direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C22×C4○D8, D8⋊7C23, C4.4C25, C8.20C24, Q16⋊7C23, D4.2C24, Q8.2C24, SD16⋊6C23, C24.145D4, D8○(C22×C4), C4○(C22×D8), Q16○(C22×C4), C4○(C22×Q16), (C2×C8)⋊15C23, (C23×C8)⋊11C2, C4○D4⋊4C23, C4○(C22×SD16), SD16○(C22×C4), (C22×D8)⋊24C2, (C2×D8)⋊59C22, C4.30(C22×D4), C2.39(D4×C23), (C2×C4).610C24, (C22×C8)⋊67C22, (C22×Q16)⋊24C2, (C2×Q16)⋊63C22, (C22×C4).630D4, C23.410(C2×D4), C22.5(C22×D4), (C22×SD16)⋊30C2, (C2×SD16)⋊82C22, (C2×D4).490C23, (C2×Q8).474C23, (C23×C4).714C22, (C22×C4).1592C23, (C22×D4).603C22, (C22×Q8).504C22, C4○(C2×C4○D8), (C2×C4)○2(C2×D8), (C2×C4)○(C4○D8), (C2×C4)○2(C2×Q16), (C2×C4)○(C22×D8), (C22×C4)○(C2×D8), (C2×C4)○2(C2×SD16), (C2×C4)○(C22×Q16), (C22×C4)○(C2×Q16), (C2×C4).883(C2×D4), (C22×C4)○(C2×SD16), (C2×C4)○(C22×SD16), (C22×C4)○(C22×D8), (C2×C4○D4)⋊76C22, (C22×C4○D4)⋊25C2, (C22×C4)○(C22×Q16), (C22×C4)○(C22×SD16), (C2×C4)○(C2×C4○D8), SmallGroup(128,2309)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22×C4○D8
G = < a,b,c,d,e | a2=b2=c4=e2=1, d4=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d3 >
Subgroups: 1148 in 752 conjugacy classes, 436 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C24, C22×C8, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C23×C8, C22×D8, C22×SD16, C22×Q16, C2×C4○D8, C22×C4○D4, C22×C4○D8
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4○D8, C22×D4, C25, C2×C4○D8, D4×C23, C22×C4○D8
►On 64 points
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 56)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 60)(18 61)(19 62)(20 63)(21 64)(22 57)(23 58)(24 59)(25 56)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 59 5 63)(2 60 6 64)(3 61 7 57)(4 62 8 58)(9 21 13 17)(10 22 14 18)(11 23 15 19)(12 24 16 20)(25 39 29 35)(26 40 30 36)(27 33 31 37)(28 34 32 38)(41 49 45 53)(42 50 46 54)(43 51 47 55)(44 52 48 56)
(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 11)(2 10)(3 9)(4 16)(5 15)(6 14)(7 13)(8 12)(17 57)(18 64)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 53)(26 52)(27 51)(28 50)(29 49)(30 56)(31 55)(32 54)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 48)
G:=sub<Sym(64)| (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,56)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(25,56)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45), (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,21,13,17)(10,22,14,18)(11,23,15,19)(12,24,16,20)(25,39,29,35)(26,40,30,36)(27,33,31,37)(28,34,32,38)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56), (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), (1,11)(2,10)(3,9)(4,16)(5,15)(6,14)(7,13)(8,12)(17,57)(18,64)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,53)(26,52)(27,51)(28,50)(29,49)(30,56)(31,55)(32,54)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,48)>;
G:=Group( (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,56)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(25,56)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45), (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,21,13,17)(10,22,14,18)(11,23,15,19)(12,24,16,20)(25,39,29,35)(26,40,30,36)(27,33,31,37)(28,34,32,38)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56), (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), (1,11)(2,10)(3,9)(4,16)(5,15)(6,14)(7,13)(8,12)(17,57)(18,64)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,53)(26,52)(27,51)(28,50)(29,49)(30,56)(31,55)(32,54)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,48) );
G=PermutationGroup([[(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(9,56),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,55),(17,48),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,57)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,60),(18,61),(19,62),(20,63),(21,64),(22,57),(23,58),(24,59),(25,56),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,46),(34,47),(35,48),(36,41),(37,42),(38,43),(39,44),(40,45)], [(1,59,5,63),(2,60,6,64),(3,61,7,57),(4,62,8,58),(9,21,13,17),(10,22,14,18),(11,23,15,19),(12,24,16,20),(25,39,29,35),(26,40,30,36),(27,33,31,37),(28,34,32,38),(41,49,45,53),(42,50,46,54),(43,51,47,55),(44,52,48,56)], [(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)], [(1,11),(2,10),(3,9),(4,16),(5,15),(6,14),(7,13),(8,12),(17,57),(18,64),(19,63),(20,62),(21,61),(22,60),(23,59),(24,58),(25,53),(26,52),(27,51),(28,50),(29,49),(30,56),(31,55),(32,54),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(40,48)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D8 |
| kernel | C22×C4○D8 | C23×C8 | C22×D8 | C22×SD16 | C22×Q16 | C2×C4○D8 | C22×C4○D4 | C22×C4 | C24 | C22 |
| # reps | 1 | 1 | 1 | 2 | 1 | 24 | 2 | 7 | 1 | 16 |
Matrix representation of C22×C4○D8 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 14 |
| 0 | 0 | 3 | 3 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 3 | 14 |
| 0 | 0 | 14 | 14 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[16,0,0,0,0,1,0,0,0,0,3,3,0,0,14,3],[16,0,0,0,0,16,0,0,0,0,3,14,0,0,14,14] >;
C22×C4○D8 in GAP, Magma, Sage, TeX
C_2^2\times C_4\circ D_8
% in TeX
G:=Group("C2^2xC4oD8"); // GroupNames label
G:=SmallGroup(128,2309);
// by ID
G=gap.SmallGroup(128,2309);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,352,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^2=1,d^4=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d^3>;
// generators/relations