p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C42⋊10Q8, C42.197D4, C23.560C24, C22.2502- (1+4), C22.3342+ (1+4), C4.9(C4⋊Q8), C42⋊4C4.28C2, C42⋊8C4.40C2, C42⋊9C4.36C2, (C2×C42).624C22, (C22×C4).859C23, C22.372(C22×D4), C22.138(C22×Q8), (C22×Q8).167C22, C2.48(C22.29C24), C23.78C23.16C2, C2.C42.274C22, C23.81C23.29C2, C2.25(C23.41C23), C2.48(C23.38C23), C2.20(C2×C4⋊Q8), (C2×C4⋊Q8).37C2, (C2×C4).406(C2×D4), (C2×C4).135(C2×Q8), (C2×C4⋊C4).383C22, (C2×C42.C2).25C2, SmallGroup(128,1392)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 388 in 224 conjugacy classes, 116 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×20], C22, C22 [×6], C2×C4 [×18], C2×C4 [×36], Q8 [×8], C23, C42 [×12], C4⋊C4 [×28], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×8], C2.C42 [×12], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×16], C42.C2 [×4], C4⋊Q8 [×4], C22×Q8 [×2], C42⋊4C4, C42⋊8C4 [×2], C42⋊9C4 [×2], C23.78C23 [×4], C23.81C23 [×4], C2×C42.C2, C2×C4⋊Q8, C42⋊10Q8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], 2+ (1+4) [×2], 2- (1+4) [×2], C2×C4⋊Q8, C22.29C24, C23.38C23, C23.41C23 [×4], C42⋊10Q8
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=c-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 25 65 30)(2 26 66 31)(3 27 67 32)(4 28 68 29)(5 48 125 41)(6 45 126 42)(7 46 127 43)(8 47 128 44)(9 84 14 52)(10 81 15 49)(11 82 16 50)(12 83 13 51)(17 60 22 92)(18 57 23 89)(19 58 24 90)(20 59 21 91)(33 117 40 122)(34 118 37 123)(35 119 38 124)(36 120 39 121)(53 93 85 106)(54 94 86 107)(55 95 87 108)(56 96 88 105)(61 114 80 109)(62 115 77 110)(63 116 78 111)(64 113 79 112)(69 100 74 101)(70 97 75 102)(71 98 76 103)(72 99 73 104)
(1 125 117 75)(2 8 118 69)(3 127 119 73)(4 6 120 71)(5 122 70 65)(7 124 72 67)(9 24 85 63)(10 18 86 77)(11 22 87 61)(12 20 88 79)(13 21 56 64)(14 19 53 78)(15 23 54 62)(16 17 55 80)(25 48 40 97)(26 44 37 101)(27 46 38 99)(28 42 39 103)(29 45 36 98)(30 41 33 102)(31 47 34 100)(32 43 35 104)(49 57 94 110)(50 92 95 114)(51 59 96 112)(52 90 93 116)(58 106 111 84)(60 108 109 82)(66 128 123 74)(68 126 121 76)(81 89 107 115)(83 91 105 113)
(1 81 117 107)(2 50 118 95)(3 83 119 105)(4 52 120 93)(5 110 70 57)(6 116 71 90)(7 112 72 59)(8 114 69 92)(9 39 85 28)(10 33 86 30)(11 37 87 26)(12 35 88 32)(13 38 56 27)(14 36 53 29)(15 40 54 25)(16 34 55 31)(17 47 80 100)(18 41 77 102)(19 45 78 98)(20 43 79 104)(21 46 64 99)(22 44 61 101)(23 48 62 97)(24 42 63 103)(49 122 94 65)(51 124 96 67)(58 126 111 76)(60 128 109 74)(66 82 123 108)(68 84 121 106)(73 91 127 113)(75 89 125 115)
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), (1,25,65,30)(2,26,66,31)(3,27,67,32)(4,28,68,29)(5,48,125,41)(6,45,126,42)(7,46,127,43)(8,47,128,44)(9,84,14,52)(10,81,15,49)(11,82,16,50)(12,83,13,51)(17,60,22,92)(18,57,23,89)(19,58,24,90)(20,59,21,91)(33,117,40,122)(34,118,37,123)(35,119,38,124)(36,120,39,121)(53,93,85,106)(54,94,86,107)(55,95,87,108)(56,96,88,105)(61,114,80,109)(62,115,77,110)(63,116,78,111)(64,113,79,112)(69,100,74,101)(70,97,75,102)(71,98,76,103)(72,99,73,104), (1,125,117,75)(2,8,118,69)(3,127,119,73)(4,6,120,71)(5,122,70,65)(7,124,72,67)(9,24,85,63)(10,18,86,77)(11,22,87,61)(12,20,88,79)(13,21,56,64)(14,19,53,78)(15,23,54,62)(16,17,55,80)(25,48,40,97)(26,44,37,101)(27,46,38,99)(28,42,39,103)(29,45,36,98)(30,41,33,102)(31,47,34,100)(32,43,35,104)(49,57,94,110)(50,92,95,114)(51,59,96,112)(52,90,93,116)(58,106,111,84)(60,108,109,82)(66,128,123,74)(68,126,121,76)(81,89,107,115)(83,91,105,113), (1,81,117,107)(2,50,118,95)(3,83,119,105)(4,52,120,93)(5,110,70,57)(6,116,71,90)(7,112,72,59)(8,114,69,92)(9,39,85,28)(10,33,86,30)(11,37,87,26)(12,35,88,32)(13,38,56,27)(14,36,53,29)(15,40,54,25)(16,34,55,31)(17,47,80,100)(18,41,77,102)(19,45,78,98)(20,43,79,104)(21,46,64,99)(22,44,61,101)(23,48,62,97)(24,42,63,103)(49,122,94,65)(51,124,96,67)(58,126,111,76)(60,128,109,74)(66,82,123,108)(68,84,121,106)(73,91,127,113)(75,89,125,115)>;
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), (1,25,65,30)(2,26,66,31)(3,27,67,32)(4,28,68,29)(5,48,125,41)(6,45,126,42)(7,46,127,43)(8,47,128,44)(9,84,14,52)(10,81,15,49)(11,82,16,50)(12,83,13,51)(17,60,22,92)(18,57,23,89)(19,58,24,90)(20,59,21,91)(33,117,40,122)(34,118,37,123)(35,119,38,124)(36,120,39,121)(53,93,85,106)(54,94,86,107)(55,95,87,108)(56,96,88,105)(61,114,80,109)(62,115,77,110)(63,116,78,111)(64,113,79,112)(69,100,74,101)(70,97,75,102)(71,98,76,103)(72,99,73,104), (1,125,117,75)(2,8,118,69)(3,127,119,73)(4,6,120,71)(5,122,70,65)(7,124,72,67)(9,24,85,63)(10,18,86,77)(11,22,87,61)(12,20,88,79)(13,21,56,64)(14,19,53,78)(15,23,54,62)(16,17,55,80)(25,48,40,97)(26,44,37,101)(27,46,38,99)(28,42,39,103)(29,45,36,98)(30,41,33,102)(31,47,34,100)(32,43,35,104)(49,57,94,110)(50,92,95,114)(51,59,96,112)(52,90,93,116)(58,106,111,84)(60,108,109,82)(66,128,123,74)(68,126,121,76)(81,89,107,115)(83,91,105,113), (1,81,117,107)(2,50,118,95)(3,83,119,105)(4,52,120,93)(5,110,70,57)(6,116,71,90)(7,112,72,59)(8,114,69,92)(9,39,85,28)(10,33,86,30)(11,37,87,26)(12,35,88,32)(13,38,56,27)(14,36,53,29)(15,40,54,25)(16,34,55,31)(17,47,80,100)(18,41,77,102)(19,45,78,98)(20,43,79,104)(21,46,64,99)(22,44,61,101)(23,48,62,97)(24,42,63,103)(49,122,94,65)(51,124,96,67)(58,126,111,76)(60,128,109,74)(66,82,123,108)(68,84,121,106)(73,91,127,113)(75,89,125,115) );
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)], [(1,25,65,30),(2,26,66,31),(3,27,67,32),(4,28,68,29),(5,48,125,41),(6,45,126,42),(7,46,127,43),(8,47,128,44),(9,84,14,52),(10,81,15,49),(11,82,16,50),(12,83,13,51),(17,60,22,92),(18,57,23,89),(19,58,24,90),(20,59,21,91),(33,117,40,122),(34,118,37,123),(35,119,38,124),(36,120,39,121),(53,93,85,106),(54,94,86,107),(55,95,87,108),(56,96,88,105),(61,114,80,109),(62,115,77,110),(63,116,78,111),(64,113,79,112),(69,100,74,101),(70,97,75,102),(71,98,76,103),(72,99,73,104)], [(1,125,117,75),(2,8,118,69),(3,127,119,73),(4,6,120,71),(5,122,70,65),(7,124,72,67),(9,24,85,63),(10,18,86,77),(11,22,87,61),(12,20,88,79),(13,21,56,64),(14,19,53,78),(15,23,54,62),(16,17,55,80),(25,48,40,97),(26,44,37,101),(27,46,38,99),(28,42,39,103),(29,45,36,98),(30,41,33,102),(31,47,34,100),(32,43,35,104),(49,57,94,110),(50,92,95,114),(51,59,96,112),(52,90,93,116),(58,106,111,84),(60,108,109,82),(66,128,123,74),(68,126,121,76),(81,89,107,115),(83,91,105,113)], [(1,81,117,107),(2,50,118,95),(3,83,119,105),(4,52,120,93),(5,110,70,57),(6,116,71,90),(7,112,72,59),(8,114,69,92),(9,39,85,28),(10,33,86,30),(11,37,87,26),(12,35,88,32),(13,38,56,27),(14,36,53,29),(15,40,54,25),(16,34,55,31),(17,47,80,100),(18,41,77,102),(19,45,78,98),(20,43,79,104),(21,46,64,99),(22,44,61,101),(23,48,62,97),(24,42,63,103),(49,122,94,65),(51,124,96,67),(58,126,111,76),(60,128,109,74),(66,82,123,108),(68,84,121,106),(73,91,127,113),(75,89,125,115)])
Matrix representation ►G ⊆ GL8(𝔽5)
| 4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 3 |
| 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 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 1 | 2 |
| 0 | 0 | 0 | 0 | 4 | 3 | 4 | 4 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 4 | 3 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 1 |
| 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 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 1 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 | 2 | 0 |
G:=sub<GL(8,GF(5))| [4,4,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,2,0,1,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,1,0,4,0,0,0,0,3,1,2,3,0,0,0,0,0,0,1,4,0,0,0,0,0,0,2,4],[2,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,4,3,2,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1],[1,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,1,4,0,0,0,0,0,0,0,2,0,0,0,0,2,3,3,2,0,0,0,0,0,3,0,0] >;
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 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | 2+ (1+4) | 2- (1+4) |
| kernel | C42⋊10Q8 | C42⋊4C4 | C42⋊8C4 | C42⋊9C4 | C23.78C23 | C23.81C23 | C2×C42.C2 | C2×C4⋊Q8 | C42 | C42 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 4 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{10}Q_8 % in TeX
G:=Group("C4^2:10Q8"); // GroupNames label
G:=SmallGroup(128,1392);
// by ID
G=gap.SmallGroup(128,1392);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,723,184,185,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations