direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C42⋊8C4, C23.163C24, C24.641C23, C42⋊44(C2×C4), (C2×C42)⋊20C4, (C22×C4).93Q8, C23.822(C2×D4), (C22×C4).593D4, C23.139(C2×Q8), C22.54(C23×C4), (C23×C4).33C22, (C22×C42).18C2, C22.63(C22×D4), C23.355(C4○D4), C22.18(C22×Q8), C23.280(C22×C4), (C22×C4).441C23, (C2×C42).1085C22, C22.72(C4.4D4), C22.24(C42.C2), C22.65(C42⋊C2), C2.C42.462C22, C4.56(C2×C4⋊C4), C2.5(C22×C4⋊C4), (C2×C4).822(C2×D4), C22.70(C2×C4⋊C4), (C2×C4).223(C2×Q8), C2.1(C2×C4.4D4), (C2×C4).146(C4⋊C4), C2.1(C2×C42.C2), (C22×C4⋊C4).19C2, C2.9(C2×C42⋊C2), C22.56(C2×C4○D4), (C2×C4⋊C4).783C22, (C2×C4).485(C22×C4), (C22×C4).451(C2×C4), (C2×C2.C42).7C2, SmallGroup(128,1013)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C42⋊8C4
G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >
Subgroups: 540 in 356 conjugacy classes, 220 normal (10 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×C42, C2×C4⋊C4, C2×C4⋊C4, C23×C4, C23×C4, C2×C2.C42, C42⋊8C4, C22×C42, C22×C4⋊C4, C2×C42⋊8C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C42⋊8C4, C22×C4⋊C4, C2×C42⋊C2, C2×C4.4D4, C2×C42.C2, C2×C42⋊8C4
►Regular action on 128 points
Generators in S128
(1 15)(2 16)(3 13)(4 14)(5 17)(6 18)(7 19)(8 20)(9 31)(10 32)(11 29)(12 30)(21 39)(22 40)(23 37)(24 38)(25 61)(26 62)(27 63)(28 64)(33 49)(34 50)(35 51)(36 52)(41 93)(42 94)(43 95)(44 96)(45 79)(46 80)(47 77)(48 78)(53 57)(54 58)(55 59)(56 60)(65 84)(66 81)(67 82)(68 83)(69 88)(70 85)(71 86)(72 87)(73 92)(74 89)(75 90)(76 91)(97 116)(98 113)(99 114)(100 115)(101 120)(102 117)(103 118)(104 119)(105 124)(106 121)(107 122)(108 123)(109 128)(110 125)(111 126)(112 127)
(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 25 18 46)(2 26 19 47)(3 27 20 48)(4 28 17 45)(5 79 14 64)(6 80 15 61)(7 77 16 62)(8 78 13 63)(9 21 36 58)(10 22 33 59)(11 23 34 60)(12 24 35 57)(29 37 50 56)(30 38 51 53)(31 39 52 54)(32 40 49 55)(41 102 125 69)(42 103 126 70)(43 104 127 71)(44 101 128 72)(65 113 106 92)(66 114 107 89)(67 115 108 90)(68 116 105 91)(73 84 98 121)(74 81 99 122)(75 82 100 123)(76 83 97 124)(85 94 118 111)(86 95 119 112)(87 96 120 109)(88 93 117 110)
(1 102 10 90)(2 70 11 116)(3 104 12 92)(4 72 9 114)(5 120 52 74)(6 88 49 100)(7 118 50 76)(8 86 51 98)(13 119 30 73)(14 87 31 99)(15 117 32 75)(16 85 29 97)(17 101 36 89)(18 69 33 115)(19 103 34 91)(20 71 35 113)(21 105 28 42)(22 65 25 127)(23 107 26 44)(24 67 27 125)(37 122 62 96)(38 82 63 110)(39 124 64 94)(40 84 61 112)(41 57 108 48)(43 59 106 46)(45 126 58 68)(47 128 60 66)(53 123 78 93)(54 83 79 111)(55 121 80 95)(56 81 77 109)
G:=sub<Sym(128)| (1,15)(2,16)(3,13)(4,14)(5,17)(6,18)(7,19)(8,20)(9,31)(10,32)(11,29)(12,30)(21,39)(22,40)(23,37)(24,38)(25,61)(26,62)(27,63)(28,64)(33,49)(34,50)(35,51)(36,52)(41,93)(42,94)(43,95)(44,96)(45,79)(46,80)(47,77)(48,78)(53,57)(54,58)(55,59)(56,60)(65,84)(66,81)(67,82)(68,83)(69,88)(70,85)(71,86)(72,87)(73,92)(74,89)(75,90)(76,91)(97,116)(98,113)(99,114)(100,115)(101,120)(102,117)(103,118)(104,119)(105,124)(106,121)(107,122)(108,123)(109,128)(110,125)(111,126)(112,127), (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), (1,25,18,46)(2,26,19,47)(3,27,20,48)(4,28,17,45)(5,79,14,64)(6,80,15,61)(7,77,16,62)(8,78,13,63)(9,21,36,58)(10,22,33,59)(11,23,34,60)(12,24,35,57)(29,37,50,56)(30,38,51,53)(31,39,52,54)(32,40,49,55)(41,102,125,69)(42,103,126,70)(43,104,127,71)(44,101,128,72)(65,113,106,92)(66,114,107,89)(67,115,108,90)(68,116,105,91)(73,84,98,121)(74,81,99,122)(75,82,100,123)(76,83,97,124)(85,94,118,111)(86,95,119,112)(87,96,120,109)(88,93,117,110), (1,102,10,90)(2,70,11,116)(3,104,12,92)(4,72,9,114)(5,120,52,74)(6,88,49,100)(7,118,50,76)(8,86,51,98)(13,119,30,73)(14,87,31,99)(15,117,32,75)(16,85,29,97)(17,101,36,89)(18,69,33,115)(19,103,34,91)(20,71,35,113)(21,105,28,42)(22,65,25,127)(23,107,26,44)(24,67,27,125)(37,122,62,96)(38,82,63,110)(39,124,64,94)(40,84,61,112)(41,57,108,48)(43,59,106,46)(45,126,58,68)(47,128,60,66)(53,123,78,93)(54,83,79,111)(55,121,80,95)(56,81,77,109)>;
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,17)(6,18)(7,19)(8,20)(9,31)(10,32)(11,29)(12,30)(21,39)(22,40)(23,37)(24,38)(25,61)(26,62)(27,63)(28,64)(33,49)(34,50)(35,51)(36,52)(41,93)(42,94)(43,95)(44,96)(45,79)(46,80)(47,77)(48,78)(53,57)(54,58)(55,59)(56,60)(65,84)(66,81)(67,82)(68,83)(69,88)(70,85)(71,86)(72,87)(73,92)(74,89)(75,90)(76,91)(97,116)(98,113)(99,114)(100,115)(101,120)(102,117)(103,118)(104,119)(105,124)(106,121)(107,122)(108,123)(109,128)(110,125)(111,126)(112,127), (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), (1,25,18,46)(2,26,19,47)(3,27,20,48)(4,28,17,45)(5,79,14,64)(6,80,15,61)(7,77,16,62)(8,78,13,63)(9,21,36,58)(10,22,33,59)(11,23,34,60)(12,24,35,57)(29,37,50,56)(30,38,51,53)(31,39,52,54)(32,40,49,55)(41,102,125,69)(42,103,126,70)(43,104,127,71)(44,101,128,72)(65,113,106,92)(66,114,107,89)(67,115,108,90)(68,116,105,91)(73,84,98,121)(74,81,99,122)(75,82,100,123)(76,83,97,124)(85,94,118,111)(86,95,119,112)(87,96,120,109)(88,93,117,110), (1,102,10,90)(2,70,11,116)(3,104,12,92)(4,72,9,114)(5,120,52,74)(6,88,49,100)(7,118,50,76)(8,86,51,98)(13,119,30,73)(14,87,31,99)(15,117,32,75)(16,85,29,97)(17,101,36,89)(18,69,33,115)(19,103,34,91)(20,71,35,113)(21,105,28,42)(22,65,25,127)(23,107,26,44)(24,67,27,125)(37,122,62,96)(38,82,63,110)(39,124,64,94)(40,84,61,112)(41,57,108,48)(43,59,106,46)(45,126,58,68)(47,128,60,66)(53,123,78,93)(54,83,79,111)(55,121,80,95)(56,81,77,109) );
G=PermutationGroup([[(1,15),(2,16),(3,13),(4,14),(5,17),(6,18),(7,19),(8,20),(9,31),(10,32),(11,29),(12,30),(21,39),(22,40),(23,37),(24,38),(25,61),(26,62),(27,63),(28,64),(33,49),(34,50),(35,51),(36,52),(41,93),(42,94),(43,95),(44,96),(45,79),(46,80),(47,77),(48,78),(53,57),(54,58),(55,59),(56,60),(65,84),(66,81),(67,82),(68,83),(69,88),(70,85),(71,86),(72,87),(73,92),(74,89),(75,90),(76,91),(97,116),(98,113),(99,114),(100,115),(101,120),(102,117),(103,118),(104,119),(105,124),(106,121),(107,122),(108,123),(109,128),(110,125),(111,126),(112,127)], [(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)], [(1,25,18,46),(2,26,19,47),(3,27,20,48),(4,28,17,45),(5,79,14,64),(6,80,15,61),(7,77,16,62),(8,78,13,63),(9,21,36,58),(10,22,33,59),(11,23,34,60),(12,24,35,57),(29,37,50,56),(30,38,51,53),(31,39,52,54),(32,40,49,55),(41,102,125,69),(42,103,126,70),(43,104,127,71),(44,101,128,72),(65,113,106,92),(66,114,107,89),(67,115,108,90),(68,116,105,91),(73,84,98,121),(74,81,99,122),(75,82,100,123),(76,83,97,124),(85,94,118,111),(86,95,119,112),(87,96,120,109),(88,93,117,110)], [(1,102,10,90),(2,70,11,116),(3,104,12,92),(4,72,9,114),(5,120,52,74),(6,88,49,100),(7,118,50,76),(8,86,51,98),(13,119,30,73),(14,87,31,99),(15,117,32,75),(16,85,29,97),(17,101,36,89),(18,69,33,115),(19,103,34,91),(20,71,35,113),(21,105,28,42),(22,65,25,127),(23,107,26,44),(24,67,27,125),(37,122,62,96),(38,82,63,110),(39,124,64,94),(40,84,61,112),(41,57,108,48),(43,59,106,46),(45,126,58,68),(47,128,60,66),(53,123,78,93),(54,83,79,111),(55,121,80,95),(56,81,77,109)]])
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 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | C4○D4 |
| kernel | C2×C42⋊8C4 | C2×C2.C42 | C42⋊8C4 | C22×C42 | C22×C4⋊C4 | C2×C42 | C22×C4 | C22×C4 | C23 |
| # reps | 1 | 4 | 8 | 1 | 2 | 16 | 4 | 4 | 16 |
Matrix representation of C2×C42⋊8C4 ►in GL6(𝔽5)
| 4 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 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 | 3 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 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 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,4,0,0,0,0,3,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,4,1] >;
C2×C42⋊8C4 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes_8C_4
% in TeX
G:=Group("C2xC4^2:8C4"); // GroupNames label
G:=SmallGroup(128,1013);
// by ID
G=gap.SmallGroup(128,1013);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,100]);
// 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*c^2,d*c*d^-1=b^2*c>;
// generators/relations