direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×C22⋊C4, C22⋊C42, C24.22C22, C23.53C23, C2.1(C4×D4), (C22×C4)⋊6C4, (C2×C42)⋊1C2, (C2×C4).142D4, (C23×C4).2C2, C2.3(C2×C42), C23.23(C2×C4), C22.27(C2×D4), C4○2(C2.C42), C2.C42⋊13C2, C2.2(C42⋊C2), C22.13(C4○D4), (C22×C4).85C22, C22.15(C22×C4), (C2×C4)⋊6(C2×C4), C2.2(C2×C22⋊C4), (C2×C22⋊C4).14C2, (C2×C4)○(C2.C42), (C22×C4)○(C2.C42), (C2×C4)○(C2×C22⋊C4), SmallGroup(64,58)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×C22⋊C4
G = < a,b,c,d | a4=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 185 in 129 conjugacy classes, 73 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C23×C4, C4×C22⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×C22⋊C4
►On 32 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)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 24)(10 21)(11 22)(12 23)(25 32)(26 29)(27 30)(28 31)
(1 5)(2 6)(3 7)(4 8)(9 30)(10 31)(11 32)(12 29)(13 17)(14 18)(15 19)(16 20)(21 28)(22 25)(23 26)(24 27)
(1 24 13 30)(2 21 14 31)(3 22 15 32)(4 23 16 29)(5 27 17 9)(6 28 18 10)(7 25 19 11)(8 26 20 12)
G:=sub<Sym(32)| (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), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,24)(10,21)(11,22)(12,23)(25,32)(26,29)(27,30)(28,31), (1,5)(2,6)(3,7)(4,8)(9,30)(10,31)(11,32)(12,29)(13,17)(14,18)(15,19)(16,20)(21,28)(22,25)(23,26)(24,27), (1,24,13,30)(2,21,14,31)(3,22,15,32)(4,23,16,29)(5,27,17,9)(6,28,18,10)(7,25,19,11)(8,26,20,12)>;
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), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,24)(10,21)(11,22)(12,23)(25,32)(26,29)(27,30)(28,31), (1,5)(2,6)(3,7)(4,8)(9,30)(10,31)(11,32)(12,29)(13,17)(14,18)(15,19)(16,20)(21,28)(22,25)(23,26)(24,27), (1,24,13,30)(2,21,14,31)(3,22,15,32)(4,23,16,29)(5,27,17,9)(6,28,18,10)(7,25,19,11)(8,26,20,12) );
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)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,24),(10,21),(11,22),(12,23),(25,32),(26,29),(27,30),(28,31)], [(1,5),(2,6),(3,7),(4,8),(9,30),(10,31),(11,32),(12,29),(13,17),(14,18),(15,19),(16,20),(21,28),(22,25),(23,26),(24,27)], [(1,24,13,30),(2,21,14,31),(3,22,15,32),(4,23,16,29),(5,27,17,9),(6,28,18,10),(7,25,19,11),(8,26,20,12)]])
C4×C22⋊C4 is a maximal subgroup of
C23.21C42 C24.46D4 C24.48D4 C24.53D4 C24.54D4 C24.55D4 C23.36C42 C23.17C42 C24.66D4 C24.70D4 C23.21M4(2) (C2×C8).195D4 C24.72D4 C22⋊C4⋊4C8 C23.9M4(2) C24.175C23 C24.176C23 D4×C42 C23⋊C42 C24.524C23 C25.85C22 C23.165C24 C42⋊42D4 C43⋊9C2 C23.179C24 C43⋊2C2 C23.194C24 C23.195C24 C24.547C23 C23.203C24 C24.195C23 C23.211C24 C23.214C24 C23.215C24 C24.203C23 C24.204C23 C24.205C23 C24.549C23 C23.223C24 C23.224C24 C23.225C24 C23.226C24 C23.227C24 C24.208C23 C23.229C24 C23.234C24 C23.235C24 C24.212C23 C23.240C24 C23.241C24 C24.558C23 C24.215C23 C23.244C24 C24.217C23 C24.218C23 C24.220C23 C23.250C24 C24.221C23 C23.255C24 C24.223C23 C23.288C24 C42⋊15D4 C23.295C24 C42.162D4 C42.163D4 C23.301C24 C24.254C23 C23.321C24 C23.322C24 C23.323C24 C24.259C23 C23.327C24 C23.328C24 C23.329C24 C24.263C23 C24.264C23 C23.333C24 C23.334C24 C23.335C24 C24.565C23 C24.267C23 C24.568C23 C24.268C23 C24.569C23 C24.278C23 C24.279C23 C23.359C24 C23.360C24 C23.364C24 C24.285C23 C24.286C23 C23.367C24 C24.289C23 C24.290C23 C23.372C24 C24.572C23 C24.293C23 C23.377C24 C23.380C24 C24.573C23 C23.388C24 C24.301C23 C23.390C24 C23.391C24 C23.392C24 C24.577C23 C24.304C23 C23.395C24 C23.396C24 C23.397C24 C24.308C23 C23.400C24 C23.401C24 C23.402C24 C24.579C23 C23.404C24 C23.405C24 C23.410C24 C24.309C23 C23.416C24 C23.417C24 C23.418C24 C24.311C23 C23.422C24 C24.313C23 C23.426C24 C24.315C23 C23.430C24 C23.431C24 C23.439C24 C23.449C24 C24.326C23 C24.327C23 C23.455C24 C23.456C24 C23.457C24 C23.458C24 C24.331C23 C24.332C23 C23.461C24 C24.584C23 C23.472C24 C23.473C24 C24.338C23 C24.339C23 C24.340C23 C24.341C23 C23.478C24 C23.483C24 C24.345C23 C24.346C23 C23.491C24 C24.347C23 C24.348C23 C23.502C24 C24.355C23 C23.508C24 C23.548C24 C24.375C23 C24.376C23 C23.553C24 C24.377C23 C24.378C23 C24.379C23 C23.567C24
D2p⋊C42: D4⋊4C42 D6⋊C42 D10⋊2C42 D14⋊C42 ...
C4×C22⋊C4 is a maximal quotient of
C24.17Q8 C24.624C23 C24.626C23 C23⋊2C42 C42.378D4 C42.379D4 C23.36C42 C23.17C42 C23.5C42 D4.C42 Q8⋊C42 Q8.C42 D4.3C42
D2p⋊C42: D4⋊C42 D6⋊C42 D10⋊2C42 D14⋊C42 ...
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 | 2 |
| type | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | C4○D4 |
| kernel | C4×C22⋊C4 | C2.C42 | C2×C42 | C2×C22⋊C4 | C23×C4 | C22⋊C4 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 16 | 8 | 4 | 4 |
Matrix representation of C4×C22⋊C4 ►in GL4(𝔽5) generated by
| 3 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 2 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 2 |
G:=sub<GL(4,GF(5))| [3,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,1,0,0,0,0,4,2,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,3,0,0,0,0,3,0,0,0,3,2] >;
C4×C22⋊C4 in GAP, Magma, Sage, TeX
C_4\times C_2^2\rtimes C_4
% in TeX
G:=Group("C4xC2^2:C4"); // GroupNames label
G:=SmallGroup(64,58);
// by ID
G=gap.SmallGroup(64,58);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,96,121,199,86]);
// Polycyclic
G:=Group<a,b,c,d|a^4=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