p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊14Q8, C43.12C2, C23.177C24, (C4×Q8)⋊17C4, C4.49(C4×Q8), C42.175(C2×C4), C42⋊4C4.11C2, C22.68(C23×C4), C4.13(C42⋊C2), C22.25(C22×Q8), C4○(C23.67C23), (C2×C42).1005C22, (C22×C4).1239C23, (C22×Q8).390C22, C4○2(C23.65C23), C4○2(C23.63C23), C2.C42.512C22, C23.65C23.93C2, C23.63C23.66C2, C23.67C23.67C2, C2.2(C23.37C23), C2.4(C23.36C23), C2.7(C2×C4×Q8), C2.7(C4×C4○D4), (C4×C4⋊C4).31C2, (C2×C4×Q8).19C2, C4⋊C4.199(C2×C4), (C2×C4).160(C2×Q8), (C2×Q8).191(C2×C4), C22.69(C2×C4○D4), (C2×C4).636(C4○D4), (C2×C4⋊C4).792C22, (C2×C4).210(C22×C4), C2.17(C2×C42⋊C2), SmallGroup(128,1027)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊14Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 348 in 250 conjugacy classes, 160 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22×Q8, C43, C42⋊4C4, C4×C4⋊C4, C4×C4⋊C4, C23.63C23, C23.65C23, C23.67C23, C2×C4×Q8, C42⋊14Q8
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, C24, C42⋊C2, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, C2×C42⋊C2, C2×C4×Q8, C4×C4○D4, C23.36C23, C23.37C23, C42⋊14Q8
►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 27 23 55)(2 28 24 56)(3 25 21 53)(4 26 22 54)(5 41 37 9)(6 42 38 10)(7 43 39 11)(8 44 40 12)(13 49 45 17)(14 50 46 18)(15 51 47 19)(16 52 48 20)(29 61 57 69)(30 62 58 70)(31 63 59 71)(32 64 60 72)(33 89 125 121)(34 90 126 122)(35 91 127 123)(36 92 128 124)(65 74 78 102)(66 75 79 103)(67 76 80 104)(68 73 77 101)(81 117 113 93)(82 118 114 94)(83 119 115 95)(84 120 116 96)(85 109 105 98)(86 110 106 99)(87 111 107 100)(88 112 108 97)
(1 13 5 59)(2 14 6 60)(3 15 7 57)(4 16 8 58)(9 63 55 17)(10 64 56 18)(11 61 53 19)(12 62 54 20)(21 47 39 29)(22 48 40 30)(23 45 37 31)(24 46 38 32)(25 51 43 69)(26 52 44 70)(27 49 41 71)(28 50 42 72)(33 75 100 93)(34 76 97 94)(35 73 98 95)(36 74 99 96)(65 106 116 124)(66 107 113 121)(67 108 114 122)(68 105 115 123)(77 85 83 91)(78 86 84 92)(79 87 81 89)(80 88 82 90)(101 109 119 127)(102 110 120 128)(103 111 117 125)(104 112 118 126)
(1 104 5 118)(2 73 6 95)(3 102 7 120)(4 75 8 93)(9 82 55 80)(10 115 56 68)(11 84 53 78)(12 113 54 66)(13 126 59 112)(14 35 60 98)(15 128 57 110)(16 33 58 100)(17 90 63 88)(18 123 64 105)(19 92 61 86)(20 121 62 107)(21 74 39 96)(22 103 40 117)(23 76 37 94)(24 101 38 119)(25 65 43 116)(26 79 44 81)(27 67 41 114)(28 77 42 83)(29 99 47 36)(30 111 48 125)(31 97 45 34)(32 109 46 127)(49 122 71 108)(50 91 72 85)(51 124 69 106)(52 89 70 87)
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), (1,27,23,55)(2,28,24,56)(3,25,21,53)(4,26,22,54)(5,41,37,9)(6,42,38,10)(7,43,39,11)(8,44,40,12)(13,49,45,17)(14,50,46,18)(15,51,47,19)(16,52,48,20)(29,61,57,69)(30,62,58,70)(31,63,59,71)(32,64,60,72)(33,89,125,121)(34,90,126,122)(35,91,127,123)(36,92,128,124)(65,74,78,102)(66,75,79,103)(67,76,80,104)(68,73,77,101)(81,117,113,93)(82,118,114,94)(83,119,115,95)(84,120,116,96)(85,109,105,98)(86,110,106,99)(87,111,107,100)(88,112,108,97), (1,13,5,59)(2,14,6,60)(3,15,7,57)(4,16,8,58)(9,63,55,17)(10,64,56,18)(11,61,53,19)(12,62,54,20)(21,47,39,29)(22,48,40,30)(23,45,37,31)(24,46,38,32)(25,51,43,69)(26,52,44,70)(27,49,41,71)(28,50,42,72)(33,75,100,93)(34,76,97,94)(35,73,98,95)(36,74,99,96)(65,106,116,124)(66,107,113,121)(67,108,114,122)(68,105,115,123)(77,85,83,91)(78,86,84,92)(79,87,81,89)(80,88,82,90)(101,109,119,127)(102,110,120,128)(103,111,117,125)(104,112,118,126), (1,104,5,118)(2,73,6,95)(3,102,7,120)(4,75,8,93)(9,82,55,80)(10,115,56,68)(11,84,53,78)(12,113,54,66)(13,126,59,112)(14,35,60,98)(15,128,57,110)(16,33,58,100)(17,90,63,88)(18,123,64,105)(19,92,61,86)(20,121,62,107)(21,74,39,96)(22,103,40,117)(23,76,37,94)(24,101,38,119)(25,65,43,116)(26,79,44,81)(27,67,41,114)(28,77,42,83)(29,99,47,36)(30,111,48,125)(31,97,45,34)(32,109,46,127)(49,122,71,108)(50,91,72,85)(51,124,69,106)(52,89,70,87)>;
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), (1,27,23,55)(2,28,24,56)(3,25,21,53)(4,26,22,54)(5,41,37,9)(6,42,38,10)(7,43,39,11)(8,44,40,12)(13,49,45,17)(14,50,46,18)(15,51,47,19)(16,52,48,20)(29,61,57,69)(30,62,58,70)(31,63,59,71)(32,64,60,72)(33,89,125,121)(34,90,126,122)(35,91,127,123)(36,92,128,124)(65,74,78,102)(66,75,79,103)(67,76,80,104)(68,73,77,101)(81,117,113,93)(82,118,114,94)(83,119,115,95)(84,120,116,96)(85,109,105,98)(86,110,106,99)(87,111,107,100)(88,112,108,97), (1,13,5,59)(2,14,6,60)(3,15,7,57)(4,16,8,58)(9,63,55,17)(10,64,56,18)(11,61,53,19)(12,62,54,20)(21,47,39,29)(22,48,40,30)(23,45,37,31)(24,46,38,32)(25,51,43,69)(26,52,44,70)(27,49,41,71)(28,50,42,72)(33,75,100,93)(34,76,97,94)(35,73,98,95)(36,74,99,96)(65,106,116,124)(66,107,113,121)(67,108,114,122)(68,105,115,123)(77,85,83,91)(78,86,84,92)(79,87,81,89)(80,88,82,90)(101,109,119,127)(102,110,120,128)(103,111,117,125)(104,112,118,126), (1,104,5,118)(2,73,6,95)(3,102,7,120)(4,75,8,93)(9,82,55,80)(10,115,56,68)(11,84,53,78)(12,113,54,66)(13,126,59,112)(14,35,60,98)(15,128,57,110)(16,33,58,100)(17,90,63,88)(18,123,64,105)(19,92,61,86)(20,121,62,107)(21,74,39,96)(22,103,40,117)(23,76,37,94)(24,101,38,119)(25,65,43,116)(26,79,44,81)(27,67,41,114)(28,77,42,83)(29,99,47,36)(30,111,48,125)(31,97,45,34)(32,109,46,127)(49,122,71,108)(50,91,72,85)(51,124,69,106)(52,89,70,87) );
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)], [(1,27,23,55),(2,28,24,56),(3,25,21,53),(4,26,22,54),(5,41,37,9),(6,42,38,10),(7,43,39,11),(8,44,40,12),(13,49,45,17),(14,50,46,18),(15,51,47,19),(16,52,48,20),(29,61,57,69),(30,62,58,70),(31,63,59,71),(32,64,60,72),(33,89,125,121),(34,90,126,122),(35,91,127,123),(36,92,128,124),(65,74,78,102),(66,75,79,103),(67,76,80,104),(68,73,77,101),(81,117,113,93),(82,118,114,94),(83,119,115,95),(84,120,116,96),(85,109,105,98),(86,110,106,99),(87,111,107,100),(88,112,108,97)], [(1,13,5,59),(2,14,6,60),(3,15,7,57),(4,16,8,58),(9,63,55,17),(10,64,56,18),(11,61,53,19),(12,62,54,20),(21,47,39,29),(22,48,40,30),(23,45,37,31),(24,46,38,32),(25,51,43,69),(26,52,44,70),(27,49,41,71),(28,50,42,72),(33,75,100,93),(34,76,97,94),(35,73,98,95),(36,74,99,96),(65,106,116,124),(66,107,113,121),(67,108,114,122),(68,105,115,123),(77,85,83,91),(78,86,84,92),(79,87,81,89),(80,88,82,90),(101,109,119,127),(102,110,120,128),(103,111,117,125),(104,112,118,126)], [(1,104,5,118),(2,73,6,95),(3,102,7,120),(4,75,8,93),(9,82,55,80),(10,115,56,68),(11,84,53,78),(12,113,54,66),(13,126,59,112),(14,35,60,98),(15,128,57,110),(16,33,58,100),(17,90,63,88),(18,123,64,105),(19,92,61,86),(20,121,62,107),(21,74,39,96),(22,103,40,117),(23,76,37,94),(24,101,38,119),(25,65,43,116),(26,79,44,81),(27,67,41,114),(28,77,42,83),(29,99,47,36),(30,111,48,125),(31,97,45,34),(32,109,46,127),(49,122,71,108),(50,91,72,85),(51,124,69,106),(52,89,70,87)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AF | 4AG | ··· | 4AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | C4○D4 |
| kernel | C42⋊14Q8 | C43 | C42⋊4C4 | C4×C4⋊C4 | C23.63C23 | C23.65C23 | C23.67C23 | C2×C4×Q8 | C4×Q8 | C42 | C2×C4 |
| # reps | 1 | 1 | 2 | 3 | 4 | 2 | 2 | 1 | 16 | 4 | 20 |
Matrix representation of C42⋊14Q8 ►in GL5(𝔽5)
| 3 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 3 | 0 |
G:=sub<GL(5,GF(5))| [3,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,2],[1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,3],[4,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,3,0] >;
C42⋊14Q8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{14}Q_8 % in TeX
G:=Group("C4^2:14Q8"); // GroupNames label
G:=SmallGroup(128,1027);
// by ID
G=gap.SmallGroup(128,1027);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,680,758,184,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations