direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×C4⋊C8, C42⋊7C8, C43.6C2, C42.67Q8, C42.459D4, C4⋊1(C4×C8), C4.41(C4×Q8), C4.165(C4×D4), (C2×C4).71C42, (C2×C42).34C4, C2.4(C4×M4(2)), C42.263(C2×C4), (C2×C4).91M4(2), C22.21(C22×C8), C22.29(C2×C42), (C22×C8).475C22, (C2×C42).992C22, C23.256(C22×C4), C22.40(C2×M4(2)), C2.3(C42.12C4), C4○4(C22.7C42), (C22×C4).1609C23, C22.51(C42⋊C2), C22.7C42.48C2, C42○3(C22.7C42), C2.8(C2×C4×C8), C2.2(C2×C4⋊C8), C2.2(C4×C4⋊C4), C42○(C2×C4⋊C8), (C2×C4×C8).14C2, (C2×C4⋊C8).59C2, (C2×C4).83(C2×C8), (C2×C8).158(C2×C4), C22.55(C2×C4⋊C4), (C2×C4).329(C2×Q8), (C2×C4).163(C4⋊C4), (C2×C4).1501(C2×D4), (C2×C4).919(C4○D4), (C22×C4).438(C2×C4), (C2×C4).599(C22×C4), SmallGroup(128,498)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×C4⋊C8
G = < a,b,c | a4=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 196 in 158 conjugacy classes, 120 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, C2×C8, C22×C4, C22×C4, C4×C8, C4⋊C8, C2×C42, C2×C42, C22×C8, C22.7C42, C43, C2×C4×C8, C2×C4⋊C8, C4×C4⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C42, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4×C8, C4⋊C8, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22×C8, C2×M4(2), C4×C4⋊C4, C2×C4×C8, C4×M4(2), C2×C4⋊C8, C42.12C4, C4×C4⋊C8
►Regular action on 128 points
(1 28 95 102)(2 29 96 103)(3 30 89 104)(4 31 90 97)(5 32 91 98)(6 25 92 99)(7 26 93 100)(8 27 94 101)(9 111 74 64)(10 112 75 57)(11 105 76 58)(12 106 77 59)(13 107 78 60)(14 108 79 61)(15 109 80 62)(16 110 73 63)(17 85 45 38)(18 86 46 39)(19 87 47 40)(20 88 48 33)(21 81 41 34)(22 82 42 35)(23 83 43 36)(24 84 44 37)(49 125 115 66)(50 126 116 67)(51 127 117 68)(52 128 118 69)(53 121 119 70)(54 122 120 71)(55 123 113 72)(56 124 114 65)
(1 77 38 66)(2 67 39 78)(3 79 40 68)(4 69 33 80)(5 73 34 70)(6 71 35 74)(7 75 36 72)(8 65 37 76)(9 92 122 82)(10 83 123 93)(11 94 124 84)(12 85 125 95)(13 96 126 86)(14 87 127 89)(15 90 128 88)(16 81 121 91)(17 49 28 59)(18 60 29 50)(19 51 30 61)(20 62 31 52)(21 53 32 63)(22 64 25 54)(23 55 26 57)(24 58 27 56)(41 119 98 110)(42 111 99 120)(43 113 100 112)(44 105 101 114)(45 115 102 106)(46 107 103 116)(47 117 104 108)(48 109 97 118)
(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)| (1,28,95,102)(2,29,96,103)(3,30,89,104)(4,31,90,97)(5,32,91,98)(6,25,92,99)(7,26,93,100)(8,27,94,101)(9,111,74,64)(10,112,75,57)(11,105,76,58)(12,106,77,59)(13,107,78,60)(14,108,79,61)(15,109,80,62)(16,110,73,63)(17,85,45,38)(18,86,46,39)(19,87,47,40)(20,88,48,33)(21,81,41,34)(22,82,42,35)(23,83,43,36)(24,84,44,37)(49,125,115,66)(50,126,116,67)(51,127,117,68)(52,128,118,69)(53,121,119,70)(54,122,120,71)(55,123,113,72)(56,124,114,65), (1,77,38,66)(2,67,39,78)(3,79,40,68)(4,69,33,80)(5,73,34,70)(6,71,35,74)(7,75,36,72)(8,65,37,76)(9,92,122,82)(10,83,123,93)(11,94,124,84)(12,85,125,95)(13,96,126,86)(14,87,127,89)(15,90,128,88)(16,81,121,91)(17