p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊9C8, C43.9C2, C42.57Q8, C42.323D4, C4⋊1(C4⋊C8), C4.44(C4⋊Q8), (C2×C42).45C4, C4.44(C4⋊1D4), (C2×C4).78M4(2), C2.2(C42⋊9C4), (C22×C8).23C22, C22.38(C22×C8), C2.5(C4⋊M4(2)), C23.267(C22×C4), C22.49(C2×M4(2)), (C22×C4).1625C23, (C2×C42).1058C22, C2.10(C2×C4⋊C8), (C2×C4⋊C8).26C2, (C2×C4).85(C2×C8), (C2×C4).88(C4⋊C4), C22.65(C2×C4⋊C4), (C2×C4).339(C2×Q8), (C2×C4).1523(C2×D4), (C22×C4).483(C2×C4), SmallGroup(128,574)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊9C8
G = < a,b,c | a4=b4=c8=1, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 204 in 150 conjugacy classes, 104 normal (9 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C42, C2×C8, C22×C4, C22×C4, C4⋊C8, C2×C42, C2×C42, C22×C8, C43, C2×C4⋊C8, C42⋊9C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4⋊C8, C2×C4⋊C4, C4⋊1D4, C4⋊Q8, C22×C8, C2×M4(2), C42⋊9C4, C2×C4⋊C8, C4⋊M4(2), C42⋊9C8
►Regular action on 128 points
Generators in S128
(1 39 19 95)(2 96 20 40)(3 33 21 89)(4 90 22 34)(5 35 23 91)(6 92 24 36)(7 37 17 93)(8 94 18 38)(9 28 78 84)(10 85 79 29)(11 30 80 86)(12 87 73 31)(13 32 74 88)(14 81 75 25)(15 26 76 82)(16 83 77 27)(41 117 97 61)(42 62 98 118)(43 119 99 63)(44 64 100 120)(45 113 101 57)(46 58 102 114)(47 115 103 59)(48 60 104 116)(49 125 105 65)(50 66 106 126)(51 127 107 67)(52 68 108 128)(53 121 109 69)(54 70 110 122)(55 123 111 71)(56 72 112 124)
(1 107 103 87)(2 88 104 108)(3 109 97 81)(4 82 98 110)(5 111 99 83)(6 84 100 112)(7 105 101 85)(8 86 102 106)(9 120 124 92)(10 93 125 113)(11 114 126 94)(12 95 127 115)(13 116 128 96)(14 89 121 117)(15 118 122 90)(16 91 123 119)(17 49 45 29)(18 30 46 50)(19 51 47 31)(20 32 48 52)(21 53 41 25)(22 26 42 54)(23 55 43 27)(24 28 44 56)(33 69 61 75)(34 76 62 70)(35 71 63 77)(36 78 64 72)(37 65 57 79)(38 80 58 66)(39 67 59 73)(40 74 60 68)
(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)| (1,39,19,95)(2,96,20,40)(3,33,21,89)(4,90,22,34)(5,35,23,91)(6,92,24,36)(7,37,17,93)(8,94,18,38)(9,28,78,84)(10,85,79,29)(11,30,80,86)(12,87,73,31)(13,32,74,88)(14,81,75,25)(15,26,76,82)(16,83,77,27)(41,117,97,61)(42,62,98,118)(43,119,99,63)(44,64,100,120)(45,113,101,57)(46,58,102,114)(47,115,103,59)(48,60,104,116)(49,125,105,65)(50,66,106,126)(51,127,107,67)(52,68,108,128)(53,121,109,69)(54,70,110,122)(55,123,111,71)(56,72,112,124), (1,107,103,87)(2,88,104,108)(3,109,97,81)(4,82,98,110)(5,111,99,83)(6,84,100,112)(7,105,101,85)(8,86,102,106)(9,120,124,92)(10,93,125,113)(11,114,126,94)(12,95,127,115)(13,116,128,96)(14,89,121,117)(15,118,122,90)(16,91,123,119)(17,49,45,29)(18,30,46,50)(19,51,47,31)(20,32,48,52)(21,53,41,25)(22,26,42,54)(23,55,43,27)(24,28,44,56)(33,69,61,75)(34,76,62,70)(35,71,63,77)(36,78,64,72)(37,65,57,79)(38,80,58,66)(39,67,59,73)(40,74,60,68), (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( (1,39,19,95)(2,96,20,40)(3,33,21,89)(4,90,22,34)(5,35,23,91)(6,92,24,36)(7,37,17,93)(8,94,18,38)(9,28,78,84)(10,85,79,29)(11,30,80,86)(12,87,73,31)(13,32,74,88)(14,81,75,25)(15,26,76,82)(16,83,77,27)(41,117,97,61)(42,62,98,118)(43,119,99,63)(44,64,100,120)(45,113,101,57)(46,58,102,114)(47,115,103,59)(48,60,104,116)(49,125,105,65)(50,66,106,126)(51,127,107,67)(52,68,108,128)(53,121,109,69)(54,70,110,122)(55,123,111,71)(56,72,112,124), (1,107,103,87)(2,88,104,108)(3,109,97,81)(4,82,98,110)(5,111,99,83)(6,84,100,112)(7,105,101,85)(8,86,102,106)(9,120,124,92)(10,93,125,113)(11,114,126,94)(12,95,127,115)(13,116,128,96)(14,89,121,117)(15,118,122,90)(16,91,123,119)(17,49,45,29)(18,30,46,50)(19,51,47,31)(20,32,48,52)(21,53,41,25)(22,26,42,54)(23,55,43,27)(24,28,44,56)(33,69,61,75)(34,76,62,70)(35,71,63,77)(36,78,64,72)(37,65,57,79)(38,80,58,66)(39,67,59,73)(40,74,60,68), (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([[(1,39,19,95),(2,96,20,40),(3,33,21,89),(4,90,22,34),(5,35,23,91),(6,92,24,36),(7,37,17,93),(8,94,18,38),(9,28,78,84),(10,85,79,29),(11,30,80,86),(12,87,73,31),(13,32,74,88),(14,81,75,25),(15,26,76,82),(16,83,77,27),(41,117,97,61),(42,62,98,118),(43,119,99,63),(44,64,100,120),(45,113,101,57),(46,58,102,114),(47,115,103,59),(48,60,104,116),(49,125,105,65),(50,66,106,126),(51,127,107,67),(52,68,108,128),(53,121,109,69),(54,70,110,122),(55,123,111,71),(56,72,112,124)], [(1,107,103,87),(2,88,104,108),(3,109,97,81),(4,82,98,110),(5,111,99,83),(6,84,100,112),(7,105,101,85),(8,86,102,106),(9,120,124,92),(10,93,125,113),(11,114,126,94),(12,95,127,115),(13,116,128,96),(14,89,121,117),(15,118,122,90),(16,91,123,119),(17,49,45,29),(18,30,46,50),(19,51,47,31),(20,32,48,52),(21,53,41,25),(22,26,42,54),(23,55,43,27),(24,28,44,56),(33,69,61,75),(34,76,62,70),(35,71,63,77),(36,78,64,72),(37,65,57,79),(38,80,58,66),(39,67,59,73),(40,74,60,68)], [(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 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C4 | C8 | D4 | Q8 | M4(2) |
| kernel | C42⋊9C8 | C43 | C2×C4⋊C8 | C2×C42 | C42 | C42 | C42 | C2×C4 |
| # reps | 1 | 1 | 6 | 8 | 16 | 6 | 6 | 12 |
Matrix representation of C42⋊9C8 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 14 | 12 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,4,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,2,13],[1,0,0,0,0,0,4,0,0,0,0,0,13,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,5,14,0,0,0,8,12] >;
C42⋊9C8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_9C_8
% in TeX
G:=Group("C4^2:9C8"); // GroupNames label
G:=SmallGroup(128,574);
// by ID
G=gap.SmallGroup(128,574);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,100,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^8=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations