p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.178D4, C23.480C24, C24.344C23, C22.2632+ 1+4, (C2×D4)⋊19Q8, C42⋊8C4⋊48C2, C23.28(C2×Q8), C23⋊Q8⋊23C2, C2.40(D4⋊3Q8), C4.103(C22⋊Q8), C23.4Q8⋊24C2, C23.8Q8⋊73C2, (C2×C42).574C22, (C22×C4).110C23, (C23×C4).125C22, C22.321(C22×D4), C22.115(C22×Q8), (C22×D4).536C22, (C22×Q8).439C22, C23.65C23⋊91C2, C2.27(C22.29C24), C2.65(C22.19C24), C24.3C22.50C2, C2.C42.214C22, C2.22(C22.53C24), (C2×C4×Q8)⋊25C2, (C2×C4×D4).67C2, (C2×C4).312(C2×Q8), C2.38(C2×C22⋊Q8), (C2×C4).1196(C2×D4), (C2×C4).155(C4○D4), (C2×C4⋊C4).326C22, C22.356(C2×C4○D4), (C2×C22⋊C4).195C22, SmallGroup(128,1312)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.178D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=dbd=b-1, dcd=a2c-1 >
Subgroups: 532 in 282 conjugacy classes, 112 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×Q8, C23×C4, C22×D4, C22×Q8, C42⋊8C4, C23.8Q8, C23.65C23, C24.3C22, C23⋊Q8, C23.4Q8, C2×C4×D4, C2×C4×Q8, C42.178D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, C2×C22⋊Q8, C22.19C24, C22.29C24, D4⋊3Q8, C22.53C24, C42.178D4
►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 12 21 43)(2 9 22 44)(3 10 23 41)(4 11 24 42)(5 56 40 26)(6 53 37 27)(7 54 38 28)(8 55 39 25)(13 64 48 36)(14 61 45 33)(15 62 46 34)(16 63 47 35)(17 30 52 60)(18 31 49 57)(19 32 50 58)(20 29 51 59)
(1 61 37 51)(2 36 38 19)(3 63 39 49)(4 34 40 17)(5 52 24 62)(6 20 21 33)(7 50 22 64)(8 18 23 35)(9 48 28 58)(10 16 25 31)(11 46 26 60)(12 14 27 29)(13 54 32 44)(15 56 30 42)(41 47 55 57)(43 45 53 59)
(1 55)(2 26)(3 53)(4 28)(5 44)(6 10)(7 42)(8 12)(9 40)(11 38)(13 64)(14 33)(15 62)(16 35)(17 60)(18 31)(19 58)(20 29)(21 25)(22 56)(23 27)(24 54)(30 52)(32 50)(34 46)(36 48)(37 41)(39 43)(45 61)(47 63)(49 57)(51 59)
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), (1,12,21,43)(2,9,22,44)(3,10,23,41)(4,11,24,42)(5,56,40,26)(6,53,37,27)(7,54,38,28)(8,55,39,25)(13,64,48,36)(14,61,45,33)(15,62,46,34)(16,63,47,35)(17,30,52,60)(18,31,49,57)(19,32,50,58)(20,29,51,59), (1,61,37,51)(2,36,38,19)(3,63,39,49)(4,34,40,17)(5,52,24,62)(6,20,21,33)(7,50,22,64)(8,18,23,35)(9,48,28,58)(10,16,25,31)(11,46,26,60)(12,14,27,29)(13,54,32,44)(15,56,30,42)(41,47,55,57)(43,45,53,59), (1,55)(2,26)(3,53)(4,28)(5,44)(6,10)(7,42)(8,12)(9,40)(11,38)(13,64)(14,33)(15,62)(16,35)(17,60)(18,31)(19,58)(20,29)(21,25)(22,56)(23,27)(24,54)(30,52)(32,50)(34,46)(36,48)(37,41)(39,43)(45,61)(47,63)(49,57)(51,59)>;
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), (1,12,21,43)(2,9,22,44)(3,10,23,41)(4,11,24,42)(5,56,40,26)(6,53,37,27)(7,54,38,28)(8,55,39,25)(13,64,48,36)(14,61,45,33)(15,62,46,34)(16,63,47,35)(17,30,52,60)(18,31,49,57)(19,32,50,58)(20,29,51,59), (1,61,37,51)(2,36,38,19)(3,63,39,49)(4,34,40,17)(5,52,24,62)(6,20,21,33)(7,50,22,64)(8,18,23,35)(9,48,28,58)(10,16,25,31)(11,46,26,60)(12,14,27,29)(13,54,32,44)(15,56,30,42)(41,47,55,57)(43,45,53,59), (1,55)(2,26)(3,53)(4,28)(5,44)(6,10)(7,42)(8,12)(9,40)(11,38)(13,64)(14,33)(15,62)(16,35)(17,60)(18,31)(19,58)(20,29)(21,25)(22,56)(23,27)(24,54)(30,52)(32,50)(34,46)(36,48)(37,41)(39,43)(45,61)(47,63)(49,57)(51,59) );
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)], [(1,12,21,43),(2,9,22,44),(3,10,23,41),(4,11,24,42),(5,56,40,26),(6,53,37,27),(7,54,38,28),(8,55,39,25),(13,64,48,36),(14,61,45,33),(15,62,46,34),(16,63,47,35),(17,30,52,60),(18,31,49,57),(19,32,50,58),(20,29,51,59)], [(1,61,37,51),(2,36,38,19),(3,63,39,49),(4,34,40,17),(5,52,24,62),(6,20,21,33),(7,50,22,64),(8,18,23,35),(9,48,28,58),(10,16,25,31),(11,46,26,60),(12,14,27,29),(13,54,32,44),(15,56,30,42),(41,47,55,57),(43,45,53,59)], [(1,55),(2,26),(3,53),(4,28),(5,44),(6,10),(7,42),(8,12),(9,40),(11,38),(13,64),(14,33),(15,62),(16,35),(17,60),(18,31),(19,58),(20,29),(21,25),(22,56),(23,27),(24,54),(30,52),(32,50),(34,46),(36,48),(37,41),(39,43),(45,61),(47,63),(49,57),(51,59)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4V | 4W | 4X | 4Y | 4Z |
| 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 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | 2+ 1+4 |
| kernel | C42.178D4 | C42⋊8C4 | C23.8Q8 | C23.65C23 | C24.3C22 | C23⋊Q8 | C23.4Q8 | C2×C4×D4 | C2×C4×Q8 | C42 | C2×D4 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 4 | 12 | 2 |
Matrix representation of C42.178D4 ►in GL6(𝔽5)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 3 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 |
G:=sub<GL(6,GF(5))| [0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,2,0,0,0,0,1,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,2,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,0,0,0,0,1,3,0,0,0,0,1,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,4,0,0,0,0,0,0,1,3,0,0,0,0,0,4] >;
C42.178D4 in GAP, Magma, Sage, TeX
C_4^2._{178}D_4 % in TeX
G:=Group("C4^2.178D4"); // GroupNames label
G:=SmallGroup(128,1312);
// by ID
G=gap.SmallGroup(128,1312);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,456,758,723,352,675,80]);
// 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*b^2,c*b*c^-1=d*b*d=b^-1,d*c*d=a^2*c^-1>;
// generators/relations