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, C23.67C23.66C2, C23.63C23.64C2, C2.C42.458C22, C23.65C23.91C2, 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
Subgroups: 324 in 204 conjugacy classes, 108 normal (9 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×20], C22, C22 [×6], C2×C4 [×18], C2×C4 [×36], Q8 [×4], C23, C42 [×4], C42 [×12], C4⋊C4 [×18], C22×C4, C22×C4 [×14], C2×Q8 [×6], C2.C42 [×18], C2×C42, C2×C42 [×6], C2×C4⋊C4 [×9], C22×Q8, C43, C42⋊5C4 [×2], C23.63C23 [×6], C23.65C23 [×3], C23.67C23 [×3], C42⋊15Q8
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×12], C24, C42⋊2C2 [×4], C22×Q8, C2×C4○D4 [×6], C2×C42⋊2C2, C23.36C23 [×3], C23.37C23 [×3], C42⋊15Q8
Generators and relations
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 >
►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 11 55 16)(2 12 56 13)(3 9 53 14)(4 10 54 15)(5 85 127 82)(6 86 128 83)(7 87 125 84)(8 88 126 81)(17 61 22 100)(18 62 23 97)(19 63 24 98)(20 64 21 99)(25 48 30 42)(26 45 31 43)(27 46 32 44)(28 47 29 41)(33 111 38 114)(34 112 39 115)(35 109 40 116)(36 110 37 113)(49 72 60 65)(50 69 57 66)(51 70 58 67)(52 71 59 68)(73 121 80 118)(74 122 77 119)(75 123 78 120)(76 124 79 117)(89 107 96 104)(90 108 93 101)(91 105 94 102)(92 106 95 103)
(1 49 41 98)(2 50 42 99)(3 51 43 100)(4 52 44 97)(5 102 110 122)(6 103 111 123)(7 104 112 124)(8 101 109 121)(9 70 26 17)(10 71 27 18)(11 72 28 19)(12 69 25 20)(13 66 30 21)(14 67 31 22)(15 68 32 23)(16 65 29 24)(33 75 83 95)(34 76 84 96)(35 73 81 93)(36 74 82 94)(37 77 85 91)(38 78 86 92)(39 79 87 89)(40 80 88 90)(45 61 53 58)(46 62 54 59)(47 63 55 60)(48 64 56 57)(105 113 119 127)(106 114 120 128)(107 115 117 125)(108 116 118 126)
(1 105 41 119)(2 103 42 123)(3 107 43 117)(4 101 44 121)(5 63 110 60)(6 99 111 50)(7 61 112 58)(8 97 109 52)(9 91 26 77)(10 95 27 75)(11 89 28 79)(12 93 25 73)(13 90 30 80)(14 94 31 74)(15 92 32 78)(16 96 29 76)(17 37 70 85)(18 33 71 83)(19 39 72 87)(20 35 69 81)(21 40 66 88)(22 36 67 82)(23 38 68 86)(24 34 65 84)(45 124 53 104)(46 118 54 108)(47 122 55 102)(48 120 56 106)(49 127 98 113)(51 125 100 115)(57 128 64 114)(59 126 62 116)
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), (1,11,55,16)(2,12,56,13)(3,9,53,14)(4,10,54,15)(5,85,127,82)(6,86,128,83)(7,87,125,84)(8,88,126,81)(17,61,22,100)(18,62,23,97)(19,63,24,98)(20,64,21,99)(25,48,30,42)(26,45,31,43)(27,46,32,44)(28,47,29,41)(33,111,38,114)(34,112,39,115)(35,109,40,116)(36,110,37,113)(49,72,60,65)(50,69,57,66)(51,70,58,67)(52,71,59,68)(73,121,80,118)(74,122,77,119)(75,123,78,120)(76,124,79,117)(89,107,96,104)(90,108,93,101)(91,105,94,102)(92,106,95,103), (1,49,41,98)(2,50,42,99)(3,51,43,100)(4,52,44,97)(5,102,110,122)(6,103,111,123)(7,104,112,124)(8,101,109,121)(9,70,26,17)(10,71,27,18)(11,72,28,19)(12,69,25,20)(13,66,30,21)(14,67,31,22)(15,68,32,23)(16,65,29,24)(33,75,83,95)(34,76,84,96)(35,73,81,93)(36,74,82,94)(37,77,85,91)(38,78,86,92)(39,79,87,89)(40,80,88,90)(45,61,53,58)(46,62,54,59)(47,63,55,60)(48,64,56,57)(105,113,119,127)(106,114,120,128)(107,115,117,125)(108,116,118,126), (1,105,41,119)(2,103,42,123)(3,107,43,117)(4,101,44,121)(5,63,110,60)(6,99,111,50)(7,61,112,58)(8,97,109,52)(9,91,26,77)(10,95,27,75)(11,89,28,79)(12,93,25,73)(13,90,30,80)(14,94,31,74)(15,92,32,78)(16,96,29,76)(17,37,70,85)(18,33,71,83)(19,39,72,87)(20,35,69,81)(21,40,66,88)(22,36,67,82)(23,38,68,86)(24,34,65,84)(45,124,53,104)(46,118,54,108)(47,122,55,102)(48,120,56,106)(49,127,98,113)(51,125,100,115)(57,128,64,114)(59,126,62,116)>;
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), (1,11,55,16)(2,12,56,13)(3,9,53,14)(4,10,54,15)(5,85,127,82)(6,86,128,83)(7,87,125,84)(8,88,126,81)(17,61,22,100)(18,62,23,97)(19,63,24,98)(20,64,21,99)(25,48,30,42)(26,45,31,43)(27,46,32,44)(28,47,29,41)(33,111,38,114)(34,112,39,115)(35,109,40,116)(36,110,37,113)(49,72,60,65)(50,69,57,66)(51,70,58,67)(52,71,59,68)(73,121,80,118)(74,122,77,119)(75,123,78,120)(76,124,79,117)(89,107,96,104)(90,108,93,101)(91,105,94,102)(92,106,95,103), (1,49,41,98)(2,50,42,99)(3,51,43,100)(4,52,44,97)(5,102,110,122)(6,103,111,123)(7,104,112,124)(8,101,109,121)(9,70,26,17)(10,71,27,18)(11,72,28,19)(12,69,25,20)(13,66,30,21)(14,67,31,22)(15,68,32,23)(16,65,29,24)(33,75,83,95)(34,76,84,96)(35,73,81,93)(36,74,82,94)(37,77,85,91)(38,78,86,92)(39,79,87,89)(40,80,88,90)(45,61,53,58)(46,62,54,59)(47,63,55,60)(48,64,56,57)(105,113,119,127)(106,114,120,128)(107,115,117,125)(108,116,118,126), (1,105,41,119)(2,103,42,123)(3,107,43,117)(4,101,44,121)(5,63,110,60)(6,99,111,50)(7,61,112,58)(8,97,109,52)(9,91,26,77)(10,95,27,75)(11,89,28,79)(12,93,25,73)(13,90,30,80)(14,94,31,74)(15,92,32,78)(16,96,29,76)(17,37,70,85)(18,33,71,83)(19,39,72,87)(20,35,69,81)(21,40,66,88)(22,36,67,82)(23,38,68,86)(24,34,65,84)(45,124,53,104)(46,118,54,108)(47,122,55,102)(48,120,56,106)(49,127,98,113)(51,125,100,115)(57,128,64,114)(59,126,62,116) );
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)], [(1,11,55,16),(2,12,56,13),(3,9,53,14),(4,10,54,15),(5,85,127,82),(6,86,128,83),(7,87,125,84),(8,88,126,81),(17,61,22,100),(18,62,23,97),(19,63,24,98),(20,64,21,99),(25,48,30,42),(26,45,31,43),(27,46,32,44),(28,47,29,41),(33,111,38,114),(34,112,39,115),(35,109,40,116),(36,110,37,113),(49,72,60,65),(50,69,57,66),(51,70,58,67),(52,71,59,68),(73,121,80,118),(74,122,77,119),(75,123,78,120),(76,124,79,117),(89,107,96,104),(90,108,93,101),(91,105,94,102),(92,106,95,103)], [(1,49,41,98),(2,50,42,99),(3,51,43,100),(4,52,44,97),(5,102,110,122),(6,103,111,123),(7,104,112,124),(8,101,109,121),(9,70,26,17),(10,71,27,18),(11,72,28,19),(12,69,25,20),(13,66,30,21),(14,67,31,22),(15,68,32,23),(16,65,29,24),(33,75,83,95),(34,76,84,96),(35,73,81,93),(36,74,82,94),(37,77,85,91),(38,78,86,92),(39,79,87,89),(40,80,88,90),(45,61,53,58),(46,62,54,59),(47,63,55,60),(48,64,56,57),(105,113,119,127),(106,114,120,128),(107,115,117,125),(108,116,118,126)], [(1,105,41,119),(2,103,42,123),(3,107,43,117),(4,101,44,121),(5,63,110,60),(6,99,111,50),(7,61,112,58),(8,97,109,52),(9,91,26,77),(10,95,27,75),(11,89,28,79),(12,93,25,73),(13,90,30,80),(14,94,31,74),(15,92,32,78),(16,96,29,76),(17,37,70,85),(18,33,71,83),(19,39,72,87),(20,35,69,81),(21,40,66,88),(22,36,67,82),(23,38,68,86),(24,34,65,84),(45,124,53,104),(46,118,54,108),(47,122,55,102),(48,120,56,106),(49,127,98,113),(51,125,100,115),(57,128,64,114),(59,126,62,116)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