direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C42⋊9C4, C23.166C24, C24.643C23, C42⋊45(C2×C4), (C2×C42)⋊22C4, (C22×C4).94Q8, C23.823(C2×D4), (C22×C4).594D4, C22.39(C4⋊Q8), C23.140(C2×Q8), C22.57(C23×C4), (C22×C42).20C2, C22.64(C22×D4), C22.19(C22×Q8), C22.43(C4⋊1D4), (C22×C4).444C23, (C23×C4).677C22, C23.282(C22×C4), (C2×C42).1086C22, C4⋊2(C2×C4⋊C4), (C2×C4)⋊9(C4⋊C4), C2.1(C2×C4⋊Q8), C2.1(C2×C4⋊1D4), C2.6(C22×C4⋊C4), (C2×C4).823(C2×D4), C22.71(C2×C4⋊C4), (C2×C4).224(C2×Q8), (C22×C4⋊C4).20C2, (C2×C4⋊C4).785C22, (C22×C4).490(C2×C4), (C2×C4).565(C22×C4), SmallGroup(128,1016)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C42⋊9C4
G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 636 in 444 conjugacy classes, 300 normal (7 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C23, C42, C4⋊C4, C22×C4, C22×C4, C24, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C23×C4, C42⋊9C4, C22×C42, C22×C4⋊C4, C2×C42⋊9C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C4⋊C4, C4⋊1D4, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C42⋊9C4, C22×C4⋊C4, C2×C4⋊1D4, C2×C4⋊Q8, C2×C42⋊9C4
►Regular action on 128 points
Generators in S128
(1 28)(2 25)(3 26)(4 27)(5 92)(6 89)(7 90)(8 91)(9 20)(10 17)(11 18)(12 19)(13 77)(14 78)(15 79)(16 80)(21 40)(22 37)(23 38)(24 39)(29 99)(30 100)(31 97)(32 98)(33 53)(34 54)(35 55)(36 56)(41 112)(42 109)(43 110)(44 111)(45 116)(46 113)(47 114)(48 115)(49 73)(50 74)(51 75)(52 76)(57 61)(58 62)(59 63)(60 64)(65 121)(66 122)(67 123)(68 124)(69 117)(70 118)(71 119)(72 120)(81 107)(82 108)(83 105)(84 106)(85 103)(86 104)(87 101)(88 102)(93 125)(94 126)(95 127)(96 128)
(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 29 22 52)(2 30 23 49)(3 31 24 50)(4 32 21 51)(5 66 48 105)(6 67 45 106)(7 68 46 107)(8 65 47 108)(9 78 35 61)(10 79 36 62)(11 80 33 63)(12 77 34 64)(13 54 60 19)(14 55 57 20)(15 56 58 17)(16 53 59 18)(25 100 38 73)(26 97 39 74)(27 98 40 75)(28 99 37 76)(41 102 125 71)(42 103 126 72)(43 104 127 69)(44 101 128 70)(81 90 124 113)(82 91 121 114)(83 92 122 115)(84 89 123 116)(85 94 120 109)(86 95 117 110)(87 96 118 111)(88 93 119 112)
(1 108 9 119)(2 107 10 118)(3 106 11 117)(4 105 12 120)(5 64 109 51)(6 63 110 50)(7 62 111 49)(8 61 112 52)(13 126 98 115)(14 125 99 114)(15 128 100 113)(16 127 97 116)(17 70 25 81)(18 69 26 84)(19 72 27 83)(20 71 28 82)(21 66 34 85)(22 65 35 88)(23 68 36 87)(24 67 33 86)(29 47 78 93)(30 46 79 96)(31 45 80 95)(32 48 77 94)(37 121 55 102)(38 124 56 101)(39 123 53 104)(40 122 54 103)(41 76 91 57)(42 75 92 60)(43 74 89 59)(44 73 90 58)
G:=sub<Sym(128)| (1,28)(2,25)(3,26)(4,27)(5,92)(6,89)(7,90)(8,91)(9,20)(10,17)(11,18)(12,19)(13,77)(14,78)(15,79)(16,80)(21,40)(22,37)(23,38)(24,39)(29,99)(30,100)(31,97)(32,98)(33,53)(34,54)(35,55)(36,56)(41,112)(42,109)(43,110)(44,111)(45,116)(46,113)(47,114)(48,115)(49,73)(50,74)(51,75)(52,76)(57,61)(58,62)(59,63)(60,64)(65,121)(66,122)(67,123)(68,124)(69,117)(70,118)(71,119)(72,120)(81,107)(82,108)(83,105)(84,106)(85,103)(86,104)(87,101)(88,102)(93,125)(94,126)(95,127)(96,128), (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), (1,29,22,52)(2,30,23,49)(3,31,24,50)(4,32,21,51)(5,66,48,105)(6,67,45,106)(7,68,46,107)(8,65,47,108)(9,78,35,61)(10,79,36,62)(11,80,33,63)(12,77,34,64)(13,54,60,19)(14,55,57,20)(15,56,58,17)(16,53,59,18)(25,100,38,73)(26,97,39,74)(27,98,40,75)(28,99,37,76)(41,102,125,71)(42,103,126,72)(43,104,127,69)(44,101,128,70)(81,90,124,113)(82,91,121,114)(83,92,122,115)(84,89,123,116)(85,94,120,109)(86,95,117,110)(87,96,118,111)(88,93,119,112), (1,108,9,119)(2,107,10,118)(3,106,11,117)(4,105,12,120)(5,64,109,51)(6,63,110,50)(7,62,111,49)(8,61,112,52)(13,126,98,115)(14,125,99,114)(15,128,100,113)(16,127,97,116)(17,70,25,81)(18,69,26,84)(19,72,27,83)(20,71,28,82)(21,66,34,85)(22,65,35,88)(23,68,36,87)(24,67,33,86)(29,47,78,93)(30,46,79,96)(31,45,80,95)(32,48,77,94)(37,121,55,102)(38,124,56,101)(39,123,53,104)(40,122,54,103)(41,76,91,57)(42,75,92,60)(43,74,89,59)(44,73,90,58)>;
G:=Group( (1,28)(2,25)(3,26)(4,27)(5,92)(6,89)(7,90)(8,91)(9,20)(10,17)(11,18)(12,19)(13,77)(14,78)(15,79)(16,80)(21,40)(22,37)(23,38)(24,39)(29,99)(30,100)(31,97)(32,98)(33,53)(34,54)(35,55)(36,56)(41,112)(42,109)(43,110)(44,111)(45,116)(46,113)(47,114)(48,115)(49,73)(50,74)(51,75)(52,76)(57,61)(58,62)(59,63)(60,64)(65,121)(66,122)(67,123)(68,124)(69,117)(70,118)(71,119)(72,120)(81,107)(82,108)(83,105)(84,106)(85,103)(86,104)(87,101)(88,102)(93,125)(94,126)(95,127)(96,128), (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), (1,29,22,52)(2,30,23,49)(3,31,24,50)(4,32,21,51)(5,66,48,105)(6,67,45,106)(7,68,46,107)(8,65,47,108)(9,78,35,61)(10,79,36,62)(11,80,33,63)(12,77,34,64)(13,54,60,19)(14,55,57,20)(15,56,58,17)(16,53,59,18)(25,100,38,73)(26,97,39,74)(27,98,40,75)(28,99,37,76)(41,102,125,71)(42,103,126,72)(43,104,127,69)(44,101,128,70)(81,90,124,113)(82,91,121,114)(83,92,122,115)(84,89,123,116)(85,94,120,109)(86,95,117,110)(87,96,118,111)(88,93,119,112), (1,108,9,119)(2,107,10,118)(3,106,11,117)(4,105,12,120)(5,64,109,51)(6,63,110,50)(7,62,111,49)(8,61,112,52)(13,126,98,115)(14,125,99,114)(15,128,100,113)(16,127,97,116)(17,70,25,81)(18,69,26,84)(19,72,27,83)(20,71,28,82)(21,66,34,85)(22,65,35,88)(23,68,36,87)(24,67,33,86)(29,47,78,93)(30,46,79,96)(31,45,80,95)(32,48,77,94)(37,121,55,102)(38,124,56,101)(39,123,53,104)(40,122,54,103)(41,76,91,57)(42,75,92,60)(43,74,89,59)(44,73,90,58) );
G=PermutationGroup([[(1,28),(2,25),(3,26),(4,27),(5,92),(6,89),(7,90),(8,91),(9,20),(10,17),(11,18),(12,19),(13,77),(14,78),(15,79),(16,80),(21,40),(22,37),(23,38),(24,39),(29,99),(30,100),(31,97),(32,98),(33,53),(34,54),(35,55),(36,56),(41,112),(42,109),(43,110),(44,111),(45,116),(46,113),(47,114),(48,115),(49,73),(50,74),(51,75),(52,76),(57,61),(58,62),(59,63),(60,64),(65,121),(66,122),(67,123),(68,124),(69,117),(70,118),(71,119),(72,120),(81,107),(82,108),(83,105),(84,106),(85,103),(86,104),(87,101),(88,102),(93,125),(94,126),(95,127),(96,128)], [(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)], [(1,29,22,52),(2,30,23,49),(3,31,24,50),(4,32,21,51),(5,66,48,105),(6,67,45,106),(7,68,46,107),(8,65,47,108),(9,78,35,61),(10,79,36,62),(11,80,33,63),(12,77,34,64),(13,54,60,19),(14,55,57,20),(15,56,58,17),(16,53,59,18),(25,100,38,73),(26,97,39,74),(27,98,40,75),(28,99,37,76),(41,102,125,71),(42,103,126,72),(43,104,127,69),(44,101,128,70),(81,90,124,113),(82,91,121,114),(83,92,122,115),(84,89,123,116),(85,94,120,109),(86,95,117,110),(87,96,118,111),(88,93,119,112)], [(1,108,9,119),(2,107,10,118),(3,106,11,117),(4,105,12,120),(5,64,109,51),(6,63,110,50),(7,62,111,49),(8,61,112,52),(13,126,98,115),(14,125,99,114),(15,128,100,113),(16,127,97,116),(17,70,25,81),(18,69,26,84),(19,72,27,83),(20,71,28,82),(21,66,34,85),(22,65,35,88),(23,68,36,87),(24,67,33,86),(29,47,78,93),(30,46,79,96),(31,45,80,95),(32,48,77,94),(37,121,55,102),(38,124,56,101),(39,123,53,104),(40,122,54,103),(41,76,91,57),(42,75,92,60),(43,74,89,59),(44,73,90,58)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4X | 4Y | ··· | 4AN |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 |
| kernel | C2×C42⋊9C4 | C42⋊9C4 | C22×C42 | C22×C4⋊C4 | C2×C42 | C22×C4 | C22×C4 |
| # reps | 1 | 8 | 1 | 6 | 16 | 12 | 12 |
Matrix representation of C2×C42⋊9C4 ►in GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0] >;
C2×C42⋊9C4 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes_9C_4
% in TeX
G:=Group("C2xC4^2:9C4"); // GroupNames label
G:=SmallGroup(128,1016);
// by ID
G=gap.SmallGroup(128,1016);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,184]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations