direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×C42⋊C2, C43⋊1C2, C23.152C24, C24.521C23, (C2×C4)⋊6C42, C42○2(C4⋊C4), (C2×C42)⋊19C4, C42⋊34(C2×C4), C4.28(C2×C42), C4○2(C42⋊4C4), C42⋊4C4⋊35C2, C42○2(C22⋊C4), C2.5(C22×C42), C42○(C42⋊4C4), C22.13(C2×C42), C22.24(C23×C4), (C22×C42).17C2, C42○(C2.C42), C23.203(C22×C4), (C23×C4).640C22, (C2×C42).1144C22, (C22×C4).1643C23, C2.C42.564C22, C4○2(C4×C4⋊C4), C42○(C4×C4⋊C4), C42○(C2×C4⋊C4), C4⋊C4⋊47(C2×C4), C4○2(C4×C22⋊C4), (C4×C4⋊C4)⋊117C2, C2.1(C4×C4○D4), C42○(C4×C22⋊C4), C42○(C2×C22⋊C4), C22⋊C4.82(C2×C4), (C4×C22⋊C4).77C2, C2.4(C2×C42⋊C2), C42○(C2×C42⋊C2), C22.50(C2×C4○D4), (C2×C4)○2(C42⋊4C4), (C2×C4).947(C4○D4), (C2×C4⋊C4).967C22, (C22×C4).450(C2×C4), (C2×C4).286(C22×C4), (C2×C42⋊C2).66C2, (C2×C42)○(C42⋊4C4), (C2×C22⋊C4).549C22, SmallGroup(128,1002)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×C42⋊C2
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >
Subgroups: 476 in 376 conjugacy classes, 276 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C23×C4, C43, C42⋊4C4, C4×C22⋊C4, C4×C4⋊C4, C22×C42, C2×C42⋊C2, C4×C42⋊C2
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C4○D4, C24, C2×C42, C42⋊C2, C23×C4, C2×C4○D4, C22×C42, C2×C42⋊C2, C4×C4○D4, C4×C42⋊C2
►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)
(1 19 13 42)(2 20 14 43)(3 17 15 44)(4 18 16 41)(5 48 29 38)(6 45 30 39)(7 46 31 40)(8 47 32 37)(9 58 35 56)(10 59 36 53)(11 60 33 54)(12 57 34 55)(21 62 27 51)(22 63 28 52)(23 64 25 49)(24 61 26 50)
(1 35 23 31)(2 36 24 32)(3 33 21 29)(4 34 22 30)(5 15 11 27)(6 16 12 28)(7 13 9 25)(8 14 10 26)(17 54 62 38)(18 55 63 39)(19 56 64 40)(20 53 61 37)(41 57 52 45)(42 58 49 46)(43 59 50 47)(44 60 51 48)
(1 23)(2 24)(3 21)(4 22)(5 11)(6 12)(7 9)(8 10)(13 25)(14 26)(15 27)(16 28)(29 33)(30 34)(31 35)(32 36)
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), (1,19,13,42)(2,20,14,43)(3,17,15,44)(4,18,16,41)(5,48,29,38)(6,45,30,39)(7,46,31,40)(8,47,32,37)(9,58,35,56)(10,59,36,53)(11,60,33,54)(12,57,34,55)(21,62,27,51)(22,63,28,52)(23,64,25,49)(24,61,26,50), (1,35,23,31)(2,36,24,32)(3,33,21,29)(4,34,22,30)(5,15,11,27)(6,16,12,28)(7,13,9,25)(8,14,10,26)(17,54,62,38)(18,55,63,39)(19,56,64,40)(20,53,61,37)(41,57,52,45)(42,58,49,46)(43,59,50,47)(44,60,51,48), (1,23)(2,24)(3,21)(4,22)(5,11)(6,12)(7,9)(8,10)(13,25)(14,26)(15,27)(16,28)(29,33)(30,34)(31,35)(32,36)>;
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), (1,19,13,42)(2,20,14,43)(3,17,15,44)(4,18,16,41)(5,48,29,38)(6,45,30,39)(7,46,31,40)(8,47,32,37)(9,58,35,56)(10,59,36,53)(11,60,33,54)(12,57,34,55)(21,62,27,51)(22,63,28,52)(23,64,25,49)(24,61,26,50), (1,35,23,31)(2,36,24,32)(3,33,21,29)(4,34,22,30)(5,15,11,27)(6,16,12,28)(7,13,9,25)(8,14,10,26)(17,54,62,38)(18,55,63,39)(19,56,64,40)(20,53,61,37)(41,57,52,45)(42,58,49,46)(43,59,50,47)(44,60,51,48), (1,23)(2,24)(3,21)(4,22)(5,11)(6,12)(7,9)(8,10)(13,25)(14,26)(15,27)(16,28)(29,33)(30,34)(31,35)(32,36) );
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)], [(1,19,13,42),(2,20,14,43),(3,17,15,44),(4,18,16,41),(5,48,29,38),(6,45,30,39),(7,46,31,40),(8,47,32,37),(9,58,35,56),(10,59,36,53),(11,60,33,54),(12,57,34,55),(21,62,27,51),(22,63,28,52),(23,64,25,49),(24,61,26,50)], [(1,35,23,31),(2,36,24,32),(3,33,21,29),(4,34,22,30),(5,15,11,27),(6,16,12,28),(7,13,9,25),(8,14,10,26),(17,54,62,38),(18,55,63,39),(19,56,64,40),(20,53,61,37),(41,57,52,45),(42,58,49,46),(43,59,50,47),(44,60,51,48)], [(1,23),(2,24),(3,21),(4,22),(5,11),(6,12),(7,9),(8,10),(13,25),(14,26),(15,27),(16,28),(29,33),(30,34),(31,35),(32,36)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4X | 4Y | ··· | 4BP |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4○D4 |
| kernel | C4×C42⋊C2 | C43 | C42⋊4C4 | C4×C22⋊C4 | C4×C4⋊C4 | C22×C42 | C2×C42⋊C2 | C2×C42 | C42⋊C2 | C2×C4 |
| # reps | 1 | 2 | 2 | 4 | 4 | 1 | 2 | 16 | 32 | 16 |
Matrix representation of C4×C42⋊C2 ►in GL4(𝔽5) generated by
| 3 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 1 |
| 0 | 0 | 2 | 3 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(5))| [3,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,3,0,0,0,0,2,2,0,0,1,3],[4,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,2,1] >;
C4×C42⋊C2 in GAP, Magma, Sage, TeX
C_4\times C_4^2\rtimes C_2
% in TeX
G:=Group("C4xC4^2:C2"); // GroupNames label
G:=SmallGroup(128,1002);
// by ID
G=gap.SmallGroup(128,1002);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,184,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,c*d=d*c>;
// generators/relations