direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C8×C22⋊C4, C23.35C42, C2.1(C8×D4), C22⋊2(C4×C8), C8○4(C22⋊C8), C22⋊C8⋊14C4, (C22×C8)⋊13C4, C4.158(C4×D4), (C2×C8).399D4, (C23×C8).6C2, C24.93(C2×C4), C23.18(C2×C8), (C2×C4).45C42, C22.83(C4×D4), C8○2(C2.C42), C22.22(C8○D4), C22.20(C22×C8), C22.28(C2×C42), C4.70(C42⋊C2), C2.4(C8○2M4(2)), C2.C42.27C4, C8○(C22.7C42), C23.253(C22×C4), (C2×C42).990C22, (C22×C8).589C22, (C23×C4).630C22, C22.7C42⋊43C2, (C22×C4).1606C23, (C2×C4×C8)⋊7C2, C2.7(C2×C4×C8), (C2×C4)⋊7(C2×C8), C8○(C2×C22⋊C8), C2.3(C4×C22⋊C4), C22⋊C8○(C22×C8), (C2×C8)○2(C22⋊C8), (C2×C8).203(C2×C4), (C2×C4).1496(C2×D4), (C2×C22⋊C8).49C2, (C4×C22⋊C4).76C2, (C2×C22⋊C4).53C4, C4.106(C2×C22⋊C4), (C2×C4).916(C4○D4), (C2×C8)○(C2.C42), (C22×C4).376(C2×C4), (C2×C4).596(C22×C4), C2.C42○(C22×C8), (C2×C8)○(C22.7C42), (C22×C8)○(C22.7C42), (C2×C8)○(C2×C22⋊C4), (C2×C8)○(C4×C22⋊C4), (C2×C8)○(C2×C22⋊C8), (C22×C8)○(C4×C22⋊C4), SmallGroup(128,483)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8×C22⋊C4
G = < a,b,c,d | a8=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 276 in 192 conjugacy classes, 108 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C2.C42, C4×C8, C22⋊C8, C2×C42, C2×C22⋊C4, C22×C8, C22×C8, C22×C8, C23×C4, C22.7C42, C4×C22⋊C4, C2×C4×C8, C2×C22⋊C8, C23×C8, C8×C22⋊C4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C42, C22⋊C4, C2×C8, C22×C4, C2×D4, C4○D4, C4×C8, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C22×C8, C8○D4, C4×C22⋊C4, C2×C4×C8, C8○2M4(2), C8×D4, C8×C22⋊C4
►On 64 points
(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)
(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 53)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 53)
(1 61 27 41)(2 62 28 42)(3 63 29 43)(4 64 30 44)(5 57 31 45)(6 58 32 46)(7 59 25 47)(8 60 26 48)(9 17 33 52)(10 18 34 53)(11 19 35 54)(12 20 36 55)(13 21 37 56)(14 22 38 49)(15 23 39 50)(16 24 40 51)
G:=sub<Sym(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), (17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53), (1,61,27,41)(2,62,28,42)(3,63,29,43)(4,64,30,44)(5,57,31,45)(6,58,32,46)(7,59,25,47)(8,60,26,48)(9,17,33,52)(10,18,34,53)(11,19,35,54)(12,20,36,55)(13,21,37,56)(14,22,38,49)(15,23,39,50)(16,24,40,51)>;
G:=Group( (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), (17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53), (1,61,27,41)(2,62,28,42)(3,63,29,43)(4,64,30,44)(5,57,31,45)(6,58,32,46)(7,59,25,47)(8,60,26,48)(9,17,33,52)(10,18,34,53)(11,19,35,54)(12,20,36,55)(13,21,37,56)(14,22,38,49)(15,23,39,50)(16,24,40,51) );
G=PermutationGroup([[(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)], [(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,57),(24,58),(41,54),(42,55),(43,56),(44,49),(45,50),(46,51),(47,52),(48,53)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,57),(24,58),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(31,39),(32,40),(41,54),(42,55),(43,56),(44,49),(45,50),(46,51),(47,52),(48,53)], [(1,61,27,41),(2,62,28,42),(3,63,29,43),(4,64,30,44),(5,57,31,45),(6,58,32,46),(7,59,25,47),(8,60,26,48),(9,17,33,52),(10,18,34,53),(11,19,35,54),(12,20,36,55),(13,21,37,56),(14,22,38,49),(15,23,39,50),(16,24,40,51)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4AB | 8A | ··· | 8P | 8Q | ··· | 8AN |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | D4 | C4○D4 | C8○D4 |
| kernel | C8×C22⋊C4 | C22.7C42 | C4×C22⋊C4 | C2×C4×C8 | C2×C22⋊C8 | C23×C8 | C2.C42 | C22⋊C8 | C2×C22⋊C4 | C22×C8 | C22⋊C4 | C2×C8 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 8 | 4 | 8 | 32 | 4 | 4 | 8 |
Matrix representation of C8×C22⋊C4 ►in GL4(𝔽17) generated by
| 2 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 16 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 16 | 1 |
| 0 | 0 | 15 | 1 |
G:=sub<GL(4,GF(17))| [2,0,0,0,0,13,0,0,0,0,2,0,0,0,0,2],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,16,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,13,0,0,0,0,16,15,0,0,1,1] >;
C8×C22⋊C4 in GAP, Magma, Sage, TeX
C_8\times C_2^2\rtimes C_4
% in TeX
G:=Group("C8xC2^2:C4"); // GroupNames label
G:=SmallGroup(128,483);
// by ID
G=gap.SmallGroup(128,483);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^2=d^4=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