p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4⋊C8⋊14C4, C8⋊9(C4⋊C4), C8⋊C4⋊15C4, C4.46(C4×Q8), (C2×C8).59Q8, C4.170(C4×D4), (C2×C8).385D4, C2.3(C8⋊9D4), (C2×C4).26C42, C2.2(C8⋊4Q8), C22.92(C4×D4), C22.24(C4×Q8), C42.132(C2×C4), C2.13(C4×M4(2)), (C2×C4).35M4(2), C22.28(C8○D4), C22.56(C2×C42), C4.75(C42⋊C2), C2.C42.16C4, (C2×C42).239C22, C23.261(C22×C4), (C22×C8).477C22, C22.43(C2×M4(2)), C2.17(C8○2M4(2)), (C22×C4).1614C23, C22.7C42.43C2, (C2×C4×C8).63C2, (C4×C4⋊C4).8C2, C4.75(C2×C4⋊C4), C2.10(C4×C4⋊C4), (C2×C4⋊C8).54C2, (C2×C4⋊C4).49C4, (C2×C8).138(C2×C4), (C2×C4).334(C2×Q8), (C2×C8⋊C4).27C2, (C2×C4).1506(C2×D4), (C2×C4).924(C4○D4), (C22×C4).111(C2×C4), (C2×C4).604(C22×C4), SmallGroup(128,503)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C8⋊14C4
G = < a,b,c | a4=b8=c4=1, bab-1=a-1, cac-1=a-1b4, bc=cb >
Subgroups: 188 in 136 conjugacy classes, 88 normal (36 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×8], C2×C4 [×14], C23, C42 [×4], C42 [×4], C4⋊C4 [×4], C2×C8 [×12], C2×C8 [×6], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C4×C8 [×2], C8⋊C4 [×4], C8⋊C4 [×2], C4⋊C8 [×4], C2×C42 [×3], C2×C4⋊C4 [×2], C22×C8 [×4], C22.7C42 [×2], C4×C4⋊C4, C2×C4×C8, C2×C8⋊C4 [×2], C2×C4⋊C8, C4⋊C8⋊14C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], M4(2) [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C2×M4(2) [×2], C8○D4 [×2], C4×C4⋊C4, C4×M4(2), C8○2M4(2), C8⋊9D4 [×2], C8⋊4Q8 [×2], C4⋊C8⋊14C4
►Regular action on 128 points
(1 80 34 65)(2 66 35 73)(3 74 36 67)(4 68 37 75)(5 76 38 69)(6 70 39 77)(7 78 40 71)(8 72 33 79)(9 81 125 94)(10 95 126 82)(11 83 127 96)(12 89 128 84)(13 85 121 90)(14 91 122 86)(15 87 123 92)(16 93 124 88)(17 57 32 51)(18 52 25 58)(19 59 26 53)(20 54 27 60)(21 61 28 55)(22 56 29 62)(23 63 30 49)(24 50 31 64)(41 116 103 110)(42 111 104 117)(43 118 97 112)(44 105 98 119)(45 120 99 106)(46 107 100 113)(47 114 101 108)(48 109 102 115)
(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)
(1 18 96 104)(2 19 89 97)(3 20 90 98)(4 21 91 99)(5 22 92 100)(6 23 93 101)(7 24 94 102)(8 17 95 103)(9 105 78 60)(10 106 79 61)(11 107 80 62)(12 108 73 63)(13 109 74 64)(14 110 75 57)(15 111 76 58)(16 112 77 59)(25 83 42 34)(26 84 43 35)(27 85 44 36)(28 86 45 37)(29 87 46 38)(30 88 47 39)(31 81 48 40)(32 82 41 33)(49 128 114 66)(50 121 115 67)(51 122 116 68)(52 123 117 69)(53 124 118 70)(54 125 119 71)(55 126 120 72)(56 127 113 65)
G:=sub<Sym(128)| (1,80,34,65)(2,66,35,73)(3,74,36,67)(4,68,37,75)(5,76,38,69)(6,70,39,77)(7,78,40,71)(8,72,33,79)(9,81,125,94)(10,95,126,82)(11,83,127,96)(12,89,128,84)(13,85,121,90)(14,91,122,86)(15,87,123,92)(16,93,124,88)(17,57,32,51)(18,52,25,58)(19,59,26,53)(20,54,27,60)(21,61,28,55)(22,56,29,62)(23,63,30,49)(24,50,31,64)(41,116,103,110)(42,111,104,117)(43,118,97,112)(44,105,98,119)(45,120,99,106)(46,107,100,113)(47,114,101,108)(48,109,102,115), (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), (1,18,96,104)(2,19,89,97)(3,20,90,98)(4,21,91,99)(5,22,92,100)(6,23,93,101)(7,24,94,102)(8,17,95,103)(9,105,78,60)(10,106,79,61)(11,107,80,62)(12,108,73,63)(13,109,74,64)(14,110,75,57)(15,111,76,58)(16,112,77,59)(25,83,42,34)(26,84,43,35)(27,85,44,36)(28,86,45,37)(29,87,46,38)(30,88,47,39)(31,81,48,40)(32,82,41,33)(49,128,114,66)(50,121,115,67)(51,122,116,68)(52,123,117,69)(53,124,118,70)(54,125,119,71)(55,126,120,72)(56,127,113,65)>;
G:=Group( (1,80,34,65)(2,66,35,73)(3,74,36,67)(4,68,37,75)(5,76,38,69)(6,70,39,77)(7,78,40,71)(8,72,33,79)(9,81,125,94)(10,95,126,82)(11,83,127,96)(12,89,128,84)(13,85,121,90)(14,91,122,86)(15,87,123,92)(16,93,124,88)(17,57,32,51)(18,52,25,58)(19,59,26,53)(20,54,27,60)(21,61,28,55)(22,56,29,62)(23,63,30,49)(24,50,31,64)(41,116,103,110)(42,111,104,117)(43,118,97,112)(44,105,98,119)(45,120,99,106)(46,107,100,113)(47,114,101,108)(48,109,102,115), (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), (1,18,96,104)(2,19,89,97)(3,20,90,98)(4,21,91,99)(5,22,92,100)(6,23,93,101)(7,24,94,102)(8,17,95,103)(9,105,78,60)(10,106,79,61)(11,107,80,62)(12,108,73,63)(13,109,74,64)(14,110,75,57)(15,111,76,58)(16,112,77,59)(25,83,42,34)(26,84,43,35)(27,85,44,36)(28,86,45,37)(29,87,46,38)(30,88,47,39)(31,81,48,40)(32,82,41,33)(49,128,114,66)(50,121,115,67)(51,122,116,68)(52,123,117,69)(53,124,118,70)(54,125,119,71)(55,126,120,72)(56,127,113,65) );
G=PermutationGroup([(1,80,34,65),(2,66,35,73),(3,74,36,67),(4,68,37,75),(5,76,38,69),(6,70,39,77),(7,78,40,71),(8,72,33,79),(9,81,125,94),(10,95,126,82),(11,83,127,96),(12,89,128,84),(13,85,121,90),(14,91,122,86),(15,87,123,92),(16,93,124,88),(17,57,32,51),(18,52,25,58),(19,59,26,53),(20,54,27,60),(21,61,28,55),(22,56,29,62),(23,63,30,49),(24,50,31,64),(41,116,103,110),(42,111,104,117),(43,118,97,112),(44,105,98,119),(45,120,99,106),(46,107,100,113),(47,114,101,108),(48,109,102,115)], [(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)], [(1,18,96,104),(2,19,89,97),(3,20,90,98),(4,21,91,99),(5,22,92,100),(6,23,93,101),(7,24,94,102),(8,17,95,103),(9,105,78,60),(10,106,79,61),(11,107,80,62),(12,108,73,63),(13,109,74,64),(14,110,75,57),(15,111,76,58),(16,112,77,59),(25,83,42,34),(26,84,43,35),(27,85,44,36),(28,86,45,37),(29,87,46,38),(30,88,47,39),(31,81,48,40),(32,82,41,33),(49,128,114,66),(50,121,115,67),(51,122,116,68),(52,123,117,69),(53,124,118,70),(54,125,119,71),(55,126,120,72),(56,127,113,65)])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4X | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | Q8 | M4(2) | C4○D4 | C8○D4 |
| kernel | C4⋊C8⋊14C4 | C22.7C42 | C4×C4⋊C4 | C2×C4×C8 | C2×C8⋊C4 | C2×C4⋊C8 | C2.C42 | C8⋊C4 | C4⋊C8 | C2×C4⋊C4 | C2×C8 | C2×C8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 8 | 8 | 4 | 2 | 2 | 8 | 4 | 8 |
Matrix representation of C4⋊C8⋊14C4 ►in GL6(𝔽17)
| 13 | 12 | 0 | 0 | 0 | 0 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 11 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 9 |
| 0 | 0 | 0 | 0 | 2 | 10 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
G:=sub<GL(6,GF(17))| [13,3,0,0,0,0,12,4,0,0,0,0,0,0,3,11,0,0,0,0,7,14,0,0,0,0,0,0,7,2,0,0,0,0,9,10],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[0,9,0,0,0,0,15,0,0,0,0,0,0,0,0,9,0,0,0,0,2,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13] >;
C4⋊C8⋊14C4 in GAP, Magma, Sage, TeX
C_4\rtimes C_8\rtimes_{14}C_4 % in TeX
G:=Group("C4:C8:14C4"); // GroupNames label
G:=SmallGroup(128,503);
// by ID
G=gap.SmallGroup(128,503);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,142,172]);
// Polycyclic
G:=Group<a,b,c|a^4=b^8=c^4=1,b*a*b^-1=a^-1,c*a*c^-1=a^-1*b^4,b*c=c*b>;
// generators/relations