122,54,74)(32,124,56,76)(73,85,121,109)(75,87,123,111)(77,81,125,105)(79,83,127,107), (1,27,23,83)(2,52,24,108)(3,29,17,85)(4,54,18,110)(5,31,19,87)(6,56,20,112)(7,25,21,81)(8,50,22,106)(9,62,74,118)(10,39,75,95)(11,64,76,120)(12,33,77,89)(13,58,78,114)(14,35,79,91)(15,60,80,116)(16,37,73,93)(26,46,82,102)(28,48,84,104)(30,42,86,98)(32,44,88,100)(34,126,90,66)(36,128,92,68)(38,122,94,70)(40,124,96,72)(41,109,97,53)(43,111,99,55)(45,105,101,49)(47,107,103,51)(57,125,113,65)(59,127,115,67)(61,121,117,69)(63,123,119,71), (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( (1,115,103,91)(2,60,104,36)(3,117,97,93)(4,62,98,38)(5,119,99,95)(6,64,100,40)(7,113,101,89)(8,58,102,34)(9,86,70,110)(10,31,71,55)(11,88,72,112)(12,25,65,49)(13,82,66,106)(14,27,67,51)(15,84,68,108)(16,29,69,53)(17,61,41,37)(18,118,42,94)(19,63,43,39)(20,120,44,96)(21,57,45,33)(22,114,46,90)(23,59,47,35)(24,116,48,92)(26,126,50,78)(28,128,52,80)(30,122,54,74)(32,124,56,76)(73,85,121,109)(75,87,123,111)(77,81,125,105)(79,83,127,107), (1,27,23,83)(2,52,24,108)(3,29,17,85)(4,54,18,110)(5,31,19,87)(6,56,20,112)(7,25,21,81)(8,50,22,106)(9,62,74,118)(10,39,75,95)(11,64,76,120)(12,33,77,89)(13,58,78,114)(14,35,79,91)(15,60,80,116)(16,37,73,93)(26,46,82,102)(28,48,84,104)(30,42,86,98)(32,44,88,100)(34,126,90,66)(36,128,92,68)(38,122,94,70)(40,124,96,72)(41,109,97,53)(43,111,99,55)(45,105,101,49)(47,107,103,51)(57,125,113,65)(59,127,115,67)(61,121,117,69)(63,123,119,71), (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([[(1,115,103,91),(2,60,104,36),(3,117,97,93),(4,62,98,38),(5,119,99,95),(6,64,100,40),(7,113,101,89),(8,58,102,34),(9,86,70,110),(10,31,71,55),(11,88,72,112),(12,25,65,49),(13,82,66,106),(14,27,67,51),(15,84,68,108),(16,29,69,53),(17,61,41,37),(18,118,42,94),(19,63,43,39),(20,120,44,96),(21,57,45,33),(22,114,46,90),(23,59,47,35),(24,116,48,92),(26,126,50,78),(28,128,52,80),(30,122,54,74),(32,124,56,76),(73,85,121,109),(75,87,123,111),(77,81,125,105),(79,83,127,107)], [(1,27,23,83),(2,52,24,108),(3,29,17,85),(4,54,18,110),(5,31,19,87),(6,56,20,112),(7,25,21,81),(8,50,22,106),(9,62,74,118),(10,39,75,95),(11,64,76,120),(12,33,77,89),(13,58,78,114),(14,35,79,91),(15,60,80,116),(16,37,73,93),(26,46,82,102),(28,48,84,104),(30,42,86,98),(32,44,88,100),(34,126,90,66),(36,128,92,68),(38,122,94,70),(40,124,96,72),(41,109,97,53),(43,111,99,55),(45,105,101,49),(47,107,103,51),(57,125,113,65),(59,127,115,67),(61,121,117,69),(63,123,119,71)], [(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 | ··· | 4AF | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C8 | D4 | Q8 | M4(2) | C4○D4 |
| kernel | C42⋊8C8 | C22.7C42 | C43 | C2×C4⋊C8 | C2×C42 | C42 | C42 | C42 | C2×C4 | C2×C4 |
| # reps | 1 | 4 | 1 | 2 | 8 | 16 | 2 | 2 | 12 | 8 |
Matrix representation of C42⋊8C8 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 16 | 0 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 6 | 6 | 0 | 0 |
| 0 | 6 | 11 | 0 | 0 |
| 0 | 0 | 0 | 15 | 5 |
| 0 | 0 | 0 | 12 | 2 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,16,0],[9,0,0,0,0,0,6,6,0,0,0,6,11,0,0,0,0,0,15,12,0,0,0,5,2] >;
C42⋊8C8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_8C_8
% in TeX
G:=Group("C4^2:8C8"); // GroupNames label
G:=SmallGroup(128,563);
// by ID
G=gap.SmallGroup(128,563);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,58,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^8=1,a*b=b*a,c*a*c^-1=a*b^2,c*b*c^-1=a^2*b>;
// generators/relations