direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C42⋊C2, C42⋊13C22, C22.7C24, C23.28C23, C24.28C22, (C2×C42)⋊3C2, C4⋊C4⋊18C22, (C22×C4)⋊10C4, C2.3(C23×C4), (C23×C4).9C2, C4○(C42⋊C2), C4.30(C22×C4), (C2×C4).49C23, C23.34(C2×C4), C22.26(C4○D4), C22⋊C4.26C22, (C22×C4).98C22, C22.11(C22×C4), C4○(C2×C4⋊C4), C4○(C2×C22⋊C4), (C2×C4⋊C4)⋊24C2, (C2×C4)○2(C4⋊C4), (C2×C4)⋊11(C2×C4), C2.1(C2×C4○D4), (C2×C4)○2(C22⋊C4), (C2×C4)○(C42⋊C2), (C2×C22⋊C4).15C2, (C2×C4)○(C2×C4⋊C4), SmallGroup(64,195)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C42⋊C2
G = < a,b,c,d | a2=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >
Subgroups: 201 in 165 conjugacy classes, 129 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C2×C42⋊C2
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C42⋊C2, C23×C4, C2×C4○D4, C2×C42⋊C2
►On 32 points
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 15)(10 16)(11 13)(12 14)(17 21)(18 22)(19 23)(20 24)
(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)
(1 31 11 19)(2 32 12 20)(3 29 9 17)(4 30 10 18)(5 15 21 27)(6 16 22 28)(7 13 23 25)(8 14 24 26)
(2 12)(4 10)(6 22)(8 24)(14 26)(16 28)(18 30)(20 32)
G:=sub<Sym(32)| (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,15)(10,16)(11,13)(12,14)(17,21)(18,22)(19,23)(20,24), (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), (1,31,11,19)(2,32,12,20)(3,29,9,17)(4,30,10,18)(5,15,21,27)(6,16,22,28)(7,13,23,25)(8,14,24,26), (2,12)(4,10)(6,22)(8,24)(14,26)(16,28)(18,30)(20,32)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,15)(10,16)(11,13)(12,14)(17,21)(18,22)(19,23)(20,24), (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), (1,31,11,19)(2,32,12,20)(3,29,9,17)(4,30,10,18)(5,15,21,27)(6,16,22,28)(7,13,23,25)(8,14,24,26), (2,12)(4,10)(6,22)(8,24)(14,26)(16,28)(18,30)(20,32) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,15),(10,16),(11,13),(12,14),(17,21),(18,22),(19,23),(20,24)], [(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)], [(1,31,11,19),(2,32,12,20),(3,29,9,17),(4,30,10,18),(5,15,21,27),(6,16,22,28),(7,13,23,25),(8,14,24,26)], [(2,12),(4,10),(6,22),(8,24),(14,26),(16,28),(18,30),(20,32)]])
C2×C42⋊C2 is a maximal subgroup of
C42.371D4 C42.42D4 C23.29C42 C24.63D4 C24.132D4 C24.152D4 C24.162C23 C23.15C42 C42.379D4 C42.95D4 C24.53(C2×C4) C24.169C23 (C22×C4).276D4 C24.69D4 C24.70D4 C24.71D4 C24.73D4 C24.74D4 C42⋊7D4 C24.174C23 C24.524C23 C25.85C22 C23.167C24 C42⋊42D4 C43⋊9C2 C23.178C24 C23.179C24 C43⋊2C2 C23.191C24 C23.192C24 C24.542C23 C24.192C23 C24.545C23 C23.199C24 C42.159D4 C42⋊13D4 C23.224C24 C23.225C24 C23.226C24 C24.208C23 C23.229C24 C23.234C24 C23.236C24 C23.241C24 C23.244C24 C24.217C23 C42⋊15D4 C23.295C24 C42.162D4 C23.311C24 C23.313C24 C24.249C23 C23.315C24 C24.567C23 C24.267C23 C24.268C23 C24.289C23 C24.290C23 C23.374C24 C23.375C24 C24.293C23 C23.377C24 C24.295C23 C23.379C24 C23.382C24 C24.576C23 C23.385C24 C23.398C24 C24.308C23 C23.400C24 C42⋊22D4 C42.183D4 C42⋊23D4 C42⋊25D4 C42⋊26D4 C42.185D4 C42⋊27D4 C42⋊28D4 C42.186D4 C23.524C24 C23.525C24 C42.187D4 C42.188D4 M4(2)○2M4(2) C24.98D4 C42.257C23 C24.100D4 C42.259C23 C42.262C23 C24.115D4 C24.116D4 C24.117D4 C24.118D4 C2×C4×C4○D4 C22.14C25 C22.38C25 C22.44C25 C22.47C25 C22.64C25 C22.80C25 C22.82C25 C22.83C25 C22.84C25
C2×C42⋊C2 is a maximal quotient of
C2×C4×C22⋊C4 C2×C4×C4⋊C4 C25.85C22 C23.165C24 C23.167C24 C42⋊42D4 C43⋊9C2 C42⋊14Q8 C43⋊2C2 C23.194C24 C23.195C24 C24.192C23 C24.547C23 C23.201C24 C23.202C24 C23.224C24 C23.225C24 C23.226C24 C23.227C24 C24.208C23 C23.229C24 C23.234C24 C23.235C24 C23.236C24 C23.237C24 C23.238C24 C24.212C23 C42.677C23 C42.259C23 C42.260C23 C42.261C23 C42.262C23 C42.678C23
40 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4AB |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4○D4 |
| kernel | C2×C42⋊C2 | C2×C42 | C2×C22⋊C4 | C2×C4⋊C4 | C42⋊C2 | C23×C4 | C22×C4 | C22 |
| # reps | 1 | 2 | 2 | 2 | 8 | 1 | 16 | 8 |
Matrix representation of C2×C42⋊C2 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,3,0,0,0,0,0,1,0,0,4,0],[1,0,0,0,0,4,0,0,0,0,3,0,0,0,0,3],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4] >;
C2×C42⋊C2 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes C_2
% in TeX
G:=Group("C2xC4^2:C2"); // GroupNames label
G:=SmallGroup(64,195);
// by ID
G=gap.SmallGroup(64,195);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,192,217,86]);
// Polycyclic
G:=Group<a,b,c,d|a^2=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