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, C2.C42.555C22, C23.83C23.24C2, C23.67C23.49C2, C2.29(C22.35C24), C2.46(C23.38C23), (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
Subgroups: 356 in 208 conjugacy classes, 100 normal (10 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×16], C22, C22 [×6], C2×C4 [×10], C2×C4 [×40], Q8 [×8], C23, C42 [×4], C42 [×4], C4⋊C4 [×20], C22×C4, C22×C4 [×14], C2×Q8 [×12], C2.C42 [×20], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×10], C42.C2 [×4], C4⋊Q8 [×4], C22×Q8 [×2], C42⋊4C4, C23.67C23 [×4], C23.83C23 [×8], C2×C42.C2, C2×C4⋊Q8, C42.195D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], 2- (1+4) [×4], C2×C4.4D4, C23.38C23 [×2], C22.35C24 [×4], C42.195D4
Generators and relations
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 >
(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 53 94 88)(6 54 95 85)(7 55 96 86)(8 56 93 87)(9 84 102 49)(10 81 103 50)(11 82 104 51)(12 83 101 52)(13 77 106 61)(14 78 107 62)(15 79 108 63)(16 80 105 64)(17 57 110 92)(18 58 111 89)(19 59 112 90)(20 60 109 91)(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 84 22 94)(2 50 23 6)(3 82 24 96)(4 52 21 8)(5 35 49 115)(7 33 51 113)(9 39 53 119)(10 67 54 27)(11 37 55 117)(12 65 56 25)(13 43 57 123)(14 71 58 31)(15 41 59 121)(16 69 60 29)(17 47 61 127)(18 75 62 99)(19 45 63 125)(20 73 64 97)(26 102 66 88)(28 104 68 86)(30 106 70 92)(32 108 72 90)(34 83 114 93)(36 81 116 95)(38 87 118 101)(40 85 120 103)(42 91 122 105)(44 89 124 107)(46 80 126 109)(48 78 128 111)(74 77 98 110)(76 79 100 112)
(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 60 23 58)(22 90 24 92)(25 64 27 62)(26 79 28 77)(29 83 31 81)(30 51 32 49)(34 105 36 107)(38 109 40 111)(41 94 43 96)(45 102 47 104)(50 122 52 124)(53 98 55 100)(54 126 56 128)(57 115 59 113)(61 119 63 117)(66 112 68 110)(69 93 71 95)(73 101 75 103)(78 118 80 120)(82 121 84 123)(85 97 87 99)(86 125 88 127)(89 114 91 116)
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), (1,26,35,119)(2,27,36,120)(3,28,33,117)(4,25,34,118)(5,53,94,88)(6,54,95,85)(7,55,96,86)(8,56,93,87)(9,84,102,49)(10,81,103,50)(11,82,104,51)(12,83,101,52)(13,77,106,61)(14,78,107,62)(15,79,108,63)(16,80,105,64)(17,57,110,92)(18,58,111,89)(19,59,112,90)(20,60,109,91)(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), (1,84,22,94)(2,50,23,6)(3,82,24,96)(4,52,21,8)(5,35,49,115)(7,33,51,113)(9,39,53,119)(10,67,54,27)(11,37,55,117)(12,65,56,25)(13,43,57,123)(14,71,58,31)(15,41,59,121)(16,69,60,29)(17,47,61,127)(18,75,62,99)(19,45,63,125)(20,73,64,97)(26,102,66,88)(28,104,68,86)(30,106,70,92)(32,108,72,90)(34,83,114,93)(36,81,116,95)(38,87,118,101)(40,85,120,103)(42,91,122,105)(44,89,124,107)(46,80,126,109)(48,78,128,111)(74,77,98,110)(76,79,100,112), (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)(21,60,23,58)(22,90,24,92)(25,64,27,62)(26,79,28,77)(29,83,31,81)(30,51,32,49)(34,105,36,107)(38,109,40,111)(41,94,43,96)(45,102,47,104)(50,122,52,124)(53,98,55,100)(54,126,56,128)(57,115,59,113)(61,119,63,117)(66,112,68,110)(69,93,71,95)(73,101,75,103)(78,118,80,120)(82,121,84,123)(85,97,87,99)(86,125,88,127)(89,114,91,116)>;
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), (1,26,35,119)(2,27,36,120)(3,28,33,117)(4,25,34,118)(5,53,94,88)(6,54,95,85)(7,55,96,86)(8,56,93,87)(9,84,102,49)(10,81,103,50)(11,82,104,51)(12,83,101,52)(13,77,106,61)(14,78,107,62)(15,79,108,63)(16,80,105,64)(17,57,110,92)(18,58,111,89)(19,59,112,90)(20,60,109,91)(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), (1,84,22,94)(2,50,23,6)(3,82,24,96)(4,52,21,8)(5,35,49,115)(7,33,51,113)(9,39,53,119)(10,67,54,27)(11,37,55,117)(12,65,56,25)(13,43,57,123)(14,71,58,31)(15,41,59,121)(16,69,60,29)(17,47,61,127)(18,75,62,99)(19,45,63,125)(20,73,64,97)(26,102,66,88)(28,104,68,86)(30,106,70,92)(32,108,72,90)(34,83,114,93)(36,81,116,95)(38,87,118,101)(40,85,120,103)(42,91,122,105)(44,89,124,107)(46,80,126,109)(48,78,128,111)(74,77,98,110)(76,79,100,112), (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)(21,60,23,58)(22,90,24,92)(25,64,27,62)(26,79,28,77)(29,83,31,81)(30,51,32,49)(34,105,36,107)(38,109,40,111)(41,94,43,96)(45,102,47,104)(50,122,52,124)(53,98,55,100)(54,126,56,128)(57,115,59,113)(61,119,63,117)(66,112,68,110)(69,93,71,95)(73,101,75,103)(78,118,80,120)(82,121,84,123)(85,97,87,99)(86,125,88,127)(89,114,91,116) );
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)], [(1,26,35,119),(2,27,36,120),(3,28,33,117),(4,25,34,118),(5,53,94,88),(6,54,95,85),(7,55,96,86),(8,56,93,87),(9,84,102,49),(10,81,103,50),(11,82,104,51),(12,83,101,52),(13,77,106,61),(14,78,107,62),(15,79,108,63),(16,80,105,64),(17,57,110,92),(18,58,111,89),(19,59,112,90),(20,60,109,91),(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)], [(1,84,22,94),(2,50,23,6),(3,82,24,96),(4,52,21,8),(5,35,49,115),(7,33,51,113),(9,39,53,119),(10,67,54,27),(11,37,55,117),(12,65,56,25),(13,43,57,123),(14,71,58,31),(15,41,59,121),(16,69,60,29),(17,47,61,127),(18,75,62,99),(19,45,63,125),(20,73,64,97),(26,102,66,88),(28,104,68,86),(30,106,70,92),(32,108,72,90),(34,83,114,93),(36,81,116,95),(38,87,118,101),(40,85,120,103),(42,91,122,105),(44,89,124,107),(46,80,126,109),(48,78,128,111),(74,77,98,110),(76,79,100,112)], [(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),(21,60,23,58),(22,90,24,92),(25,64,27,62),(26,79,28,77),(29,83,31,81),(30,51,32,49),(34,105,36,107),(38,109,40,111),(41,94,43,96),(45,102,47,104),(50,122,52,124),(53,98,55,100),(54,126,56,128),(57,115,59,113),(61,119,63,117),(66,112,68,110),(69,93,71,95),(73,101,75,103),(78,118,80,120),(82,121,84,123),(85,97,87,99),(86,125,88,127),(89,114,91,116)])
Matrix representation ►G ⊆ 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] >;
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 |
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