direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C2×Q32⋊C2, C16.1C23, C8.14C24, Q32⋊3C22, D8.3C23, C23.54D8, SD32⋊2C22, Q16.3C23, M5(2)⋊6C22, (C2×C4).55D8, C4.76(C2×D8), C8.38(C2×D4), (C2×Q32)⋊12C2, (C2×SD32)⋊5C2, (C2×C8).148D4, (C2×M5(2))⋊4C2, C2.29(C22×D8), C4.20(C22×D4), C22.79(C2×D8), (C2×C8).292C23, (C2×C16).34C22, (C22×Q16)⋊21C2, (C2×Q16)⋊56C22, C4○D8.31C22, (C22×C4).533D4, (C2×D8).152C22, (C22×C8).295C22, (C2×C4○D8).20C2, (C2×C4).877(C2×D4), SmallGroup(128,2145)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 372 in 180 conjugacy classes, 92 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×7], Q8 [×13], C23, C23, C16 [×4], C2×C8 [×2], C2×C8 [×4], D8 [×2], D8, SD16 [×4], Q16 [×6], Q16 [×7], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8 [×10], C4○D4 [×6], C2×C16 [×2], M5(2) [×4], SD32 [×8], Q32 [×8], C22×C8, C2×D8, C2×SD16, C2×Q16, C2×Q16 [×6], C2×Q16 [×3], C4○D8 [×4], C4○D8 [×2], C22×Q8, C2×C4○D4, C2×M5(2), C2×SD32 [×2], C2×Q32 [×2], Q32⋊C2 [×8], C22×Q16, C2×C4○D8, C2×Q32⋊C2
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, Q32⋊C2 [×2], C22×D8, C2×Q32⋊C2
Generators and relations
G = < a,b,c,d | a2=b16=d2=1, c2=b8, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, cd=dc >
►On 64 points
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 44)(18 45)(19 46)(20 47)(21 48)(22 33)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)
(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 25 9 17)(2 24 10 32)(3 23 11 31)(4 22 12 30)(5 21 13 29)(6 20 14 28)(7 19 15 27)(8 18 16 26)(33 53 41 61)(34 52 42 60)(35 51 43 59)(36 50 44 58)(37 49 45 57)(38 64 46 56)(39 63 47 55)(40 62 48 54)
(1 32)(2 25)(3 18)(4 27)(5 20)(6 29)(7 22)(8 31)(9 24)(10 17)(11 26)(12 19)(13 28)(14 21)(15 30)(16 23)(33 64)(34 57)(35 50)(36 59)(37 52)(38 61)(39 54)(40 63)(41 56)(42 49)(43 58)(44 51)(45 60)(46 53)(47 62)(48 55)
G:=sub<Sym(64)| (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,44)(18,45)(19,46)(20,47)(21,48)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43), (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), (1,25,9,17)(2,24,10,32)(3,23,11,31)(4,22,12,30)(5,21,13,29)(6,20,14,28)(7,19,15,27)(8,18,16,26)(33,53,41,61)(34,52,42,60)(35,51,43,59)(36,50,44,58)(37,49,45,57)(38,64,46,56)(39,63,47,55)(40,62,48,54), (1,32)(2,25)(3,18)(4,27)(5,20)(6,29)(7,22)(8,31)(9,24)(10,17)(11,26)(12,19)(13,28)(14,21)(15,30)(16,23)(33,64)(34,57)(35,50)(36,59)(37,52)(38,61)(39,54)(40,63)(41,56)(42,49)(43,58)(44,51)(45,60)(46,53)(47,62)(48,55)>;
G:=Group( (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,44)(18,45)(19,46)(20,47)(21,48)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43), (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), (1,25,9,17)(2,24,10,32)(3,23,11,31)(4,22,12,30)(5,21,13,29)(6,20,14,28)(7,19,15,27)(8,18,16,26)(33,53,41,61)(34,52,42,60)(35,51,43,59)(36,50,44,58)(37,49,45,57)(38,64,46,56)(39,63,47,55)(40,62,48,54), (1,32)(2,25)(3,18)(4,27)(5,20)(6,29)(7,22)(8,31)(9,24)(10,17)(11,26)(12,19)(13,28)(14,21)(15,30)(16,23)(33,64)(34,57)(35,50)(36,59)(37,52)(38,61)(39,54)(40,63)(41,56)(42,49)(43,58)(44,51)(45,60)(46,53)(47,62)(48,55) );
G=PermutationGroup([(1,58),(2,59),(3,60),(4,61),(5,62),(6,63),(7,64),(8,49),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,44),(18,45),(19,46),(20,47),(21,48),(22,33),(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43)], [(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)], [(1,25,9,17),(2,24,10,32),(3,23,11,31),(4,22,12,30),(5,21,13,29),(6,20,14,28),(7,19,15,27),(8,18,16,26),(33,53,41,61),(34,52,42,60),(35,51,43,59),(36,50,44,58),(37,49,45,57),(38,64,46,56),(39,63,47,55),(40,62,48,54)], [(1,32),(2,25),(3,18),(4,27),(5,20),(6,29),(7,22),(8,31),(9,24),(10,17),(11,26),(12,19),(13,28),(14,21),(15,30),(16,23),(33,64),(34,57),(35,50),(36,59),(37,52),(38,61),(39,54),(40,63),(41,56),(42,49),(43,58),(44,51),(45,60),(46,53),(47,62),(48,55)])
Matrix representation ►G ⊆ 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 14 | 13 | 0 | 0 | 0 | 0 |
| 11 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 9 | 5 |
| 0 | 0 | 2 | 15 | 12 | 9 |
| 0 | 0 | 4 | 8 | 2 | 10 |
| 0 | 0 | 8 | 4 | 15 | 14 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 15 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 3 | 1 | 12 |
| 0 | 0 | 2 | 2 | 12 | 16 |
| 0 | 0 | 9 | 4 | 15 | 3 |
| 0 | 0 | 13 | 8 | 2 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 | 0 |
| 0 | 0 | 1 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 16 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[14,11,0,0,0,0,13,3,0,0,0,0,0,0,3,2,4,8,0,0,7,15,8,4,0,0,9,12,2,15,0,0,5,9,10,14],[3,15,0,0,0,0,4,14,0,0,0,0,0,0,10,2,9,13,0,0,3,2,4,8,0,0,1,12,15,2,0,0,12,16,3,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,16,0,0,0,2,16,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | D8 | Q32⋊C2 |
| kernel | C2×Q32⋊C2 | C2×M5(2) | C2×SD32 | C2×Q32 | Q32⋊C2 | C22×Q16 | C2×C4○D8 | C2×C8 | C22×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 3 | 1 | 6 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2\times Q_{32}\rtimes C_2 % in TeX
G:=Group("C2xQ32:C2"); // GroupNames label
G:=SmallGroup(128,2145);
// by ID
G=gap.SmallGroup(128,2145);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,456,1430,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^16=d^2=1,c^2=b^8,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^9,c*d=d*c>;
// generators/relations