p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊4C8, C43.2C2, (C4×C8)⋊11C4, C4.14(C4×C8), (C2×C4).89C42, (C2×C42).30C4, C4.17(C8⋊C4), C42.339(C2×C4), (C2×C4).89M4(2), C2.2(C42⋊4C4), C22.18(C22×C8), C22.26(C2×C42), C4.67(C42⋊C2), (C2×C42).984C22, (C22×C8).468C22, C23.247(C22×C4), C22.34(C2×M4(2)), C2.1(C42.12C4), C4○2(C22.7C42), (C22×C4).1600C23, C22.45(C42⋊C2), C22.7C42.47C2, C42○(C22.7C42), C2.5(C2×C4×C8), (C2×C4×C8).11C2, C2.4(C2×C8⋊C4), (C2×C4).58(C2×C8), (C2×C8).202(C2×C4), (C2×C4).910(C4○D4), (C22×C4).504(C2×C4), (C2×C4).590(C22×C4), (C2×C42)○(C22.7C42), SmallGroup(128,476)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊4C8
G = < a,b,c | a4=b4=c8=1, ab=ba, cac-1=ab2, bc=cb >
Subgroups: 188 in 148 conjugacy classes, 108 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C42, C2×C8, C2×C8, C22×C4, C22×C4, C4×C8, C2×C42, C2×C42, C22×C8, C22.7C42, C43, C2×C4×C8, C42⋊4C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, M4(2), C22×C4, C4○D4, C4×C8, C8⋊C4, C2×C42, C42⋊C2, C22×C8, C2×M4(2), C42⋊4C4, C2×C4×C8, C2×C8⋊C4, C42.12C4, C42⋊4C8
►Regular action on 128 points
Generators in S128
(1 117 99 33)(2 62 100 18)(3 119 101 35)(4 64 102 20)(5 113 103 37)(6 58 104 22)(7 115 97 39)(8 60 98 24)(9 27 121 111)(10 92 122 56)(11 29 123 105)(12 94 124 50)(13 31 125 107)(14 96 126 52)(15 25 127 109)(16 90 128 54)(17 87 61 43)(19 81 63 45)(21 83 57 47)(23 85 59 41)(26 70 110 75)(28 72 112 77)(30 66 106 79)(32 68 108 73)(34 88 118 44)(36 82 120 46)(38 84 114 48)(40 86 116 42)(49 78 93 65)(51 80 95 67)(53 74 89 69)(55 76 91 71)
(1 91 87 27)(2 92 88 28)(3 93 81 29)(4 94 82 30)(5 95 83 31)(6 96 84 32)(7 89 85 25)(8 90 86 26)(9 33 76 17)(10 34 77 18)(11 35 78 19)(12 36 79 20)(13 37 80 21)(14 38 73 22)(15 39 74 23)(16 40 75 24)(41 109 97 53)(42 110 98 54)(43 111 99 55)(44 112 100 56)(45 105 101 49)(46 106 102 50)(47 107 103 51)(48 108 104 52)(57 125 113 67)(58 126 114 68)(59 127 115 69)(60 128 116 70)(61 121 117 71)(62 122 118 72)(63 123 119 65)(64 124 120 66)
(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)| (1,117,99,33)(2,62,100,18)(3,119,101,35)(4,64,102,20)(5,113,103,37)(6,58,104,22)(7,115,97,39)(8,60,98,24)(9,27,121,111)(10,92,122,56)(11,29,123,105)(12,94,124,50)(13,31,125,107)(14,96,126,52)(15,25,127,109)(16,90,128,54)(17,87,61,43)(19,81,63,45)(21,83,57,47)(23,85,59,41)(26,70,110,75)(28,72,112,77)(30,66,106,79)(32,68,108,73)(34,88,118,44)(36,82,120,46)(38,84,114,48)(40,86,116,42)(49,78,93,65)(51,80,95,67)(53,74,89,69)(55,76,91,71), (1,91,87,27)(2,92,88,28)(3,93,81,29)(4,94,82,30)(5,95,83,31)(6,96,84,32)(7,89,85,25)(8,90,86,26)(9,33,76,17)(10,34,77,18)(11,35,78,19)(12,36,79,20)(13,37,80,21)(14,38,73,22)(15,39,74,23)(16,40,75,24)(41,109,97,53)(42,110,98,54)(43,111,99,55)(44,112,100,56)(45,105,101,49)(46,106,102,50)(47,107,103,51)(48,108,104,52)(57,125,113,67)(58,126,114,68)(59,127,115,69)(60,128,116,70)(61,121,117,71)(62,122,118,72)(63,123,119,65)(64,124,120,66), (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( (1,117,99,33)(2,62,100,18)(3,119,101,35)(4,64,102,20)(5,113,103,37)(6,58,104,22)(7,115,97,39)(8,60,98,24)(9,27,121,111)(10,92,122,56)(11,29,123,105)(12,94,124,50)(13,31,125,107)(14,96,126,52)(15,25,127,109)(16,90,128,54)(17,87,61,43)(19,81,63,45)(21,83,57,47)(23,85,59,41)(26,70,110,75)(28,72,112,77)(30,66,106,79)(32,68,108,73)(34,88,118,44)(36,82,120,46)(38,84,114,48)(40,86,116,42)(49,78,93,65)(51,80,95,67)(53,74,89,69)(55,76,91,71), (1,91,87,27)(2,92,88,28)(3,93,81,29)(4,94,82,30)(5,95,83,31)(6,96,84,32)(7,89,85,25)(8,90,86,26)(9,33,76,17)(10,34,77,18)(11,35,78,19)(12,36,79,20)(13,37,80,21)(14,38,73,22)(15,39,74,23)(16,40,75,24)(41,109,97,53)(42,110,98,54)(43,111,99,55)(44,112,100,56)(45,105,101,49)(46,106,102,50)(47,107,103,51)(48,108,104,52)(57,125,113,67)(58,126,114,68)(59,127,115,69)(60,128,116,70)(61,121,117,71)(62,122,118,72)(63,123,119,65)(64,124,120,66), (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([[(1,117,99,33),(2,62,100,18),(3,119,101,35),(4,64,102,20),(5,113,103,37),(6,58,104,22),(7,115,97,39),(8,60,98,24),(9,27,121,111),(10,92,122,56),(11,29,123,105),(12,94,124,50),(13,31,125,107),(14,96,126,52),(15,25,127,109),(16,90,128,54),(17,87,61,43),(19,81,63,45),(21,83,57,47),(23,85,59,41),(26,70,110,75),(28,72,112,77),(30,66,106,79),(32,68,108,73),(34,88,118,44),(36,82,120,46),(38,84,114,48),(40,86,116,42),(49,78,93,65),(51,80,95,67),(53,74,89,69),(55,76,91,71)], [(1,91,87,27),(2,92,88,28),(3,93,81,29),(4,94,82,30),(5,95,83,31),(6,96,84,32),(7,89,85,25),(8,90,86,26),(9,33,76,17),(10,34,77,18),(11,35,78,19),(12,36,79,20),(13,37,80,21),(14,38,73,22),(15,39,74,23),(16,40,75,24),(41,109,97,53),(42,110,98,54),(43,111,99,55),(44,112,100,56),(45,105,101,49),(46,106,102,50),(47,107,103,51),(48,108,104,52),(57,125,113,67),(58,126,114,68),(59,127,115,69),(60,128,116,70),(61,121,117,71),(62,122,118,72),(63,123,119,65),(64,124,120,66)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4X | 4Y | ··· | 4AN | 8A | ··· | 8AF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | M4(2) | C4○D4 |
| kernel | C42⋊4C8 | C22.7C42 | C43 | C2×C4×C8 | C4×C8 | C2×C42 | C42 | C2×C4 | C2×C4 |
| # reps | 1 | 4 | 1 | 2 | 16 | 8 | 32 | 8 | 8 |
Matrix representation of C42⋊4C8 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 13 |
| 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 15 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 16 | 1 |
| 0 | 0 | 15 | 1 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,13,0,0,0,0,4,0,0,0,13,13],[16,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[15,0,0,0,0,13,0,0,0,0,16,15,0,0,1,1] >;
C42⋊4C8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_4C_8
% in TeX
G:=Group("C4^2:4C8"); // GroupNames label
G:=SmallGroup(128,476);
// by ID
G=gap.SmallGroup(128,476);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,58,172]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^8=1,a*b=b*a,c*a*c^-1=a*b^2,b*c=c*b>;
// generators/relations