p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.183D4, C24.350C23, C23.499C24, C22.2052- 1+4, C42⋊5C4⋊23C2, C23⋊Q8.12C2, C23.160(C4○D4), (C23×C4).411C22, (C22×C4).121C23, (C2×C42).586C22, C22.329(C22×D4), C23.8Q8.39C2, C23.11D4.24C2, (C22×Q8).444C22, C23.81C23⋊51C2, C23.78C23⋊21C2, C2.72(C22.19C24), C24.C22.40C2, C23.63C23⋊103C2, C2.C42.229C22, C2.28(C23.38C23), C2.50(C22.50C24), C2.73(C22.46C24), (C2×C4×Q8)⋊27C2, (C2×C4).368(C2×D4), (C2×C4).408(C4○D4), (C2×C4⋊C4).339C22, C22.375(C2×C4○D4), (C2×C42⋊C2).43C2, (C2×C22⋊C4).514C22, SmallGroup(128,1331)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.183D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=a2c-1 >
Subgroups: 420 in 246 conjugacy classes, 100 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×Q8, C23×C4, C22×Q8, C42⋊5C4, C23.8Q8, C23.63C23, C24.C22, C23⋊Q8, C23.78C23, C23.11D4, C23.81C23, C2×C42⋊C2, C2×C4×Q8, C42.183D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, C22.19C24, C23.38C23, C22.46C24, C22.50C24, C42.183D4
►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 49 37 10)(2 50 38 11)(3 51 39 12)(4 52 40 9)(5 19 45 42)(6 20 46 43)(7 17 47 44)(8 18 48 41)(13 33 29 27)(14 34 30 28)(15 35 31 25)(16 36 32 26)(21 62 55 60)(22 63 56 57)(23 64 53 58)(24 61 54 59)
(1 59 15 43)(2 64 16 19)(3 57 13 41)(4 62 14 17)(5 9 23 28)(6 51 24 33)(7 11 21 26)(8 49 22 35)(10 56 25 48)(12 54 27 46)(18 39 63 29)(20 37 61 31)(30 44 40 60)(32 42 38 58)(34 45 52 53)(36 47 50 55)
(1 41 3 43)(2 44 4 42)(5 52 7 50)(6 51 8 49)(9 47 11 45)(10 46 12 48)(13 59 15 57)(14 58 16 60)(17 40 19 38)(18 39 20 37)(21 36 23 34)(22 35 24 33)(25 54 27 56)(26 53 28 55)(29 61 31 63)(30 64 32 62)
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), (1,49,37,10)(2,50,38,11)(3,51,39,12)(4,52,40,9)(5,19,45,42)(6,20,46,43)(7,17,47,44)(8,18,48,41)(13,33,29,27)(14,34,30,28)(15,35,31,25)(16,36,32,26)(21,62,55,60)(22,63,56,57)(23,64,53,58)(24,61,54,59), (1,59,15,43)(2,64,16,19)(3,57,13,41)(4,62,14,17)(5,9,23,28)(6,51,24,33)(7,11,21,26)(8,49,22,35)(10,56,25,48)(12,54,27,46)(18,39,63,29)(20,37,61,31)(30,44,40,60)(32,42,38,58)(34,45,52,53)(36,47,50,55), (1,41,3,43)(2,44,4,42)(5,52,7,50)(6,51,8,49)(9,47,11,45)(10,46,12,48)(13,59,15,57)(14,58,16,60)(17,40,19,38)(18,39,20,37)(21,36,23,34)(22,35,24,33)(25,54,27,56)(26,53,28,55)(29,61,31,63)(30,64,32,62)>;
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), (1,49,37,10)(2,50,38,11)(3,51,39,12)(4,52,40,9)(5,19,45,42)(6,20,46,43)(7,17,47,44)(8,18,48,41)(13,33,29,27)(14,34,30,28)(15,35,31,25)(16,36,32,26)(21,62,55,60)(22,63,56,57)(23,64,53,58)(24,61,54,59), (1,59,15,43)(2,64,16,19)(3,57,13,41)(4,62,14,17)(5,9,23,28)(6,51,24,33)(7,11,21,26)(8,49,22,35)(10,56,25,48)(12,54,27,46)(18,39,63,29)(20,37,61,31)(30,44,40,60)(32,42,38,58)(34,45,52,53)(36,47,50,55), (1,41,3,43)(2,44,4,42)(5,52,7,50)(6,51,8,49)(9,47,11,45)(10,46,12,48)(13,59,15,57)(14,58,16,60)(17,40,19,38)(18,39,20,37)(21,36,23,34)(22,35,24,33)(25,54,27,56)(26,53,28,55)(29,61,31,63)(30,64,32,62) );
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)], [(1,49,37,10),(2,50,38,11),(3,51,39,12),(4,52,40,9),(5,19,45,42),(6,20,46,43),(7,17,47,44),(8,18,48,41),(13,33,29,27),(14,34,30,28),(15,35,31,25),(16,36,32,26),(21,62,55,60),(22,63,56,57),(23,64,53,58),(24,61,54,59)], [(1,59,15,43),(2,64,16,19),(3,57,13,41),(4,62,14,17),(5,9,23,28),(6,51,24,33),(7,11,21,26),(8,49,22,35),(10,56,25,48),(12,54,27,46),(18,39,63,29),(20,37,61,31),(30,44,40,60),(32,42,38,58),(34,45,52,53),(36,47,50,55)], [(1,41,3,43),(2,44,4,42),(5,52,7,50),(6,51,8,49),(9,47,11,45),(10,46,12,48),(13,59,15,57),(14,58,16,60),(17,40,19,38),(18,39,20,37),(21,36,23,34),(22,35,24,33),(25,54,27,56),(26,53,28,55),(29,61,31,63),(30,64,32,62)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4X | 4Y | 4Z | 4AA | 4AB |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | C4○D4 | 2- 1+4 |
| kernel | C42.183D4 | C42⋊5C4 | C23.8Q8 | C23.63C23 | C24.C22 | C23⋊Q8 | C23.78C23 | C23.11D4 | C23.81C23 | C2×C42⋊C2 | C2×C4×Q8 | C42 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 2 | 4 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 12 | 4 | 2 |
Matrix representation of C42.183D4 ►in GL6(𝔽5)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,1,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,2,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,4,0,0,0,0,3,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,3,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0] >;
C42.183D4 in GAP, Magma, Sage, TeX
C_4^2._{183}D_4 % in TeX
G:=Group("C4^2.183D4"); // GroupNames label
G:=SmallGroup(128,1331);
// by ID
G=gap.SmallGroup(128,1331);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,675,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^-1>;
// generators/relations