direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C42⋊5C4, C24.642C23, C23.164C24, (C2×C42)⋊15C4, C42⋊41(C2×C4), C22.55(C23×C4), (C22×C42).10C2, C23.356(C4○D4), (C22×C4).442C23, (C23×C4).644C22, C23.281(C22×C4), (C2×C42).1000C22, C22.66(C42⋊C2), C22.29(C42⋊2C2), C2.C42.463C22, C2.1(C2×C42⋊2C2), C22.57(C2×C4○D4), (C2×C4).486(C22×C4), (C22×C4).452(C2×C4), C2.10(C2×C42⋊C2), (C2×C2.C42).8C2, SmallGroup(128,1014)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C42⋊5C4
G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c-1 >
Subgroups: 492 in 312 conjugacy classes, 180 normal (6 characteristic)
C1, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C42, C22×C4, C22×C4, C24, C2.C42, C2×C42, C23×C4, C2×C2.C42, C42⋊5C4, C22×C42, C2×C42⋊5C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C42⋊C2, C42⋊2C2, C23×C4, C2×C4○D4, C42⋊5C4, C2×C42⋊C2, C2×C42⋊2C2, C2×C42⋊5C4
►Regular action on 128 points
Generators in S128
(1 15)(2 16)(3 13)(4 14)(5 90)(6 91)(7 92)(8 89)(9 20)(10 17)(11 18)(12 19)(21 77)(22 78)(23 79)(24 80)(25 54)(26 55)(27 56)(28 53)(29 76)(30 73)(31 74)(32 75)(33 60)(34 57)(35 58)(36 59)(37 64)(38 61)(39 62)(40 63)(41 112)(42 109)(43 110)(44 111)(45 116)(46 113)(47 114)(48 115)(49 100)(50 97)(51 98)(52 99)(65 123)(66 124)(67 121)(68 122)(69 119)(70 120)(71 117)(72 118)(81 105)(82 106)(83 107)(84 108)(85 101)(86 102)(87 103)(88 104)(93 127)(94 128)(95 125)(96 126)
(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 52 57 29)(2 49 58 30)(3 50 59 31)(4 51 60 32)(5 105 46 66)(6 106 47 67)(7 107 48 68)(8 108 45 65)(9 61 26 78)(10 62 27 79)(11 63 28 80)(12 64 25 77)(13 97 36 74)(14 98 33 75)(15 99 34 76)(16 100 35 73)(17 39 56 23)(18 40 53 24)(19 37 54 21)(20 38 55 22)(41 69 127 104)(42 70 128 101)(43 71 125 102)(44 72 126 103)(81 113 124 90)(82 114 121 91)(83 115 122 92)(84 116 123 89)(85 109 120 94)(86 110 117 95)(87 111 118 96)(88 112 119 93)
(1 65 11 119)(2 105 12 85)(3 67 9 117)(4 107 10 87)(5 62 94 51)(6 80 95 29)(7 64 96 49)(8 78 93 31)(13 121 20 71)(14 83 17 103)(15 123 18 69)(16 81 19 101)(21 44 73 115)(22 127 74 89)(23 42 75 113)(24 125 76 91)(25 120 58 66)(26 86 59 106)(27 118 60 68)(28 88 57 108)(30 48 77 111)(32 46 79 109)(33 122 56 72)(34 84 53 104)(35 124 54 70)(36 82 55 102)(37 126 100 92)(38 41 97 116)(39 128 98 90)(40 43 99 114)(45 61 112 50)(47 63 110 52)
G:=sub<Sym(128)| (1,15)(2,16)(3,13)(4,14)(5,90)(6,91)(7,92)(8,89)(9,20)(10,17)(11,18)(12,19)(21,77)(22,78)(23,79)(24,80)(25,54)(26,55)(27,56)(28,53)(29,76)(30,73)(31,74)(32,75)(33,60)(34,57)(35,58)(36,59)(37,64)(38,61)(39,62)(40,63)(41,112)(42,109)(43,110)(44,111)(45,116)(46,113)(47,114)(48,115)(49,100)(50,97)(51,98)(52,99)(65,123)(66,124)(67,121)(68,122)(69,119)(70,120)(71,117)(72,118)(81,105)(82,106)(83,107)(84,108)(85,101)(86,102)(87,103)(88,104)(93,127)(94,128)(95,125)(96,126), (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), (1,52,57,29)(2,49,58,30)(3,50,59,31)(4,51,60,32)(5,105,46,66)(6,106,47,67)(7,107,48,68)(8,108,45,65)(9,61,26,78)(10,62,27,79)(11,63,28,80)(12,64,25,77)(13,97,36,74)(14,98,33,75)(15,99,34,76)(16,100,35,73)(17,39,56,23)(18,40,53,24)(19,37,54,21)(20,38,55,22)(41,69,127,104)(42,70,128,101)(43,71,125,102)(44,72,126,103)(81,113,124,90)(82,114,121,91)(83,115,122,92)(84,116,123,89)(85,109,120,94)(86,110,117,95)(87,111,118,96)(88,112,119,93), (1,65,11,119)(2,105,12,85)(3,67,9,117)(4,107,10,87)(5,62,94,51)(6,80,95,29)(7,64,96,49)(8,78,93,31)(13,121,20,71)(14,83,17,103)(15,123,18,69)(16,81,19,101)(21,44,73,115)(22,127,74,89)(23,42,75,113)(24,125,76,91)(25,120,58,66)(26,86,59,106)(27,118,60,68)(28,88,57,108)(30,48,77,111)(32,46,79,109)(33,122,56,72)(34,84,53,104)(35,124,54,70)(36,82,55,102)(37,126,100,92)(38,41,97,116)(39,128,98,90)(40,43,99,114)(45,61,112,50)(47,63,110,52)>;
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,90)(6,91)(7,92)(8,89)(9,20)(10,17)(11,18)(12,19)(21,77)(22,78)(23,79)(24,80)(25,54)(26,55)(27,56)(28,53)(29,76)(30,73)(31,74)(32,75)(33,60)(34,57)(35,58)(36,59)(37,64)(38,61)(39,62)(40,63)(41,112)(42,109)(43,110)(44,111)(45,116)(46,113)(47,114)(48,115)(49,100)(50,97)(51,98)(52,99)(65,123)(66,124)(67,121)(68,122)(69,119)(70,120)(71,117)(72,118)(81,105)(82,106)(83,107)(84,108)(85,101)(86,102)(87,103)(88,104)(93,127)(94,128)(95,125)(96,126), (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), (1,52,57,29)(2,49,58,30)(3,50,59,31)(4,51,60,32)(5,105,46,66)(6,106,47,67)(7,107,48,68)(8,108,45,65)(9,61,26,78)(10,62,27,79)(11,63,28,80)(12,64,25,77)(13,97,36,74)(14,98,33,75)(15,99,34,76)(16,100,35,73)(17,39,56,23)(18,40,53,24)(19,37,54,21)(20,38,55,22)(41,69,127,104)(42,70,128,101)(43,71,125,102)(44,72,126,103)(81,113,124,90)(82,114,121,91)(83,115,122,92)(84,116,123,89)(85,109,120,94)(86,110,117,95)(87,111,118,96)(88,112,119,93), (1,65,11,119)(2,105,12,85)(3,67,9,117)(4,107,10,87)(5,62,94,51)(6,80,95,29)(7,64,96,49)(8,78,93,31)(13,121,20,71)(14,83,17,103)(15,123,18,69)(16,81,19,101)(21,44,73,115)(22,127,74,89)(23,42,75,113)(24,125,76,91)(25,120,58,66)(26,86,59,106)(27,118,60,68)(28,88,57,108)(30,48,77,111)(32,46,79,109)(33,122,56,72)(34,84,53,104)(35,124,54,70)(36,82,55,102)(37,126,100,92)(38,41,97,116)(39,128,98,90)(40,43,99,114)(45,61,112,50)(47,63,110,52) );
G=PermutationGroup([[(1,15),(2,16),(3,13),(4,14),(5,90),(6,91),(7,92),(8,89),(9,20),(10,17),(11,18),(12,19),(21,77),(22,78),(23,79),(24,80),(25,54),(26,55),(27,56),(28,53),(29,76),(30,73),(31,74),(32,75),(33,60),(34,57),(35,58),(36,59),(37,64),(38,61),(39,62),(40,63),(41,112),(42,109),(43,110),(44,111),(45,116),(46,113),(47,114),(48,115),(49,100),(50,97),(51,98),(52,99),(65,123),(66,124),(67,121),(68,122),(69,119),(70,120),(71,117),(72,118),(81,105),(82,106),(83,107),(84,108),(85,101),(86,102),(87,103),(88,104),(93,127),(94,128),(95,125),(96,126)], [(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)], [(1,52,57,29),(2,49,58,30),(3,50,59,31),(4,51,60,32),(5,105,46,66),(6,106,47,67),(7,107,48,68),(8,108,45,65),(9,61,26,78),(10,62,27,79),(11,63,28,80),(12,64,25,77),(13,97,36,74),(14,98,33,75),(15,99,34,76),(16,100,35,73),(17,39,56,23),(18,40,53,24),(19,37,54,21),(20,38,55,22),(41,69,127,104),(42,70,128,101),(43,71,125,102),(44,72,126,103),(81,113,124,90),(82,114,121,91),(83,115,122,92),(84,116,123,89),(85,109,120,94),(86,110,117,95),(87,111,118,96),(88,112,119,93)], [(1,65,11,119),(2,105,12,85),(3,67,9,117),(4,107,10,87),(5,62,94,51),(6,80,95,29),(7,64,96,49),(8,78,93,31),(13,121,20,71),(14,83,17,103),(15,123,18,69),(16,81,19,101),(21,44,73,115),(22,127,74,89),(23,42,75,113),(24,125,76,91),(25,120,58,66),(26,86,59,106),(27,118,60,68),(28,88,57,108),(30,48,77,111),(32,46,79,109),(33,122,56,72),(34,84,53,104),(35,124,54,70),(36,82,55,102),(37,126,100,92),(38,41,97,116),(39,128,98,90),(40,43,99,114),(45,61,112,50),(47,63,110,52)]])
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 |
| type | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | C4○D4 |
| kernel | C2×C42⋊5C4 | C2×C2.C42 | C42⋊5C4 | C22×C42 | C2×C42 | C23 |
| # reps | 1 | 6 | 8 | 1 | 16 | 24 |
Matrix representation of C2×C42⋊5C4 ►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 | 0 | 2 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 4 | 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 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 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],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,4,4,0,0,0,0,2,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,2,1] >;
C2×C42⋊5C4 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes_5C_4
% in TeX
G:=Group("C2xC4^2:5C4"); // GroupNames label
G:=SmallGroup(128,1014);
// by ID
G=gap.SmallGroup(128,1014);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,100]);
// 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^-1>;
// generators/relations