direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C22⋊C16, C23⋊2C16, C24.4C8, C22.9M5(2), C4○(C22⋊C16), C8○(C22⋊C16), (C22×C16)⋊1C2, C22⋊2(C2×C16), C8.130(C2×D4), (C2×C8).402D4, (C23×C8).9C2, (C2×C16)⋊14C22, C2.1(C22×C16), (C23×C4).31C4, C23.35(C2×C8), (C22×C4).11C8, (C22×C8).29C4, C2.3(C2×M5(2)), C8.62(C22⋊C4), C4.37(C22⋊C8), (C2×C8).623C23, C4.61(C2×M4(2)), (C2×C4).92M4(2), C22.26(C22×C8), C22.43(C22⋊C8), (C22×C8).592C22, (C2×C4).64(C2×C8), C2.3(C2×C22⋊C8), (C2×C8)○(C22⋊C16), (C2×C4)○(C22⋊C16), (C2×C8).214(C2×C4), C4.112(C2×C22⋊C4), (C2×C4).608(C22×C4), (C22×C4).445(C2×C4), (C2×C4).402(C22⋊C4), SmallGroup(128,843)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C22⋊C16
G = < a,b,c,d | a2=b2=c2=d16=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 196 in 136 conjugacy classes, 76 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C16, C2×C8, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C2×C16, C2×C16, C22×C8, C22×C8, C22×C8, C23×C4, C22⋊C16, C22×C16, C23×C8, C2×C22⋊C16
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C16, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C2×C16, M5(2), C2×C22⋊C4, C22×C8, C2×M4(2), C22⋊C16, C2×C22⋊C8, C22×C16, C2×M5(2), C2×C22⋊C16
►On 64 points
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(33 62)(34 63)(35 64)(36 49)(37 50)(38 51)(39 52)(40 53)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 61)
(1 9)(2 39)(3 11)(4 41)(5 13)(6 43)(7 15)(8 45)(10 47)(12 33)(14 35)(16 37)(17 60)(18 26)(19 62)(20 28)(21 64)(22 30)(23 50)(24 32)(25 52)(27 54)(29 56)(31 58)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)
(1 46)(2 47)(3 48)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 52)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 49)(31 50)(32 51)
(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)
G:=sub<Sym(64)| (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(33,62)(34,63)(35,64)(36,49)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,61), (1,9)(2,39)(3,11)(4,41)(5,13)(6,43)(7,15)(8,45)(10,47)(12,33)(14,35)(16,37)(17,60)(18,26)(19,62)(20,28)(21,64)(22,30)(23,50)(24,32)(25,52)(27,54)(29,56)(31,58)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63), (1,46)(2,47)(3,48)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,49)(31,50)(32,51), (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)>;
G:=Group( (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(33,62)(34,63)(35,64)(36,49)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,61), (1,9)(2,39)(3,11)(4,41)(5,13)(6,43)(7,15)(8,45)(10,47)(12,33)(14,35)(16,37)(17,60)(18,26)(19,62)(20,28)(21,64)(22,30)(23,50)(24,32)(25,52)(27,54)(29,56)(31,58)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63), (1,46)(2,47)(3,48)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,49)(31,50)(32,51), (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) );
G=PermutationGroup([[(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23),(33,62),(34,63),(35,64),(36,49),(37,50),(38,51),(39,52),(40,53),(41,54),(42,55),(43,56),(44,57),(45,58),(46,59),(47,60),(48,61)], [(1,9),(2,39),(3,11),(4,41),(5,13),(6,43),(7,15),(8,45),(10,47),(12,33),(14,35),(16,37),(17,60),(18,26),(19,62),(20,28),(21,64),(22,30),(23,50),(24,32),(25,52),(27,54),(29,56),(31,58),(34,42),(36,44),(38,46),(40,48),(49,57),(51,59),(53,61),(55,63)], [(1,46),(2,47),(3,48),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,43),(15,44),(16,45),(17,52),(18,53),(19,54),(20,55),(21,56),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,64),(30,49),(31,50),(32,51)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8P | 8Q | ··· | 8X | 16A | ··· | 16AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D4 | M4(2) | M5(2) |
| kernel | C2×C22⋊C16 | C22⋊C16 | C22×C16 | C23×C8 | C22×C8 | C23×C4 | C22×C4 | C24 | C23 | C2×C8 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 6 | 2 | 12 | 4 | 32 | 4 | 4 | 8 |
Matrix representation of C2×C22⋊C16 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 5 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 12 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 13 | 10 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,5,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[12,0,0,0,0,16,0,0,0,0,7,13,0,0,0,10] >;
C2×C22⋊C16 in GAP, Magma, Sage, TeX
C_2\times C_2^2\rtimes C_{16} % in TeX
G:=Group("C2xC2^2:C16"); // GroupNames label
G:=SmallGroup(128,843);
// by ID
G=gap.SmallGroup(128,843);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^2=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations