direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22×C8○D4, C8.21C24, C4.20C25, M4(2)⋊13C23, D4○(C22×C8), C8○(C22×D4), Q8○(C22×C8), C8○(C22×Q8), (C23×C8)⋊17C2, (C2×C8)⋊18C23, C4.66(C23×C4), C2.14(C24×C4), C24.102(C2×C4), (C2×C4).604C24, (C22×C8)⋊72C22, C4○D4.35C23, (C22×D4).47C4, D4.26(C22×C4), M4(2)○2(C22×C8), C8○2(C22×M4(2)), C22.7(C23×C4), (C22×Q8).37C4, Q8.27(C22×C4), (C2×M4(2))⋊82C22, (C22×M4(2))⋊29C2, (C23×C4).710C22, C23.156(C22×C4), (C22×C4).1588C23, C4○(C2×C8○D4), C8○2(C2×C4○D4), C8○2(C2×C8○D4), (C2×C8)○2(C2×D4), (C2×C8)○2(C2×Q8), (C2×C4)○(C8○D4), C8○(C22×C4○D4), C4○D4○(C22×C8), (C2×C8)○2(C8○D4), (C2×C8)○2(C4○D4), (C2×C8)○(C22×Q8), (C2×Q8)○(C22×C8), C4○D4.40(C2×C4), (C2×C4○D4).36C4, (C2×C8)○3(C2×M4(2)), (C2×D4).254(C2×C4), (C22×C8)○(C22×Q8), (C2×Q8).232(C2×C4), (C22×C4).423(C2×C4), (C2×C4).479(C22×C4), (C22×C4○D4).31C2, (C22×C8)○2(C2×M4(2)), (C2×C8)○2(C22×M4(2)), (C2×C4○D4).342C22, (C22×C8)○(C22×M4(2)), (C2×C4)○(C2×C8○D4), (C2×C8)○(C2×C8○D4), (C2×C8)○2(C2×C4○D4), (C22×C8)○(C2×C4○D4), (C2×C8)○(C22×C4○D4), (C22×C8)○(C22×C4○D4), SmallGroup(128,2303)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22×C8○D4
G = < a,b,c,d,e | a2=b2=c8=e2=1, d2=c4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c4d >
Subgroups: 812 in 752 conjugacy classes, 692 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, D4, Q8, C23, C23, C23, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C22×C8, C22×C8, C2×M4(2), C8○D4, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C23×C8, C22×M4(2), C2×C8○D4, C22×C4○D4, C22×C8○D4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C8○D4, C23×C4, C25, C2×C8○D4, C24×C4, C22×C8○D4
►On 64 points
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 57)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(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 59 5 63)(2 60 6 64)(3 61 7 57)(4 62 8 58)(9 18 13 22)(10 19 14 23)(11 20 15 24)(12 21 16 17)(25 47 29 43)(26 48 30 44)(27 41 31 45)(28 42 32 46)(33 51 37 55)(34 52 38 56)(35 53 39 49)(36 54 40 50)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(41 53)(42 54)(43 55)(44 56)(45 49)(46 50)(47 51)(48 52)
G:=sub<Sym(64)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,57)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (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), (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,18,13,22)(10,19,14,23)(11,20,15,24)(12,21,16,17)(25,47,29,43)(26,48,30,44)(27,41,31,45)(28,42,32,46)(33,51,37,55)(34,52,38,56)(35,53,39,49)(36,54,40,50), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(41,53)(42,54)(43,55)(44,56)(45,49)(46,50)(47,51)(48,52)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,57)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (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), (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,18,13,22)(10,19,14,23)(11,20,15,24)(12,21,16,17)(25,47,29,43)(26,48,30,44)(27,41,31,45)(28,42,32,46)(33,51,37,55)(34,52,38,56)(35,53,39,49)(36,54,40,50), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(41,53)(42,54)(43,55)(44,56)(45,49)(46,50)(47,51)(48,52) );
G=PermutationGroup([[(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,49),(8,50),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,57),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(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)], [(1,59,5,63),(2,60,6,64),(3,61,7,57),(4,62,8,58),(9,18,13,22),(10,19,14,23),(11,20,15,24),(12,21,16,17),(25,47,29,43),(26,48,30,44),(27,41,31,45),(28,42,32,46),(33,51,37,55),(34,52,38,56),(35,53,39,49),(36,54,40,50)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,37),(26,38),(27,39),(28,40),(29,33),(30,34),(31,35),(32,36),(41,53),(42,54),(43,55),(44,56),(45,49),(46,50),(47,51),(48,52)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2S | 4A | ··· | 4H | 4I | ··· | 4T | 8A | ··· | 8P | 8Q | ··· | 8AN |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8○D4 |
| kernel | C22×C8○D4 | C23×C8 | C22×M4(2) | C2×C8○D4 | C22×C4○D4 | C22×D4 | C22×Q8 | C2×C4○D4 | C22 |
| # reps | 1 | 3 | 3 | 24 | 1 | 6 | 2 | 24 | 16 |
Matrix representation of C22×C8○D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,13,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,16,0] >;
C22×C8○D4 in GAP, Magma, Sage, TeX
C_2^2\times C_8\circ D_4
% in TeX
G:=Group("C2^2xC8oD4"); // GroupNames label
G:=SmallGroup(128,2303);
// by ID
G=gap.SmallGroup(128,2303);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,-2,224,723,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^8=e^2=1,d^2=c^4,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^4*d>;
// generators/relations