p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4).24D8, C42⋊9C4⋊6C2, (C2×D4).117D4, (C2×C4).39SD16, C22.88(C2×D8), C2.16(C4⋊D8), C2.7(C4.4D8), (C22×C4).317D4, C23.926(C2×D4), C22.4Q16⋊27C2, (C22×C8).84C22, C4.37(C42⋊2C2), C2.16(D4.D4), (C2×C42).380C22, C22.103(C2×SD16), C2.7(C22.D8), (C22×D4).92C22, C22.247(C4⋊D4), C22.148(C8⋊C22), C22.7C42⋊13C2, (C22×C4).1460C23, C4.26(C22.D4), C2.6(C23.11D4), C22.93(C4.4D4), C2.7(C23.46D4), C22.137(C8.C22), C24.3C22.19C2, C2.9(C42.28C22), C22.116(C22.D4), (C2×C4).1059(C2×D4), (C2×D4⋊C4).17C2, (C2×C4).884(C4○D4), (C2×C4⋊C4).145C22, SmallGroup(128,803)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4).24D8
G = < a,b,c,d | a2=b4=c8=d2=1, dbd=ab=ba, ac=ca, ad=da, cbc-1=ab-1, dcd=ac-1 >
Subgroups: 344 in 137 conjugacy classes, 50 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, D4⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C22.7C42, C22.4Q16, C42⋊9C4, C24.3C22, C2×D4⋊C4, (C2×C4).24D8
Quotients: C1, C2, C22, D4, C23, D8, SD16, C2×D4, C4○D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C2×D8, C2×SD16, C8⋊C22, C8.C22, C23.11D4, C4⋊D8, D4.D4, C22.D8, C23.46D4, C4.4D8, C42.28C22, (C2×C4).24D8
►On 64 points
(1 48)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 61)(18 62)(19 63)(20 64)(21 57)(22 58)(23 59)(24 60)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)
(1 22 51 30)(2 9 52 59)(3 24 53 32)(4 11 54 61)(5 18 55 26)(6 13 56 63)(7 20 49 28)(8 15 50 57)(10 42 60 33)(12 44 62 35)(14 46 64 37)(16 48 58 39)(17 43 25 34)(19 45 27 36)(21 47 29 38)(23 41 31 40)
(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)
(2 47)(3 7)(4 45)(6 43)(8 41)(9 15)(10 28)(11 13)(12 26)(14 32)(16 30)(17 19)(18 62)(20 60)(21 23)(22 58)(24 64)(25 27)(29 31)(33 37)(34 56)(36 54)(38 52)(40 50)(42 46)(49 53)(57 59)(61 63)
G:=sub<Sym(64)| (1,48)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,31)(10,32)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52), (1,22,51,30)(2,9,52,59)(3,24,53,32)(4,11,54,61)(5,18,55,26)(6,13,56,63)(7,20,49,28)(8,15,50,57)(10,42,60,33)(12,44,62,35)(14,46,64,37)(16,48,58,39)(17,43,25,34)(19,45,27,36)(21,47,29,38)(23,41,31,40), (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), (2,47)(3,7)(4,45)(6,43)(8,41)(9,15)(10,28)(11,13)(12,26)(14,32)(16,30)(17,19)(18,62)(20,60)(21,23)(22,58)(24,64)(25,27)(29,31)(33,37)(34,56)(36,54)(38,52)(40,50)(42,46)(49,53)(57,59)(61,63)>;
G:=Group( (1,48)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,31)(10,32)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52), (1,22,51,30)(2,9,52,59)(3,24,53,32)(4,11,54,61)(5,18,55,26)(6,13,56,63)(7,20,49,28)(8,15,50,57)(10,42,60,33)(12,44,62,35)(14,46,64,37)(16,48,58,39)(17,43,25,34)(19,45,27,36)(21,47,29,38)(23,41,31,40), (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), (2,47)(3,7)(4,45)(6,43)(8,41)(9,15)(10,28)(11,13)(12,26)(14,32)(16,30)(17,19)(18,62)(20,60)(21,23)(22,58)(24,64)(25,27)(29,31)(33,37)(34,56)(36,54)(38,52)(40,50)(42,46)(49,53)(57,59)(61,63) );
G=PermutationGroup([[(1,48),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,31),(10,32),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,61),(18,62),(19,63),(20,64),(21,57),(22,58),(23,59),(24,60),(33,53),(34,54),(35,55),(36,56),(37,49),(38,50),(39,51),(40,52)], [(1,22,51,30),(2,9,52,59),(3,24,53,32),(4,11,54,61),(5,18,55,26),(6,13,56,63),(7,20,49,28),(8,15,50,57),(10,42,60,33),(12,44,62,35),(14,46,64,37),(16,48,58,39),(17,43,25,34),(19,45,27,36),(21,47,29,38),(23,41,31,40)], [(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)], [(2,47),(3,7),(4,45),(6,43),(8,41),(9,15),(10,28),(11,13),(12,26),(14,32),(16,30),(17,19),(18,62),(20,60),(21,23),(22,58),(24,64),(25,27),(29,31),(33,37),(34,56),(36,54),(38,52),(40,50),(42,46),(49,53),(57,59),(61,63)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | SD16 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | (C2×C4).24D8 | C22.7C42 | C22.4Q16 | C42⋊9C4 | C24.3C22 | C2×D4⋊C4 | C22×C4 | C2×D4 | C2×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 10 | 1 | 1 |
Matrix representation of (C2×C4).24D8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 8 |
| 0 | 0 | 0 | 0 | 13 | 13 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,13,0,0,0,0,8,13],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,1,0,0,0,0,0,2,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;
(C2×C4).24D8 in GAP, Magma, Sage, TeX
(C_2\times C_4)._{24}D_8 % in TeX
G:=Group("(C2xC4).24D8"); // GroupNames label
G:=SmallGroup(128,803);
// by ID
G=gap.SmallGroup(128,803);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,422,387,58,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,d*b*d=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*c*d=a*c^-1>;
// generators/relations