,49,28,59)(18,60,29,50)(19,51,30,61)(20,62,31,52)(21,53,32,63)(22,64,25,54)(23,55,26,57)(24,58,27,56)(41,119,98,110)(42,111,99,120)(43,113,100,112)(44,105,101,114)(45,115,102,106)(46,107,103,116)(47,117,104,108)(48,109,97,118), (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( (1,28,95,102)(2,29,96,103)(3,30,89,104)(4,31,90,97)(5,32,91,98)(6,25,92,99)(7,26,93,100)(8,27,94,101)(9,111,74,64)(10,112,75,57)(11,105,76,58)(12,106,77,59)(13,107,78,60)(14,108,79,61)(15,109,80,62)(16,110,73,63)(17,85,45,38)(18,86,46,39)(19,87,47,40)(20,88,48,33)(21,81,41,34)(22,82,42,35)(23,83,43,36)(24,84,44,37)(49,125,115,66)(50,126,116,67)(51,127,117,68)(52,128,118,69)(53,121,119,70)(54,122,120,71)(55,123,113,72)(56,124,114,65), (1,77,38,66)(2,67,39,78)(3,79,40,68)(4,69,33,80)(5,73,34,70)(6,71,35,74)(7,75,36,72)(8,65,37,76)(9,92,122,82)(10,83,123,93)(11,94,124,84)(12,85,125,95)(13,96,126,86)(14,87,127,89)(15,90,128,88)(16,81,121,91)(17,49,28,59)(18,60,29,50)(19,51,30,61)(20,62,31,52)(21,53,32,63)(22,64,25,54)(23,55,26,57)(24,58,27,56)(41,119,98,110)(42,111,99,120)(43,113,100,112)(44,105,101,114)(45,115,102,106)(46,107,103,116)(47,117,104,108)(48,109,97,118), (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([[(1,28,95,102),(2,29,96,103),(3,30,89,104),(4,31,90,97),(5,32,91,98),(6,25,92,99),(7,26,93,100),(8,27,94,101),(9,111,74,64),(10,112,75,57),(11,105,76,58),(12,106,77,59),(13,107,78,60),(14,108,79,61),(15,109,80,62),(16,110,73,63),(17,85,45,38),(18,86,46,39),(19,87,47,40),(20,88,48,33),(21,81,41,34),(22,82,42,35),(23,83,43,36),(24,84,44,37),(49,125,115,66),(50,126,116,67),(51,127,117,68),(52,128,118,69),(53,121,119,70),(54,122,120,71),(55,123,113,72),(56,124,114,65)], [(1,77,38,66),(2,67,39,78),(3,79,40,68),(4,69,33,80),(5,73,34,70),(6,71,35,74),(7,75,36,72),(8,65,37,76),(9,92,122,82),(10,83,123,93),(11,94,124,84),(12,85,125,95),(13,96,126,86),(14,87,127,89),(15,90,128,88),(16,81,121,91),(17,49,28,59),(18,60,29,50),(19,51,30,61),(20,62,31,52),(21,53,32,63),(22,64,25,54),(23,55,26,57),(24,58,27,56),(41,119,98,110),(42,111,99,120),(43,113,100,112),(44,105,101,114),(45,115,102,106),(46,107,103,116),(47,117,104,108),(48,109,97,118)], [(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 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | M4(2) | C4○D4 |
| kernel | C4×C4⋊C8 | C22.7C42 | C43 | C2×C4×C8 | C2×C4⋊C8 | C4⋊C8 | C2×C42 | C42 | C42 | C42 | C2×C4 | C2×C4 |
| # reps | 1 | 2 | 1 | 2 | 2 | 16 | 8 | 32 | 2 | 2 | 8 | 4 |
Matrix representation of C4×C4⋊C8 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[16,0,0,0,0,1,0,0,0,0,0,4,0,0,4,0],[2,0,0,0,0,4,0,0,0,0,9,0,0,0,0,8] >;
C4×C4⋊C8 in GAP, Magma, Sage, TeX
C_4\times C_4\rtimes C_8
% in TeX
G:=Group("C4xC4:C8"); // GroupNames label
G:=SmallGroup(128,498);
// by ID
G=gap.SmallGroup(128,498);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,142,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations