p-group, metabelian, nilpotent (class 2), monomial
Aliases: C43.15C2, C42.64Q8, C42.345D4, C23.759C24, C4.10(C4⋊Q8), C4⋊1(C42.C2), C42⋊8C4.53C2, C42⋊9C4.41C2, (C22×C4).265C23, C22.469(C22×D4), C22.182(C22×Q8), (C2×C42).1094C22, C23.65C23.90C2, C2.C42.454C22, C2.59(C22.26C24), C2.48(C23.37C23), C2.22(C2×C4⋊Q8), (C2×C4).688(C2×D4), (C2×C4).233(C2×Q8), C2.22(C2×C42.C2), (C2×C4).675(C4○D4), (C2×C4⋊C4).562C22, C22.600(C2×C4○D4), (C2×C42.C2).30C2, SmallGroup(128,1591)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C43.15C2
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=ac2, bc=cb, dbd-1=b-1, dcd-1=a2c >
Subgroups: 356 in 236 conjugacy classes, 132 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C42.C2, C43, C42⋊8C4, C42⋊9C4, C23.65C23, C2×C42.C2, C43.15C2
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C42.C2, C4⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C2×C42.C2, C2×C4⋊Q8, C22.26C24, C23.37C23, C43.15C2
►Regular action on 128 points
Generators in S128
(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 59 51 45)(2 60 52 46)(3 57 49 47)(4 58 50 48)(5 44 62 54)(6 41 63 55)(7 42 64 56)(8 43 61 53)(9 31 23 17)(10 32 24 18)(11 29 21 19)(12 30 22 20)(13 71 27 39)(14 72 28 40)(15 69 25 37)(16 70 26 38)(33 105 98 95)(34 106 99 96)(35 107 100 93)(36 108 97 94)(65 117 125 75)(66 118 126 76)(67 119 127 73)(68 120 128 74)(77 85 115 123)(78 86 116 124)(79 87 113 121)(80 88 114 122)(81 89 103 111)(82 90 104 112)(83 91 101 109)(84 92 102 110)
(1 13 9 41)(2 14 10 42)(3 15 11 43)(4 16 12 44)(5 48 38 20)(6 45 39 17)(7 46 40 18)(8 47 37 19)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 69)(30 62 58 70)(31 63 59 71)(32 64 60 72)(33 89 127 121)(34 90 128 122)(35 91 125 123)(36 92 126 124)(65 85 100 109)(66 86 97 110)(67 87 98 111)(68 88 99 112)(73 79 105 103)(74 80 106 104)(75 77 107 101)(76 78 108 102)(81 119 113 95)(82 120 114 96)(83 117 115 93)(84 118 116 94)
(1 77 9 101)(2 102 10 78)(3 79 11 103)(4 104 12 80)(5 97 38 66)(6 67 39 98)(7 99 40 68)(8 65 37 100)(13 105 41 73)(14 74 42 106)(15 107 43 75)(16 76 44 108)(17 109 45 85)(18 86 46 110)(19 111 47 87)(20 88 48 112)(21 81 49 113)(22 114 50 82)(23 83 51 115)(24 116 52 84)(25 93 53 117)(26 118 54 94)(27 95 55 119)(28 120 56 96)(29 89 57 121)(30 122 58 90)(31 91 59 123)(32 124 60 92)(33 63 127 71)(34 72 128 64)(35 61 125 69)(36 70 126 62)
G:=sub<Sym(128)| (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), (1,59,51,45)(2,60,52,46)(3,57,49,47)(4,58,50,48)(5,44,62,54)(6,41,63,55)(7,42,64,56)(8,43,61,53)(9,31,23,17)(10,32,24,18)(11,29,21,19)(12,30,22,20)(13,71,27,39)(14,72,28,40)(15,69,25,37)(16,70,26,38)(33,105,98,95)(34,106,99,96)(35,107,100,93)(36,108,97,94)(65,117,125,75)(66,118,126,76)(67,119,127,73)(68,120,128,74)(77,85,115,123)(78,86,116,124)(79,87,113,121)(80,88,114,122)(81,89,103,111)(82,90,104,112)(83,91,101,109)(84,92,102,110), (1,13,9,41)(2,14,10,42)(3,15,11,43)(4,16,12,44)(5,48,38,20)(6,45,39,17)(7,46,40,18)(8,47,37,19)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,69)(30,62,58,70)(31,63,59,71)(32,64,60,72)(33,89,127,121)(34,90,128,122)(35,91,125,123)(36,92,126,124)(65,85,100,109)(66,86,97,110)(67,87,98,111)(68,88,99,112)(73,79,105,103)(74,80,106,104)(75,77,107,101)(76,78,108,102)(81,119,113,95)(82,120,114,96)(83,117,115,93)(84,118,116,94), (1,77,9,101)(2,102,10,78)(3,79,11,103)(4,104,12,80)(5,97,38,66)(6,67,39,98)(7,99,40,68)(8,65,37,100)(13,105,41,73)(14,74,42,106)(15,107,43,75)(16,76,44,108)(17,109,45,85)(18,86,46,110)(19,111,47,87)(20,88,48,112)(21,81,49,113)(22,114,50,82)(23,83,51,115)(24,116,52,84)(25,93,53,117)(26,118,54,94)(27,95,55,119)(28,120,56,96)(29,89,57,121)(30,122,58,90)(31,91,59,123)(32,124,60,92)(33,63,127,71)(34,72,128,64)(35,61,125,69)(36,70,126,62)>;
G:=Group( (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), (1,59,51,45)(2,60,52,46)(3,57,49,47)(4,58,50,48)(5,44,62,54)(6,41,63,55)(7,42,64,56)(8,43,61,53)(9,31,23,17)(10,32,24,18)(11,29,21,19)(12,30,22,20)(13,71,27,39)(14,72,28,40)(15,69,25,37)(16,70,26,38)(33,105,98,95)(34,106,99,96)(35,107,100,93)(36,108,97,94)(65,117,125,75)(66,118,126,76)(67,119,127,73)(68,120,128,74)(77,85,115,123)(78,86,116,124)(79,87,113,121)(80,88,114,122)(81,89,103,111)(82,90,104,112)(83,91,101,109)(84,92,102,110), (1,13,9,41)(2,14,10,42)(3,15,11,43)(4,16,12,44)(5,48,38,20)(6,45,39,17)(7,46,40,18)(8,47,37,19)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,69)(30,62,58,70)(31,63,59,71)(32,64,60,72)(33,89,127,121)(34,90,128,122)(35,91,125,123)(36,92,126,124)(65,85,100,109)(66,86,97,110)(67,87,98,111)(68,88,99,112)(73,79,105,103)(74,80,106,104)(75,77,107,101)(76,78,108,102)(81,119,113,95)(82,120,114,96)(83,117,115,93)(84,118,116,94), (1,77,9,101)(2,102,10,78)(3,79,11,103)(4,104,12,80)(5,97,38,66)(6,67,39,98)(7,99,40,68)(8,65,37,100)(13,105,41,73)(14,74,42,106)(15,107,43,75)(16,76,44,108)(17,109,45,85)(18,86,46,110)(19,111,47,87)(20,88,48,112)(21,81,49,113)(22,114,50,82)(23,83,51,115)(24,116,52,84)(25,93,53,117)(26,118,54,94)(27,95,55,119)(28,120,56,96)(29,89,57,121)(30,122,58,90)(31,91,59,123)(32,124,60,92)(33,63,127,71)(34,72,128,64)(35,61,125,69)(36,70,126,62) );
G=PermutationGroup([[(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)], [(1,59,51,45),(2,60,52,46),(3,57,49,47),(4,58,50,48),(5,44,62,54),(6,41,63,55),(7,42,64,56),(8,43,61,53),(9,31,23,17),(10,32,24,18),(11,29,21,19),(12,30,22,20),(13,71,27,39),(14,72,28,40),(15,69,25,37),(16,70,26,38),(33,105,98,95),(34,106,99,96),(35,107,100,93),(36,108,97,94),(65,117,125,75),(66,118,126,76),(67,119,127,73),(68,120,128,74),(77,85,115,123),(78,86,116,124),(79,87,113,121),(80,88,114,122),(81,89,103,111),(82,90,104,112),(83,91,101,109),(84,92,102,110)], [(1,13,9,41),(2,14,10,42),(3,15,11,43),(4,16,12,44),(5,48,38,20),(6,45,39,17),(7,46,40,18),(8,47,37,19),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,61,57,69),(30,62,58,70),(31,63,59,71),(32,64,60,72),(33,89,127,121),(34,90,128,122),(35,91,125,123),(36,92,126,124),(65,85,100,109),(66,86,97,110),(67,87,98,111),(68,88,99,112),(73,79,105,103),(74,80,106,104),(75,77,107,101),(76,78,108,102),(81,119,113,95),(82,120,114,96),(83,117,115,93),(84,118,116,94)], [(1,77,9,101),(2,102,10,78),(3,79,11,103),(4,104,12,80),(5,97,38,66),(6,67,39,98),(7,99,40,68),(8,65,37,100),(13,105,41,73),(14,74,42,106),(15,107,43,75),(16,76,44,108),(17,109,45,85),(18,86,46,110),(19,111,47,87),(20,88,48,112),(21,81,49,113),(22,114,50,82),(23,83,51,115),(24,116,52,84),(25,93,53,117),(26,118,54,94),(27,95,55,119),(28,120,56,96),(29,89,57,121),(30,122,58,90),(31,91,59,123),(32,124,60,92),(33,63,127,71),(34,72,128,64),(35,61,125,69),(36,70,126,62)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4AB | 4AC | ··· | 4AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 |
| kernel | C43.15C2 | C43 | C42⋊8C4 | C42⋊9C4 | C23.65C23 | C2×C42.C2 | C42 | C42 | C2×C4 |
| # reps | 1 | 1 | 2 | 2 | 8 | 2 | 4 | 8 | 16 |
Matrix representation of C43.15C2 ►in GL6(𝔽5)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
G:=sub<GL(6,GF(5))| [3,2,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[3,2,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[2,0,0,0,0,0,4,3,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0] >;
C43.15C2 in GAP, Magma, Sage, TeX
C_4^3._{15}C_2 % in TeX
G:=Group("C4^3.15C2"); // GroupNames label
G:=SmallGroup(128,1591);
// by ID
G=gap.SmallGroup(128,1591);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,456,758,184,2019,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*c^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*c>;
// generators/relations