p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.190D4, C23.533C24, C24.371C23, C22.2272- 1+4, C22.3102+ 1+4, C42⋊5C4⋊26C2, C23.66(C4○D4), C23.8Q8⋊86C2, C23.4Q8⋊30C2, C23.11D4⋊61C2, (C22×C4).143C23, (C23×C4).138C22, (C2×C42).610C22, C22.358(C22×D4), C23.10D4.32C2, (C22×D4).542C22, C23.81C23⋊63C2, C2.84(C22.19C24), C2.42(C22.29C24), C2.C42.258C22, C2.40(C22.33C24), C2.41(C23.38C23), (C2×C4×D4).68C2, (C2×C4).392(C2×D4), (C2×C42.C2)⋊17C2, (C2×C4⋊C4).360C22, C22.405(C2×C4○D4), (C2×C22⋊C4).222C22, SmallGroup(128,1365)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.190D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=a-1, cbc-1=a2b, bd=db, dcd=a2c-1 >
Subgroups: 500 in 256 conjugacy classes, 96 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C42.C2, C23×C4, C22×D4, C42⋊5C4, C23.8Q8, C23.10D4, C23.11D4, C23.81C23, C23.4Q8, C2×C4×D4, C2×C42.C2, C42.190D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.19C24, C22.29C24, C23.38C23, C22.33C24, C42.190D4
►On 64 points
(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)
(1 23 31 45)(2 24 32 46)(3 21 29 47)(4 22 30 48)(5 52 36 18)(6 49 33 19)(7 50 34 20)(8 51 35 17)(9 63 55 39)(10 64 56 40)(11 61 53 37)(12 62 54 38)(13 28 42 60)(14 25 43 57)(15 26 44 58)(16 27 41 59)
(1 62 14 50)(2 37 15 19)(3 64 16 52)(4 39 13 17)(5 45 10 57)(6 22 11 28)(7 47 12 59)(8 24 9 26)(18 29 40 41)(20 31 38 43)(21 54 27 34)(23 56 25 36)(30 63 42 51)(32 61 44 49)(33 48 53 60)(35 46 55 58)
(1 13)(2 16)(3 15)(4 14)(5 35)(6 34)(7 33)(8 36)(9 56)(10 55)(11 54)(12 53)(17 52)(18 51)(19 50)(20 49)(21 26)(22 25)(23 28)(24 27)(29 44)(30 43)(31 42)(32 41)(37 62)(38 61)(39 64)(40 63)(45 60)(46 59)(47 58)(48 57)
G:=sub<Sym(64)| (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), (1,23,31,45)(2,24,32,46)(3,21,29,47)(4,22,30,48)(5,52,36,18)(6,49,33,19)(7,50,34,20)(8,51,35,17)(9,63,55,39)(10,64,56,40)(11,61,53,37)(12,62,54,38)(13,28,42,60)(14,25,43,57)(15,26,44,58)(16,27,41,59), (1,62,14,50)(2,37,15,19)(3,64,16,52)(4,39,13,17)(5,45,10,57)(6,22,11,28)(7,47,12,59)(8,24,9,26)(18,29,40,41)(20,31,38,43)(21,54,27,34)(23,56,25,36)(30,63,42,51)(32,61,44,49)(33,48,53,60)(35,46,55,58), (1,13)(2,16)(3,15)(4,14)(5,35)(6,34)(7,33)(8,36)(9,56)(10,55)(11,54)(12,53)(17,52)(18,51)(19,50)(20,49)(21,26)(22,25)(23,28)(24,27)(29,44)(30,43)(31,42)(32,41)(37,62)(38,61)(39,64)(40,63)(45,60)(46,59)(47,58)(48,57)>;
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), (1,23,31,45)(2,24,32,46)(3,21,29,47)(4,22,30,48)(5,52,36,18)(6,49,33,19)(7,50,34,20)(8,51,35,17)(9,63,55,39)(10,64,56,40)(11,61,53,37)(12,62,54,38)(13,28,42,60)(14,25,43,57)(15,26,44,58)(16,27,41,59), (1,62,14,50)(2,37,15,19)(3,64,16,52)(4,39,13,17)(5,45,10,57)(6,22,11,28)(7,47,12,59)(8,24,9,26)(18,29,40,41)(20,31,38,43)(21,54,27,34)(23,56,25,36)(30,63,42,51)(32,61,44,49)(33,48,53,60)(35,46,55,58), (1,13)(2,16)(3,15)(4,14)(5,35)(6,34)(7,33)(8,36)(9,56)(10,55)(11,54)(12,53)(17,52)(18,51)(19,50)(20,49)(21,26)(22,25)(23,28)(24,27)(29,44)(30,43)(31,42)(32,41)(37,62)(38,61)(39,64)(40,63)(45,60)(46,59)(47,58)(48,57) );
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)], [(1,23,31,45),(2,24,32,46),(3,21,29,47),(4,22,30,48),(5,52,36,18),(6,49,33,19),(7,50,34,20),(8,51,35,17),(9,63,55,39),(10,64,56,40),(11,61,53,37),(12,62,54,38),(13,28,42,60),(14,25,43,57),(15,26,44,58),(16,27,41,59)], [(1,62,14,50),(2,37,15,19),(3,64,16,52),(4,39,13,17),(5,45,10,57),(6,22,11,28),(7,47,12,59),(8,24,9,26),(18,29,40,41),(20,31,38,43),(21,54,27,34),(23,56,25,36),(30,63,42,51),(32,61,44,49),(33,48,53,60),(35,46,55,58)], [(1,13),(2,16),(3,15),(4,14),(5,35),(6,34),(7,33),(8,36),(9,56),(10,55),(11,54),(12,53),(17,52),(18,51),(19,50),(20,49),(21,26),(22,25),(23,28),(24,27),(29,44),(30,43),(31,42),(32,41),(37,62),(38,61),(39,64),(40,63),(45,60),(46,59),(47,58),(48,57)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4T |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42.190D4 | C42⋊5C4 | C23.8Q8 | C23.10D4 | C23.11D4 | C23.81C23 | C23.4Q8 | C2×C4×D4 | C2×C42.C2 | C42 | C23 | C22 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 8 | 2 | 2 |
Matrix representation of C42.190D4 ►in GL8(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 2 |
| 2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,1,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,4,2],[2,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,1],[1,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0] >;
C42.190D4 in GAP, Magma, Sage, TeX
C_4^2._{190}D_4 % in TeX
G:=Group("C4^2.190D4"); // GroupNames label
G:=SmallGroup(128,1365);
// by ID
G=gap.SmallGroup(128,1365);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,100,185,136]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=a^2*c^-1>;
// generators/relations