p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊5Q8, C42.164D4, C23.299C24, C4.56(C4⋊Q8), C42⋊4C4.18C2, C22.61(C22×Q8), (C2×C42).459C22, (C22×C4).784C23, C22.182(C22×D4), (C22×Q8).412C22, C4○3(C23.78C23), C4○4(C23.81C23), C2.15(C22.19C24), C2.C42.535C22, C23.78C23.33C2, C23.81C23.55C2, C2.8(C23.37C23), C2.7(C2×C4⋊Q8), (C4×C4⋊C4).46C2, (C2×C4×Q8).28C2, (C2×C4).299(C2×D4), (C2×C4).122(C2×Q8), (C2×C4).91(C4○D4), (C2×C4⋊C4).842C22, C22.179(C2×C4○D4), SmallGroup(128,1131)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊5Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 388 in 252 conjugacy classes, 124 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C4×Q8, C22×Q8, C42⋊4C4, C4×C4⋊C4, C23.78C23, C23.81C23, C2×C4×Q8, C42⋊5Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C4⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C22.19C24, C2×C4⋊Q8, C23.37C23, C42⋊5Q8
►Regular action on 128 points
Generators in S128
(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 56 16 127)(2 53 13 128)(3 54 14 125)(4 55 15 126)(5 120 59 73)(6 117 60 74)(7 118 57 75)(8 119 58 76)(9 69 63 112)(10 70 64 109)(11 71 61 110)(12 72 62 111)(17 90 44 96)(18 91 41 93)(19 92 42 94)(20 89 43 95)(21 100 48 85)(22 97 45 86)(23 98 46 87)(24 99 47 88)(25 34 84 104)(26 35 81 101)(27 36 82 102)(28 33 83 103)(29 108 38 79)(30 105 39 80)(31 106 40 77)(32 107 37 78)(49 67 124 113)(50 68 121 114)(51 65 122 115)(52 66 123 116)
(1 5 50 110)(2 8 51 109)(3 7 52 112)(4 6 49 111)(9 54 118 66)(10 53 119 65)(11 56 120 68)(12 55 117 67)(13 58 122 70)(14 57 123 69)(15 60 124 72)(16 59 121 71)(17 26 98 107)(18 25 99 106)(19 28 100 105)(20 27 97 108)(21 30 94 103)(22 29 95 102)(23 32 96 101)(24 31 93 104)(33 48 39 92)(34 47 40 91)(35 46 37 90)(36 45 38 89)(41 84 88 77)(42 83 85 80)(43 82 86 79)(44 81 87 78)(61 127 73 114)(62 126 74 113)(63 125 75 116)(64 128 76 115)
(1 103 50 30)(2 36 51 38)(3 101 52 32)(4 34 49 40)(5 94 110 21)(6 91 111 47)(7 96 112 23)(8 89 109 45)(9 98 118 17)(10 86 119 43)(11 100 120 19)(12 88 117 41)(13 102 122 29)(14 35 123 37)(15 104 124 31)(16 33 121 39)(18 62 99 74)(20 64 97 76)(22 58 95 70)(24 60 93 72)(25 113 106 126)(26 66 107 54)(27 115 108 128)(28 68 105 56)(42 61 85 73)(44 63 87 75)(46 57 90 69)(48 59 92 71)(53 82 65 79)(55 84 67 77)(78 125 81 116)(80 127 83 114)
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), (1,56,16,127)(2,53,13,128)(3,54,14,125)(4,55,15,126)(5,120,59,73)(6,117,60,74)(7,118,57,75)(8,119,58,76)(9,69,63,112)(10,70,64,109)(11,71,61,110)(12,72,62,111)(17,90,44,96)(18,91,41,93)(19,92,42,94)(20,89,43,95)(21,100,48,85)(22,97,45,86)(23,98,46,87)(24,99,47,88)(25,34,84,104)(26,35,81,101)(27,36,82,102)(28,33,83,103)(29,108,38,79)(30,105,39,80)(31,106,40,77)(32,107,37,78)(49,67,124,113)(50,68,121,114)(51,65,122,115)(52,66,123,116), (1,5,50,110)(2,8,51,109)(3,7,52,112)(4,6,49,111)(9,54,118,66)(10,53,119,65)(11,56,120,68)(12,55,117,67)(13,58,122,70)(14,57,123,69)(15,60,124,72)(16,59,121,71)(17,26,98,107)(18,25,99,106)(19,28,100,105)(20,27,97,108)(21,30,94,103)(22,29,95,102)(23,32,96,101)(24,31,93,104)(33,48,39,92)(34,47,40,91)(35,46,37,90)(36,45,38,89)(41,84,88,77)(42,83,85,80)(43,82,86,79)(44,81,87,78)(61,127,73,114)(62,126,74,113)(63,125,75,116)(64,128,76,115), (1,103,50,30)(2,36,51,38)(3,101,52,32)(4,34,49,40)(5,94,110,21)(6,91,111,47)(7,96,112,23)(8,89,109,45)(9,98,118,17)(10,86,119,43)(11,100,120,19)(12,88,117,41)(13,102,122,29)(14,35,123,37)(15,104,124,31)(16,33,121,39)(18,62,99,74)(20,64,97,76)(22,58,95,70)(24,60,93,72)(25,113,106,126)(26,66,107,54)(27,115,108,128)(28,68,105,56)(42,61,85,73)(44,63,87,75)(46,57,90,69)(48,59,92,71)(53,82,65,79)(55,84,67,77)(78,125,81,116)(80,127,83,114)>;
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), (1,56,16,127)(2,53,13,128)(3,54,14,125)(4,55,15,126)(5,120,59,73)(6,117,60,74)(7,118,57,75)(8,119,58,76)(9,69,63,112)(10,70,64,109)(11,71,61,110)(12,72,62,111)(17,90,44,96)(18,91,41,93)(19,92,42,94)(20,89,43,95)(21,100,48,85)(22,97,45,86)(23,98,46,87)(24,99,47,88)(25,34,84,104)(26,35,81,101)(27,36,82,102)(28,33,83,103)(29,108,38,79)(30,105,39,80)(31,106,40,77)(32,107,37,78)(49,67,124,113)(50,68,121,114)(51,65,122,115)(52,66,123,116), (1,5,50,110)(2,8,51,109)(3,7,52,112)(4,6,49,111)(9,54,118,66)(10,53,119,65)(11,56,120,68)(12,55,117,67)(13,58,122,70)(14,57,123,69)(15,60,124,72)(16,59,121,71)(17,26,98,107)(18,25,99,106)(19,28,100,105)(20,27,97,108)(21,30,94,103)(22,29,95,102)(23,32,96,101)(24,31,93,104)(33,48,39,92)(34,47,40,91)(35,46,37,90)(36,45,38,89)(41,84,88,77)(42,83,85,80)(43,82,86,79)(44,81,87,78)(61,127,73,114)(62,126,74,113)(63,125,75,116)(64,128,76,115), (1,103,50,30)(2,36,51,38)(3,101,52,32)(4,34,49,40)(5,94,110,21)(6,91,111,47)(7,96,112,23)(8,89,109,45)(9,98,118,17)(10,86,119,43)(11,100,120,19)(12,88,117,41)(13,102,122,29)(14,35,123,37)(15,104,124,31)(16,33,121,39)(18,62,99,74)(20,64,97,76)(22,58,95,70)(24,60,93,72)(25,113,106,126)(26,66,107,54)(27,115,108,128)(28,68,105,56)(42,61,85,73)(44,63,87,75)(46,57,90,69)(48,59,92,71)(53,82,65,79)(55,84,67,77)(78,125,81,116)(80,127,83,114) );
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)], [(1,56,16,127),(2,53,13,128),(3,54,14,125),(4,55,15,126),(5,120,59,73),(6,117,60,74),(7,118,57,75),(8,119,58,76),(9,69,63,112),(10,70,64,109),(11,71,61,110),(12,72,62,111),(17,90,44,96),(18,91,41,93),(19,92,42,94),(20,89,43,95),(21,100,48,85),(22,97,45,86),(23,98,46,87),(24,99,47,88),(25,34,84,104),(26,35,81,101),(27,36,82,102),(28,33,83,103),(29,108,38,79),(30,105,39,80),(31,106,40,77),(32,107,37,78),(49,67,124,113),(50,68,121,114),(51,65,122,115),(52,66,123,116)], [(1,5,50,110),(2,8,51,109),(3,7,52,112),(4,6,49,111),(9,54,118,66),(10,53,119,65),(11,56,120,68),(12,55,117,67),(13,58,122,70),(14,57,123,69),(15,60,124,72),(16,59,121,71),(17,26,98,107),(18,25,99,106),(19,28,100,105),(20,27,97,108),(21,30,94,103),(22,29,95,102),(23,32,96,101),(24,31,93,104),(33,48,39,92),(34,47,40,91),(35,46,37,90),(36,45,38,89),(41,84,88,77),(42,83,85,80),(43,82,86,79),(44,81,87,78),(61,127,73,114),(62,126,74,113),(63,125,75,116),(64,128,76,115)], [(1,103,50,30),(2,36,51,38),(3,101,52,32),(4,34,49,40),(5,94,110,21),(6,91,111,47),(7,96,112,23),(8,89,109,45),(9,98,118,17),(10,86,119,43),(11,100,120,19),(12,88,117,41),(13,102,122,29),(14,35,123,37),(15,104,124,31),(16,33,121,39),(18,62,99,74),(20,64,97,76),(22,58,95,70),(24,60,93,72),(25,113,106,126),(26,66,107,54),(27,115,108,128),(28,68,105,56),(42,61,85,73),(44,63,87,75),(46,57,90,69),(48,59,92,71),(53,82,65,79),(55,84,67,77),(78,125,81,116),(80,127,83,114)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 |
| kernel | C42⋊5Q8 | C42⋊4C4 | C4×C4⋊C4 | C23.78C23 | C23.81C23 | C2×C4×Q8 | C42 | C42 | C2×C4 |
| # reps | 1 | 1 | 4 | 4 | 4 | 2 | 4 | 8 | 16 |
Matrix representation of C42⋊5Q8 ►in GL6(𝔽5)
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 2 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 3 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 |
G:=sub<GL(6,GF(5))| [0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,2,2],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,3,0,0,0,0,1,1],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,4,0,0,0,0,0,3] >;
C42⋊5Q8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_5Q_8
% in TeX
G:=Group("C4^2:5Q8"); // GroupNames label
G:=SmallGroup(128,1131);
// by ID
G=gap.SmallGroup(128,1131);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,100,192]);
// 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,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations