p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊15Q8, C43.17C2, C23.763C24, C42⋊5C4.20C2, C4.11(C42⋊2C2), (C22×C4).268C23, C22.184(C22×Q8), (C2×C42).1097C22, (C22×Q8).253C22, C2.C42.458C22, C23.65C23.91C2, C23.63C23.64C2, C23.67C23.66C2, C2.50(C23.37C23), C2.116(C23.36C23), (C2×C4).175(C2×Q8), (C2×C4).532(C4○D4), (C2×C4⋊C4).566C22, C2.27(C2×C42⋊2C2), C22.604(C2×C4○D4), SmallGroup(128,1595)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊15Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 324 in 204 conjugacy classes, 108 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C22×Q8, C43, C42⋊5C4, C23.63C23, C23.65C23, C23.67C23, C42⋊15Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C42⋊2C2, C22×Q8, C2×C4○D4, C2×C42⋊2C2, C23.36C23, C23.37C23, C42⋊15Q8
►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 41 55 16)(2 42 56 13)(3 43 53 14)(4 44 54 15)(5 112 39 85)(6 109 40 86)(7 110 37 87)(8 111 38 88)(9 29 18 58)(10 30 19 59)(11 31 20 60)(12 32 17 57)(21 98 46 65)(22 99 47 66)(23 100 48 67)(24 97 45 68)(25 69 50 64)(26 70 51 61)(27 71 52 62)(28 72 49 63)(33 114 126 81)(34 115 127 82)(35 116 128 83)(36 113 125 84)(73 95 106 118)(74 96 107 119)(75 93 108 120)(76 94 105 117)(77 122 104 89)(78 123 101 90)(79 124 102 91)(80 121 103 92)
(1 21 18 72)(2 22 19 69)(3 23 20 70)(4 24 17 71)(5 76 84 122)(6 73 81 123)(7 74 82 124)(8 75 83 121)(9 63 55 46)(10 64 56 47)(11 61 53 48)(12 62 54 45)(13 66 30 25)(14 67 31 26)(15 68 32 27)(16 65 29 28)(33 101 109 95)(34 102 110 96)(35 103 111 93)(36 104 112 94)(37 107 115 91)(38 108 116 92)(39 105 113 89)(40 106 114 90)(41 98 58 49)(42 99 59 50)(43 100 60 51)(44 97 57 52)(77 85 117 125)(78 86 118 126)(79 87 119 127)(80 88 120 128)
(1 79 18 119)(2 103 19 93)(3 77 20 117)(4 101 17 95)(5 65 84 28)(6 99 81 50)(7 67 82 26)(8 97 83 52)(9 96 55 102)(10 120 56 80)(11 94 53 104)(12 118 54 78)(13 90 30 106)(14 124 31 74)(15 92 32 108)(16 122 29 76)(21 127 72 87)(22 35 69 111)(23 125 70 85)(24 33 71 109)(25 40 66 114)(27 38 68 116)(34 63 110 46)(36 61 112 48)(37 100 115 51)(39 98 113 49)(41 89 58 105)(42 123 59 73)(43 91 60 107)(44 121 57 75)(45 126 62 86)(47 128 64 88)
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), (1,41,55,16)(2,42,56,13)(3,43,53,14)(4,44,54,15)(5,112,39,85)(6,109,40,86)(7,110,37,87)(8,111,38,88)(9,29,18,58)(10,30,19,59)(11,31,20,60)(12,32,17,57)(21,98,46,65)(22,99,47,66)(23,100,48,67)(24,97,45,68)(25,69,50,64)(26,70,51,61)(27,71,52,62)(28,72,49,63)(33,114,126,81)(34,115,127,82)(35,116,128,83)(36,113,125,84)(73,95,106,118)(74,96,107,119)(75,93,108,120)(76,94,105,117)(77,122,104,89)(78,123,101,90)(79,124,102,91)(80,121,103,92), (1,21,18,72)(2,22,19,69)(3,23,20,70)(4,24,17,71)(5,76,84,122)(6,73,81,123)(7,74,82,124)(8,75,83,121)(9,63,55,46)(10,64,56,47)(11,61,53,48)(12,62,54,45)(13,66,30,25)(14,67,31,26)(15,68,32,27)(16,65,29,28)(33,101,109,95)(34,102,110,96)(35,103,111,93)(36,104,112,94)(37,107,115,91)(38,108,116,92)(39,105,113,89)(40,106,114,90)(41,98,58,49)(42,99,59,50)(43,100,60,51)(44,97,57,52)(77,85,117,125)(78,86,118,126)(79,87,119,127)(80,88,120,128), (1,79,18,119)(2,103,19,93)(3,77,20,117)(4,101,17,95)(5,65,84,28)(6,99,81,50)(7,67,82,26)(8,97,83,52)(9,96,55,102)(10,120,56,80)(11,94,53,104)(12,118,54,78)(13,90,30,106)(14,124,31,74)(15,92,32,108)(16,122,29,76)(21,127,72,87)(22,35,69,111)(23,125,70,85)(24,33,71,109)(25,40,66,114)(27,38,68,116)(34,63,110,46)(36,61,112,48)(37,100,115,51)(39,98,113,49)(41,89,58,105)(42,123,59,73)(43,91,60,107)(44,121,57,75)(45,126,62,86)(47,128,64,88)>;
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), (1,41,55,16)(2,42,56,13)(3,43,53,14)(4,44,54,15)(5,112,39,85)(6,109,40,86)(7,110,37,87)(8,111,38,88)(9,29,18,58)(10,30,19,59)(11,31,20,60)(12,32,17,57)(21,98,46,65)(22,99,47,66)(23,100,48,67)(24,97,45,68)(25,69,50,64)(26,70,51,61)(27,71,52,62)(28,72,49,63)(33,114,126,81)(34,115,127,82)(35,116,128,83)(36,113,125,84)(73,95,106,118)(74,96,107,119)(75,93,108,120)(76,94,105,117)(77,122,104,89)(78,123,101,90)(79,124,102,91)(80,121,103,92), (1,21,18,72)(2,22,19,69)(3,23,20,70)(4,24,17,71)(5,76,84,122)(6,73,81,123)(7,74,82,124)(8,75,83,121)(9,63,55,46)(10,64,56,47)(11,61,53,48)(12,62,54,45)(13,66,30,25)(14,67,31,26)(15,68,32,27)(16,65,29,28)(33,101,109,95)(34,102,110,96)(35,103,111,93)(36,104,112,94)(37,107,115,91)(38,108,116,92)(39,105,113,89)(40,106,114,90)(41,98,58,49)(42,99,59,50)(43,100,60,51)(44,97,57,52)(77,85,117,125)(78,86,118,126)(79,87,119,127)(80,88,120,128), (1,79,18,119)(2,103,19,93)(3,77,20,117)(4,101,17,95)(5,65,84,28)(6,99,81,50)(7,67,82,26)(8,97,83,52)(9,96,55,102)(10,120,56,80)(11,94,53,104)(12,118,54,78)(13,90,30,106)(14,124,31,74)(15,92,32,108)(16,122,29,76)(21,127,72,87)(22,35,69,111)(23,125,70,85)(24,33,71,109)(25,40,66,114)(27,38,68,116)(34,63,110,46)(36,61,112,48)(37,100,115,51)(39,98,113,49)(41,89,58,105)(42,123,59,73)(43,91,60,107)(44,121,57,75)(45,126,62,86)(47,128,64,88) );
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)], [(1,41,55,16),(2,42,56,13),(3,43,53,14),(4,44,54,15),(5,112,39,85),(6,109,40,86),(7,110,37,87),(8,111,38,88),(9,29,18,58),(10,30,19,59),(11,31,20,60),(12,32,17,57),(21,98,46,65),(22,99,47,66),(23,100,48,67),(24,97,45,68),(25,69,50,64),(26,70,51,61),(27,71,52,62),(28,72,49,63),(33,114,126,81),(34,115,127,82),(35,116,128,83),(36,113,125,84),(73,95,106,118),(74,96,107,119),(75,93,108,120),(76,94,105,117),(77,122,104,89),(78,123,101,90),(79,124,102,91),(80,121,103,92)], [(1,21,18,72),(2,22,19,69),(3,23,20,70),(4,24,17,71),(5,76,84,122),(6,73,81,123),(7,74,82,124),(8,75,83,121),(9,63,55,46),(10,64,56,47),(11,61,53,48),(12,62,54,45),(13,66,30,25),(14,67,31,26),(15,68,32,27),(16,65,29,28),(33,101,109,95),(34,102,110,96),(35,103,111,93),(36,104,112,94),(37,107,115,91),(38,108,116,92),(39,105,113,89),(40,106,114,90),(41,98,58,49),(42,99,59,50),(43,100,60,51),(44,97,57,52),(77,85,117,125),(78,86,118,126),(79,87,119,127),(80,88,120,128)], [(1,79,18,119),(2,103,19,93),(3,77,20,117),(4,101,17,95),(5,65,84,28),(6,99,81,50),(7,67,82,26),(8,97,83,52),(9,96,55,102),(10,120,56,80),(11,94,53,104),(12,118,54,78),(13,90,30,106),(14,124,31,74),(15,92,32,108),(16,122,29,76),(21,127,72,87),(22,35,69,111),(23,125,70,85),(24,33,71,109),(25,40,66,114),(27,38,68,116),(34,63,110,46),(36,61,112,48),(37,100,115,51),(39,98,113,49),(41,89,58,105),(42,123,59,73),(43,91,60,107),(44,121,57,75),(45,126,62,86),(47,128,64,88)]])
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 |
| type | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | C4○D4 |
| kernel | C42⋊15Q8 | C43 | C42⋊5C4 | C23.63C23 | C23.65C23 | C23.67C23 | C42 | C2×C4 |
| # reps | 1 | 1 | 2 | 6 | 3 | 3 | 4 | 24 |
Matrix representation of C42⋊15Q8 ►in GL6(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 |
| 0 | 0 | 0 | 0 | 4 | 3 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,3,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,4,0,0,0,0,3,3] >;
C42⋊15Q8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{15}Q_8 % in TeX
G:=Group("C4^2:15Q8"); // GroupNames label
G:=SmallGroup(128,1595);
// by ID
G=gap.SmallGroup(128,1595);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,568,758,184,2019,80]);
// 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*b^2,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations