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, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2.C42, C4×C8, C8⋊C4, C8⋊C4, C4⋊C8, C2×C42, C2×C4⋊C4, C22×C8, C22.7C42, C4×C4⋊C4, C2×C4×C8, C2×C8⋊C4, C2×C4⋊C8, C4⋊C8⋊14C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×M4(2), C8○D4, C4×C4⋊C4, C4×M4(2), C8○2M4(2), C8⋊9D4, C8⋊4Q8, C4⋊C8⋊14C4
►Regular action on 128 points
Generators in S128
(1 114 32 69)(2 70 25 115)(3 116 26 71)(4 72 27 117)(5 118 28 65)(6 66 29 119)(7 120 30 67)(8 68 31 113)(9 36 59 21)(10 22 60 37)(11 38 61 23)(12 24 62 39)(13 40 63 17)(14 18 64 33)(15 34 57 19)(16 20 58 35)(41 77 85 110)(42 111 86 78)(43 79 87 112)(44 105 88 80)(45 73 81 106)(46 107 82 74)(47 75 83 108)(48 109 84 76)(49 103 121 90)(50 91 122 104)(51 97 123 92)(52 93 124 98)(53 99 125 94)(54 95 126 100)(55 101 127 96)(56 89 128 102)
(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 35 96 86)(2 36 89 87)(3 37 90 88)(4 38 91 81)(5 39 92 82)(6 40 93 83)(7 33 94 84)(8 34 95 85)(9 124 79 66)(10 125 80 67)(11 126 73 68)(12 127 74 69)(13 128 75 70)(14 121 76 71)(15 122 77 72)(16 123 78 65)(17 98 47 29)(18 99 48 30)(19 100 41 31)(20 101 42 32)(21 102 43 25)(22 103 44 26)(23 104 45 27)(24 97 46 28)(49 109 116 64)(50 110 117 57)(51 111 118 58)(52 112 119 59)(53 105 120 60)(54 106 113 61)(55 107 114 62)(56 108 115 63)
G:=sub<Sym(128)| (1,114,32,69)(2,70,25,115)(3,116,26,71)(4,72,27,117)(5,118,28,65)(6,66,29,119)(7,120,30,67)(8,68,31,113)(9,36,59,21)(10,22,60,37)(11,38,61,23)(12,24,62,39)(13,40,63,17)(14,18,64,33)(15,34,57,19)(16,20,58,35)(41,77,85,110)(42,111,86,78)(43,79,87,112)(44,105,88,80)(45,73,81,106)(46,107,82,74)(47,75,83,108)(48,109,84,76)(49,103,121,90)(50,91,122,104)(51,97,123,92)(52,93,124,98)(53,99,125,94)(54,95,126,100)(55,101,127,96)(56,89,128,102), (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), (1,35,96,86)(2,36,89,87)(3,37,90,88)(4,38,91,81)(5,39,92,82)(6,40,93,83)(7,33,94,84)(8,34,95,85)(9,124,79,66)(10,125,80,67)(11,126,73,68)(12,127,74,69)(13,128,75,70)(14,121,76,71)(15,122,77,72)(16,123,78,65)(17,98,47,29)(18,99,48,30)(19,100,41,31)(20,101,42,32)(21,102,43,25)(22,103,44,26)(23,104,45,27)(24,97,46,28)(49,109,116,64)(50,110,117,57)(51,111,118,58)(52,112,119,59)(53,105,120,60)(54,106,113,61)(55,107,114,62)(56,108,115,63)>;
G:=Group( (1,114,32,69)(2,70,25,115)(3,116,26,71)(4,72,27,117)(5,118,28,65)(6,66,29,119)(7,120,30,67)(8,68,31,113)(9,36,59,21)(10,22,60,37)(11,38,61,23)(12,24,62,39)(13,40,63,17)(14,18,64,33)(15,34,57,19)(16,20,58,35)(41,77,85,110)(42,111,86,78)(43,79,87,112)(44,105,88,80)(45,73,81,106)(46,107,82,74)(47,75,83,108)(48,109,84,76)(49,103,121,90)(50,91,122,104)(51,97,123,92)(52,93,124,98)(53,99,125,94)(54,95,126,100)(55,101,127,96)(56,89,128,102), (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), (1,35,96,86)(2,36,89,87)(3,37,90,88)(4,38,91,81)(5,39,92,82)(6,40,93,83)(7,33,94,84)(8,34,95,85)(9,124,79,66)(10,125,80,67)(11,126,73,68)(12,127,74,69)(13,128,75,70)(14,121,76,71)(15,122,77,72)(16,123,78,65)(17,98,47,29)(18,99,48,30)(19,100,41,31)(20,101,42,32)(21,102,43,25)(22,103,44,26)(23,104,45,27)(24,97,46,28)(49,109,116,64)(50,110,117,57)(51,111,118,58)(52,112,119,59)(53,105,120,60)(54,106,113,61)(55,107,114,62)(56,108,115,63) );
G=PermutationGroup([[(1,114,32,69),(2,70,25,115),(3,116,26,71),(4,72,27,117),(5,118,28,65),(6,66,29,119),(7,120,30,67),(8,68,31,113),(9,36,59,21),(10,22,60,37),(11,38,61,23),(12,24,62,39),(13,40,63,17),(14,18,64,33),(15,34,57,19),(16,20,58,35),(41,77,85,110),(42,111,86,78),(43,79,87,112),(44,105,88,80),(45,73,81,106),(46,107,82,74),(47,75,83,108),(48,109,84,76),(49,103,121,90),(50,91,122,104),(51,97,123,92),(52,93,124,98),(53,99,125,94),(54,95,126,100),(55,101,127,96),(56,89,128,102)], [(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)], [(1,35,96,86),(2,36,89,87),(3,37,90,88),(4,38,91,81),(5,39,92,82),(6,40,93,83),(7,33,94,84),(8,34,95,85),(9,124,79,66),(10,125,80,67),(11,126,73,68),(12,127,74,69),(13,128,75,70),(14,121,76,71),(15,122,77,72),(16,123,78,65),(17,98,47,29),(18,99,48,30),(19,100,41,31),(20,101,42,32),(21,102,43,25),(22,103,44,26),(23,104,45,27),(24,97,46,28),(49,109,116,64),(50,110,117,57),(51,111,118,58),(52,112,119,59),(53,105,120,60),(54,106,113,61),(55,107,114,62),(56,108,115,63)]])
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