Aliases: C4.A4⋊4C4, (C2×C4).26S4, C4.8(A4⋊C4), Q8⋊Dic3⋊9C2, C4○D4⋊1Dic3, (C2×Q8).16D6, C22.19(C2×S4), Q8.4(C2×Dic3), C2.3(C4.6S4), SL2(𝔽3).7(C2×C4), (C2×SL2(𝔽3)).16C22, C2.9(C2×A4⋊C4), (C2×C4.A4).6C2, (C2×C4○D4).3S3, SmallGroup(192,983)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — C4.A4⋊C4 |
Generators and relations for C4.A4⋊C4
G = < a,b,c,d,e | a4=d3=e4=1, b2=c2=a2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=a2b, dbd-1=a2bc, ebe-1=a2c, dcd-1=b, ede-1=d-1 >
Subgroups: 267 in 79 conjugacy classes, 21 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, SL2(𝔽3), C2×Dic3, C2×C12, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C4×Dic3, C2×SL2(𝔽3), C4.A4, C23.24D4, Q8⋊Dic3, C2×C4.A4, C4.A4⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C2×Dic3, S4, A4⋊C4, C2×S4, C4.6S4, C2×A4⋊C4, C4.A4⋊C4
►On 64 points
Generators in S64
(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 31 3 29)(2 32 4 30)(5 16 7 14)(6 13 8 15)(9 43 11 41)(10 44 12 42)(17 23 19 21)(18 24 20 22)(25 61 27 63)(26 62 28 64)(33 54 35 56)(34 55 36 53)(37 59 39 57)(38 60 40 58)(45 50 47 52)(46 51 48 49)
(1 9 3 11)(2 10 4 12)(5 21 7 23)(6 22 8 24)(13 20 15 18)(14 17 16 19)(25 52 27 50)(26 49 28 51)(29 43 31 41)(30 44 32 42)(33 38 35 40)(34 39 36 37)(45 61 47 63)(46 62 48 64)(53 57 55 59)(54 58 56 60)
(5 21 14)(6 22 15)(7 23 16)(8 24 13)(9 43 31)(10 44 32)(11 41 29)(12 42 30)(25 52 63)(26 49 64)(27 50 61)(28 51 62)(33 60 40)(34 57 37)(35 58 38)(36 59 39)
(1 53 17 45)(2 54 18 46)(3 55 19 47)(4 56 20 48)(5 27 41 37)(6 28 42 38)(7 25 43 39)(8 26 44 40)(9 36 16 52)(10 33 13 49)(11 34 14 50)(12 35 15 51)(21 61 29 57)(22 62 30 58)(23 63 31 59)(24 64 32 60)
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), (1,31,3,29)(2,32,4,30)(5,16,7,14)(6,13,8,15)(9,43,11,41)(10,44,12,42)(17,23,19,21)(18,24,20,22)(25,61,27,63)(26,62,28,64)(33,54,35,56)(34,55,36,53)(37,59,39,57)(38,60,40,58)(45,50,47,52)(46,51,48,49), (1,9,3,11)(2,10,4,12)(5,21,7,23)(6,22,8,24)(13,20,15,18)(14,17,16,19)(25,52,27,50)(26,49,28,51)(29,43,31,41)(30,44,32,42)(33,38,35,40)(34,39,36,37)(45,61,47,63)(46,62,48,64)(53,57,55,59)(54,58,56,60), (5,21,14)(6,22,15)(7,23,16)(8,24,13)(9,43,31)(10,44,32)(11,41,29)(12,42,30)(25,52,63)(26,49,64)(27,50,61)(28,51,62)(33,60,40)(34,57,37)(35,58,38)(36,59,39), (1,53,17,45)(2,54,18,46)(3,55,19,47)(4,56,20,48)(5,27,41,37)(6,28,42,38)(7,25,43,39)(8,26,44,40)(9,36,16,52)(10,33,13,49)(11,34,14,50)(12,35,15,51)(21,61,29,57)(22,62,30,58)(23,63,31,59)(24,64,32,60)>;
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), (1,31,3,29)(2,32,4,30)(5,16,7,14)(6,13,8,15)(9,43,11,41)(10,44,12,42)(17,23,19,21)(18,24,20,22)(25,61,27,63)(26,62,28,64)(33,54,35,56)(34,55,36,53)(37,59,39,57)(38,60,40,58)(45,50,47,52)(46,51,48,49), (1,9,3,11)(2,10,4,12)(5,21,7,23)(6,22,8,24)(13,20,15,18)(14,17,16,19)(25,52,27,50)(26,49,28,51)(29,43,31,41)(30,44,32,42)(33,38,35,40)(34,39,36,37)(45,61,47,63)(46,62,48,64)(53,57,55,59)(54,58,56,60), (5,21,14)(6,22,15)(7,23,16)(8,24,13)(9,43,31)(10,44,32)(11,41,29)(12,42,30)(25,52,63)(26,49,64)(27,50,61)(28,51,62)(33,60,40)(34,57,37)(35,58,38)(36,59,39), (1,53,17,45)(2,54,18,46)(3,55,19,47)(4,56,20,48)(5,27,41,37)(6,28,42,38)(7,25,43,39)(8,26,44,40)(9,36,16,52)(10,33,13,49)(11,34,14,50)(12,35,15,51)(21,61,29,57)(22,62,30,58)(23,63,31,59)(24,64,32,60) );
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)], [(1,31,3,29),(2,32,4,30),(5,16,7,14),(6,13,8,15),(9,43,11,41),(10,44,12,42),(17,23,19,21),(18,24,20,22),(25,61,27,63),(26,62,28,64),(33,54,35,56),(34,55,36,53),(37,59,39,57),(38,60,40,58),(45,50,47,52),(46,51,48,49)], [(1,9,3,11),(2,10,4,12),(5,21,7,23),(6,22,8,24),(13,20,15,18),(14,17,16,19),(25,52,27,50),(26,49,28,51),(29,43,31,41),(30,44,32,42),(33,38,35,40),(34,39,36,37),(45,61,47,63),(46,62,48,64),(53,57,55,59),(54,58,56,60)], [(5,21,14),(6,22,15),(7,23,16),(8,24,13),(9,43,31),(10,44,32),(11,41,29),(12,42,30),(25,52,63),(26,49,64),(27,50,61),(28,51,62),(33,60,40),(34,57,37),(35,58,38),(36,59,39)], [(1,53,17,45),(2,54,18,46),(3,55,19,47),(4,56,20,48),(5,27,41,37),(6,28,42,38),(7,25,43,39),(8,26,44,40),(9,36,16,52),(10,33,13,49),(11,34,14,50),(12,35,15,51),(21,61,29,57),(22,62,30,58),(23,63,31,59),(24,64,32,60)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | ··· | 8H | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 8 | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 6 | ··· | 6 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 |
| type | + | + | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C4 | S3 | D6 | Dic3 | C4.6S4 | S4 | A4⋊C4 | C2×S4 | C4.6S4 |
| kernel | C4.A4⋊C4 | Q8⋊Dic3 | C2×C4.A4 | C4.A4 | C2×C4○D4 | C2×Q8 | C4○D4 | C2 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 2 | 8 | 2 | 4 | 2 | 4 |
Matrix representation of C4.A4⋊C4 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 66 | 58 |
| 0 | 0 | 52 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 22 | 65 |
| 0 | 0 | 15 | 51 |
| 72 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 72 | 0 |
| 0 | 27 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 28 | 42 |
| 0 | 0 | 70 | 45 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,46,0,0,0,0,46],[1,0,0,0,0,1,0,0,0,0,66,52,0,0,58,7],[1,0,0,0,0,1,0,0,0,0,22,15,0,0,65,51],[72,72,0,0,1,0,0,0,0,0,72,72,0,0,1,0],[0,27,0,0,27,0,0,0,0,0,28,70,0,0,42,45] >;
C4.A4⋊C4 in GAP, Magma, Sage, TeX
C_4.A_4\rtimes C_4
% in TeX
G:=Group("C4.A4:C4"); // GroupNames label
G:=SmallGroup(192,983);
// by ID
G=gap.SmallGroup(192,983);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,28,680,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=d^3=e^4=1,b^2=c^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*c*e^-1=a^2*b,d*b*d^-1=a^2*b*c,e*b*e^-1=a^2*c,d*c*d^-1=b,e*d*e^-1=d^-1>;
// generators/relations