p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊5C8, C43.4C2, (C2×C42).38C4, (C2×C4).62M4(2), C2.2(C42⋊5C4), (C22×C8).21C22, C22.37(C22×C8), C4.44(C42⋊2C2), C2.7(C42.6C4), C23.266(C22×C4), (C2×C42).997C22, C22.48(C2×M4(2)), C2.8(C42.12C4), (C22×C4).1623C23, C22.7C42.5C2, C22.57(C42⋊C2), (C2×C4).62(C2×C8), (C2×C4).931(C4○D4), (C22×C4).444(C2×C4), SmallGroup(128,571)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊5C8
G = < a,b,c | a4=b4=c8=1, ab=ba, cac-1=ab2, cbc-1=a2b-1 >
Subgroups: 180 in 120 conjugacy classes, 68 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C42, C2×C8, C22×C4, C22×C4, C2×C42, C2×C42, C22×C8, C22.7C42, C43, C42⋊5C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, M4(2), C22×C4, C4○D4, C42⋊C2, C42⋊2C2, C22×C8, C2×M4(2), C42⋊5C4, C42.12C4, C42.6C4, C42⋊5C8
►Regular action on 128 points
Generators in S128
(1 59 47 93)(2 116 48 38)(3 61 41 95)(4 118 42 40)(5 63 43 89)(6 120 44 34)(7 57 45 91)(8 114 46 36)(9 30 68 108)(10 87 69 53)(11 32 70 110)(12 81 71 55)(13 26 72 112)(14 83 65 49)(15 28 66 106)(16 85 67 51)(17 115 103 37)(18 60 104 94)(19 117 97 39)(20 62 98 96)(21 119 99 33)(22 64 100 90)(23 113 101 35)(24 58 102 92)(25 123 111 73)(27 125 105 75)(29 127 107 77)(31 121 109 79)(50 76 84 126)(52 78 86 128)(54 80 88 122)(56 74 82 124)
(1 29 17 85)(2 52 18 108)(3 31 19 87)(4 54 20 110)(5 25 21 81)(6 56 22 112)(7 27 23 83)(8 50 24 106)(9 116 78 60)(10 95 79 39)(11 118 80 62)(12 89 73 33)(13 120 74 64)(14 91 75 35)(15 114 76 58)(16 93 77 37)(26 44 82 100)(28 46 84 102)(30 48 86 104)(32 42 88 98)(34 124 90 72)(36 126 92 66)(38 128 94 68)(40 122 96 70)(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)| (1,59,47,93)(2,116,48,38)(3,61,41,95)(4,118,42,40)(5,63,43,89)(6,120,44,34)(7,57,45,91)(8,114,46,36)(9,30,68,108)(10,87,69,53)(11,32,70,110)(12,81,71,55)(13,26,72,112)(14,83,65,49)(15,28,66,106)(16,85,67,51)(17,115,103,37)(18,60,104,94)(19,117,97,39)(20,62,98,96)(21,119,99,33)(22,64,100,90)(23,113,101,35)(24,58,102,92)(25,123,111,73)(27,125,105,75)(29,127,107,77)(31,121,109,79)(50,76,84,126)(52,78,86,128)(54,80,88,122)(56,74,82,124), (1,29,17,85)(2,52,18,108)(3,31,19,87)(4,54,20,110)(5,25,21,81)(6,56,22,112)(7,27,23,83)(8,50,24,106)(9,116,78,60)(10,95,79,39)(11,118,80,62)(12,89,73,33)(13,120,74,64)(14,91,75,35)(15,114,76,58)(16,93,77,37)(26,44,82,100)(28,46,84,102)(30,48,86,104)(32,42,88,98)(34,124,90,72)(36,126,92,66)(38,128,94,68)(40,122,96,70)(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( (1,59,47,93)(2,116,48,38)(3,61,41,95)(4,118,42,40)(5,63,43,89)(6,120,44,34)(7,57,45,91)(8,114,46,36)(9,30,68,108)(10,87,69,53)(11,32,70,110)(12,81,71,55)(13,26,72,112)(14,83,65,49)(15,28,66,106)(16,85,67,51)(17,115,103,37)(18,60,104,94)(19,117,97,39)(20,62,98,96)(21,119,99,33)(22,64,100,90)(23,113,101,35)(24,58,102,92)(25,123,111,73)(27,125,105,75)(29,127,107,77)(31,121,109,79)(50,76,84,126)(52,78,86,128)(54,80,88,122)(56,74,82,124), (1,29,17,85)(2,52,18,108)(3,31,19,87)(4,54,20,110)(5,25,21,81)(6,56,22,112)(7,27,23,83)(8,50,24,106)(9,116,78,60)(10,95,79,39)(11,118,80,62)(12,89,73,33)(13,120,74,64)(14,91,75,35)(15,114,76,58)(16,93,77,37)(26,44,82,100)(28,46,84,102)(30,48,86,104)(32,42,88,98)(34,124,90,72)(36,126,92,66)(38,128,94,68)(40,122,96,70)(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([[(1,59,47,93),(2,116,48,38),(3,61,41,95),(4,118,42,40),(5,63,43,89),(6,120,44,34),(7,57,45,91),(8,114,46,36),(9,30,68,108),(10,87,69,53),(11,32,70,110),(12,81,71,55),(13,26,72,112),(14,83,65,49),(15,28,66,106),(16,85,67,51),(17,115,103,37),(18,60,104,94),(19,117,97,39),(20,62,98,96),(21,119,99,33),(22,64,100,90),(23,113,101,35),(24,58,102,92),(25,123,111,73),(27,125,105,75),(29,127,107,77),(31,121,109,79),(50,76,84,126),(52,78,86,128),(54,80,88,122),(56,74,82,124)], [(1,29,17,85),(2,52,18,108),(3,31,19,87),(4,54,20,110),(5,25,21,81),(6,56,22,112),(7,27,23,83),(8,50,24,106),(9,116,78,60),(10,95,79,39),(11,118,80,62),(12,89,73,33),(13,120,74,64),(14,91,75,35),(15,114,76,58),(16,93,77,37),(26,44,82,100),(28,46,84,102),(30,48,86,104),(32,42,88,98),(34,124,90,72),(36,126,92,66),(38,128,94,68),(40,122,96,70),(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 | 2 | 2 |
| type | + | + | + | ||||
| image | C1 | C2 | C2 | C4 | C8 | M4(2) | C4○D4 |
| kernel | C42⋊5C8 | C22.7C42 | C43 | C2×C42 | C42 | C2×C4 | C2×C4 |
| # reps | 1 | 6 | 1 | 8 | 16 | 12 | 12 |
Matrix representation of C42⋊5C8 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 13 | 2 |
| 0 | 0 | 0 | 1 | 4 |
| 15 | 0 | 0 | 0 | 0 |
| 0 | 10 | 7 | 0 | 0 |
| 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 5 | 0 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,13,1,0,0,0,2,4],[15,0,0,0,0,0,10,7,0,0,0,7,7,0,0,0,0,0,0,5,0,0,0,7,0] >;
C42⋊5C8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_5C_8
% in TeX
G:=Group("C4^2:5C8"); // GroupNames label
G:=SmallGroup(128,571);
// by ID
G=gap.SmallGroup(128,571);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,176,422,58,124]);
// 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^-1>;
// generators/relations