direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22×C8○D4, C8.21C24, C4.20C25, M4(2)⋊13C23, C8○(C22×D4), D4○(C22×C8), C8○(C22×Q8), Q8○(C22×C8), (C2×C8)⋊18C23, (C23×C8)⋊17C2, C2.14(C24×C4), C4.66(C23×C4), C24.102(C2×C4), (C2×C4).604C24, (C22×C8)⋊72C22, C4○D4.35C23, (C22×D4).47C4, D4.26(C22×C4), C8○2(C22×M4(2)), M4(2)○2(C22×C8), C22.7(C23×C4), Q8.27(C22×C4), (C22×Q8).37C4, (C22×M4(2))⋊29C2, (C2×M4(2))⋊82C22, C23.156(C22×C4), (C23×C4).710C22, (C22×C4).1588C23, C4○(C2×C8○D4), C8○2(C2×C8○D4), C8○2(C2×C4○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×Q8)○(C22×C8), (C2×C8)○(C22×Q8), (C2×C4○D4).36C4, C4○D4.40(C2×C4), (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, (C2×C8)○2(C22×M4(2)), (C22×C8)○2(C2×M4(2)), (C2×C4○D4).342C22, (C22×C8)○(C22×M4(2)), (C2×C8)○(C2×C8○D4), (C2×C4)○(C2×C8○D4), (C2×C8)○2(C2×C4○D4), (C2×C8)○(C22×C4○D4), (C22×C8)○(C2×C4○D4), (C22×C8)○(C22×C4○D4), SmallGroup(128,2303)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 812 in 752 conjugacy classes, 692 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×12], C4, C4 [×15], C22 [×19], C22 [×36], C8 [×16], C2×C4 [×72], D4 [×48], Q8 [×16], C23, C23 [×18], C23 [×12], C2×C8 [×72], M4(2) [×48], C22×C4, C22×C4 [×39], C2×D4 [×36], C2×Q8 [×12], C4○D4 [×64], C24 [×3], C22×C8, C22×C8 [×39], C2×M4(2) [×36], C8○D4 [×64], C23×C4 [×3], C22×D4 [×3], C22×Q8, C2×C4○D4 [×24], C23×C8 [×3], C22×M4(2) [×3], C2×C8○D4 [×24], C22×C4○D4, C22×C8○D4
Quotients:
C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C24 [×31], C8○D4 [×4], C23×C4 [×30], C25, C2×C8○D4 [×6], C24×C4, C22×C8○D4
Generators and relations
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 >
►On 64 points
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)
(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 53 13 49)(10 54 14 50)(11 55 15 51)(12 56 16 52)(17 37 21 33)(18 38 22 34)(19 39 23 35)(20 40 24 36)(25 41 29 45)(26 42 30 46)(27 43 31 47)(28 44 32 48)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(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)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64), (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)(9,53,13,49)(10,54,14,50)(11,55,15,51)(12,56,16,52)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,41,29,45)(26,42,30,46)(27,43,31,47)(28,44,32,48), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(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)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64), (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)(9,53,13,49)(10,54,14,50)(11,55,15,51)(12,56,16,52)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,41,29,45)(26,42,30,46)(27,43,31,47)(28,44,32,48), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(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),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64)], [(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),(9,53,13,49),(10,54,14,50),(11,55,15,51),(12,56,16,52),(17,37,21,33),(18,38,22,34),(19,39,23,35),(20,40,24,36),(25,41,29,45),(26,42,30,46),(27,43,31,47),(28,44,32,48)], [(1,39),(2,40),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,29),(10,30),(11,31),(12,32),(13,25),(14,26),(15,27),(16,28),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(41,53),(42,54),(43,55),(44,56),(45,49),(46,50),(47,51),(48,52)])
Matrix representation ►G ⊆ 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] >;
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 |
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