p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.195D4, C23.542C24, C22.2342- 1+4, C42⋊4C4.25C2, C4.15(C4.4D4), (C2×C42).618C22, (C22×C4).152C23, C22.367(C22×D4), (C22×Q8).160C22, C23.67C23.49C2, C23.83C23.24C2, C2.C42.555C22, C2.46(C23.38C23), C2.29(C22.35C24), (C2×C4⋊Q8).36C2, (C2×C4).401(C2×D4), C2.32(C2×C4.4D4), (C2×C4).664(C4○D4), (C2×C4⋊C4).368C22, C22.414(C2×C4○D4), (C2×C42.C2).24C2, SmallGroup(128,1374)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.195D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=a2b2c-1 >
Subgroups: 356 in 208 conjugacy classes, 100 normal (10 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, C42.C2, C4⋊Q8, C22×Q8, C42⋊4C4, C23.67C23, C23.83C23, C2×C42.C2, C2×C4⋊Q8, C42.195D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C22×D4, C2×C4○D4, 2- 1+4, C2×C4.4D4, C23.38C23, C22.35C24, C42.195D4
►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 26 35 119)(2 27 36 120)(3 28 33 117)(4 25 34 118)(5 88 94 57)(6 85 95 58)(7 86 96 59)(8 87 93 60)(9 53 102 84)(10 54 103 81)(11 55 104 82)(12 56 101 83)(13 50 106 78)(14 51 107 79)(15 52 108 80)(16 49 105 77)(17 92 110 61)(18 89 111 62)(19 90 112 63)(20 91 109 64)(21 65 114 38)(22 66 115 39)(23 67 116 40)(24 68 113 37)(29 46 122 73)(30 47 123 74)(31 48 124 75)(32 45 121 76)(41 100 72 125)(42 97 69 126)(43 98 70 127)(44 99 71 128)
(1 53 22 94)(2 81 23 6)(3 55 24 96)(4 83 21 8)(5 35 84 115)(7 33 82 113)(9 39 88 119)(10 67 85 27)(11 37 86 117)(12 65 87 25)(13 43 92 123)(14 71 89 31)(15 41 90 121)(16 69 91 29)(17 47 78 127)(18 75 79 99)(19 45 80 125)(20 73 77 97)(26 102 66 57)(28 104 68 59)(30 106 70 61)(32 108 72 63)(34 56 114 93)(36 54 116 95)(38 60 118 101)(40 58 120 103)(42 64 122 105)(44 62 124 107)(46 49 126 109)(48 51 128 111)(50 98 110 74)(52 100 112 76)
(1 108 3 106)(2 14 4 16)(5 70 7 72)(6 42 8 44)(9 74 11 76)(10 46 12 48)(13 35 15 33)(17 39 19 37)(18 65 20 67)(21 91 23 89)(22 63 24 61)(25 77 27 79)(26 52 28 50)(29 56 31 54)(30 82 32 84)(34 105 36 107)(38 109 40 111)(41 94 43 96)(45 102 47 104)(49 120 51 118)(53 123 55 121)(57 127 59 125)(58 97 60 99)(62 114 64 116)(66 112 68 110)(69 93 71 95)(73 101 75 103)(78 119 80 117)(81 122 83 124)(85 126 87 128)(86 100 88 98)(90 113 92 115)
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), (1,26,35,119)(2,27,36,120)(3,28,33,117)(4,25,34,118)(5,88,94,57)(6,85,95,58)(7,86,96,59)(8,87,93,60)(9,53,102,84)(10,54,103,81)(11,55,104,82)(12,56,101,83)(13,50,106,78)(14,51,107,79)(15,52,108,80)(16,49,105,77)(17,92,110,61)(18,89,111,62)(19,90,112,63)(20,91,109,64)(21,65,114,38)(22,66,115,39)(23,67,116,40)(24,68,113,37)(29,46,122,73)(30,47,123,74)(31,48,124,75)(32,45,121,76)(41,100,72,125)(42,97,69,126)(43,98,70,127)(44,99,71,128), (1,53,22,94)(2,81,23,6)(3,55,24,96)(4,83,21,8)(5,35,84,115)(7,33,82,113)(9,39,88,119)(10,67,85,27)(11,37,86,117)(12,65,87,25)(13,43,92,123)(14,71,89,31)(15,41,90,121)(16,69,91,29)(17,47,78,127)(18,75,79,99)(19,45,80,125)(20,73,77,97)(26,102,66,57)(28,104,68,59)(30,106,70,61)(32,108,72,63)(34,56,114,93)(36,54,116,95)(38,60,118,101)(40,58,120,103)(42,64,122,105)(44,62,124,107)(46,49,126,109)(48,51,128,111)(50,98,110,74)(52,100,112,76), (1,108,3,106)(2,14,4,16)(5,70,7,72)(6,42,8,44)(9,74,11,76)(10,46,12,48)(13,35,15,33)(17,39,19,37)(18,65,20,67)(21,91,23,89)(22,63,24,61)(25,77,27,79)(26,52,28,50)(29,56,31,54)(30,82,32,84)(34,105,36,107)(38,109,40,111)(41,94,43,96)(45,102,47,104)(49,120,51,118)(53,123,55,121)(57,127,59,125)(58,97,60,99)(62,114,64,116)(66,112,68,110)(69,93,71,95)(73,101,75,103)(78,119,80,117)(81,122,83,124)(85,126,87,128)(86,100,88,98)(90,113,92,115)>;
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), (1,26,35,119)(2,27,36,120)(3,28,33,117)(4,25,34,118)(5,88,94,57)(6,85,95,58)(7,86,96,59)(8,87,93,60)(9,53,102,84)(10,54,103,81)(11,55,104,82)(12,56,101,83)(13,50,106,78)(14,51,107,79)(15,52,108,80)(16,49,105,77)(17,92,110,61)(18,89,111,62)(19,90,112,63)(20,91,109,64)(21,65,114,38)(22,66,115,39)(23,67,116,40)(24,68,113,37)(29,46,122,73)(30,47,123,74)(31,48,124,75)(32,45,121,76)(41,100,72,125)(42,97,69,126)(43,98,70,127)(44,99,71,128), (1,53,22,94)(2,81,23,6)(3,55,24,96)(4,83,21,8)(5,35,84,115)(7,33,82,113)(9,39,88,119)(10,67,85,27)(11,37,86,117)(12,65,87,25)(13,43,92,123)(14,71,89,31)(15,41,90,121)(16,69,91,29)(17,47,78,127)(18,75,79,99)(19,45,80,125)(20,73,77,97)(26,102,66,57)(28,104,68,59)(30,106,70,61)(32,108,72,63)(34,56,114,93)(36,54,116,95)(38,60,118,101)(40,58,120,103)(42,64,122,105)(44,62,124,107)(46,49,126,109)(48,51,128,111)(50,98,110,74)(52,100,112,76), (1,108,3,106)(2,14,4,16)(5,70,7,72)(6,42,8,44)(9,74,11,76)(10,46,12,48)(13,35,15,33)(17,39,19,37)(18,65,20,67)(21,91,23,89)(22,63,24,61)(25,77,27,79)(26,52,28,50)(29,56,31,54)(30,82,32,84)(34,105,36,107)(38,109,40,111)(41,94,43,96)(45,102,47,104)(49,120,51,118)(53,123,55,121)(57,127,59,125)(58,97,60,99)(62,114,64,116)(66,112,68,110)(69,93,71,95)(73,101,75,103)(78,119,80,117)(81,122,83,124)(85,126,87,128)(86,100,88,98)(90,113,92,115) );
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)], [(1,26,35,119),(2,27,36,120),(3,28,33,117),(4,25,34,118),(5,88,94,57),(6,85,95,58),(7,86,96,59),(8,87,93,60),(9,53,102,84),(10,54,103,81),(11,55,104,82),(12,56,101,83),(13,50,106,78),(14,51,107,79),(15,52,108,80),(16,49,105,77),(17,92,110,61),(18,89,111,62),(19,90,112,63),(20,91,109,64),(21,65,114,38),(22,66,115,39),(23,67,116,40),(24,68,113,37),(29,46,122,73),(30,47,123,74),(31,48,124,75),(32,45,121,76),(41,100,72,125),(42,97,69,126),(43,98,70,127),(44,99,71,128)], [(1,53,22,94),(2,81,23,6),(3,55,24,96),(4,83,21,8),(5,35,84,115),(7,33,82,113),(9,39,88,119),(10,67,85,27),(11,37,86,117),(12,65,87,25),(13,43,92,123),(14,71,89,31),(15,41,90,121),(16,69,91,29),(17,47,78,127),(18,75,79,99),(19,45,80,125),(20,73,77,97),(26,102,66,57),(28,104,68,59),(30,106,70,61),(32,108,72,63),(34,56,114,93),(36,54,116,95),(38,60,118,101),(40,58,120,103),(42,64,122,105),(44,62,124,107),(46,49,126,109),(48,51,128,111),(50,98,110,74),(52,100,112,76)], [(1,108,3,106),(2,14,4,16),(5,70,7,72),(6,42,8,44),(9,74,11,76),(10,46,12,48),(13,35,15,33),(17,39,19,37),(18,65,20,67),(21,91,23,89),(22,63,24,61),(25,77,27,79),(26,52,28,50),(29,56,31,54),(30,82,32,84),(34,105,36,107),(38,109,40,111),(41,94,43,96),(45,102,47,104),(49,120,51,118),(53,123,55,121),(57,127,59,125),(58,97,60,99),(62,114,64,116),(66,112,68,110),(69,93,71,95),(73,101,75,103),(78,119,80,117),(81,122,83,124),(85,126,87,128),(86,100,88,98),(90,113,92,115)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | ··· | 4X |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2- 1+4 |
| kernel | C42.195D4 | C42⋊4C4 | C23.67C23 | C23.83C23 | C2×C42.C2 | C2×C4⋊Q8 | C42 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 8 | 1 | 1 | 4 | 8 | 4 |
Matrix representation of C42.195D4 ►in GL8(𝔽5)
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 2 | 0 | 0 |
G:=sub<GL(8,GF(5))| [3,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,3,4],[1,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0],[1,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,2,0,0,0,0,2,2,0,0,0,0,0,0,0,3,0,0] >;
C42.195D4 in GAP, Magma, Sage, TeX
C_4^2._{195}D_4 % in TeX
G:=Group("C4^2.195D4"); // GroupNames label
G:=SmallGroup(128,1374);
// by ID
G=gap.SmallGroup(128,1374);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,100,185,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^-1>;
// generators/relations