direct product, non-abelian, soluble
Aliases: C22×C4.A4, SL2(𝔽3)⋊3C23, C2.5(C23×A4), C4.16(C22×A4), (C22×C4).11A4, C23.32(C2×A4), Q8.1(C22×C6), (C22×Q8).6C6, C22.20(C22×A4), (C2×SL2(𝔽3))⋊8C22, (C22×SL2(𝔽3))⋊7C2, (C2×C4○D4)⋊4C6, C4○D4⋊4(C2×C6), (C2×C4).21(C2×A4), (C22×C4○D4)⋊1C3, (C2×Q8).43(C2×C6), SmallGroup(192,1500)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C22×SL2(𝔽3) — C22×C4.A4 |
| Q8 — C22×C4.A4 |
Subgroups: 629 in 220 conjugacy classes, 59 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×4], C4 [×4], C22 [×7], C22 [×16], C6 [×7], C2×C4 [×6], C2×C4 [×22], D4 [×16], Q8, Q8 [×5], C23, C23 [×10], C12 [×4], C2×C6 [×7], C22×C4, C22×C4 [×13], C2×D4 [×12], C2×Q8 [×3], C2×Q8 [×3], C4○D4 [×4], C4○D4 [×20], C24, SL2(𝔽3), C2×C12 [×6], C22×C6, C23×C4, C22×D4, C22×Q8, C2×C4○D4 [×6], C2×C4○D4 [×6], C2×SL2(𝔽3) [×3], C4.A4 [×4], C22×C12, C22×C4○D4, C22×SL2(𝔽3), C2×C4.A4 [×6], C22×C4.A4
Quotients:
C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, A4, C2×C6 [×7], C2×A4 [×7], C22×C6, C4.A4 [×4], C22×A4 [×7], C2×C4.A4 [×6], C23×A4, C22×C4.A4
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c4=f3=1, d2=e2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=c2d, fdf-1=c2de, fef-1=d >
►On 64 points
Generators in S64
(1 15)(2 16)(3 13)(4 14)(5 38)(6 39)(7 40)(8 37)(9 47)(10 48)(11 45)(12 46)(17 29)(18 30)(19 31)(20 32)(21 42)(22 43)(23 44)(24 41)(25 34)(26 35)(27 36)(28 33)(49 56)(50 53)(51 54)(52 55)(57 64)(58 61)(59 62)(60 63)
(1 34)(2 35)(3 36)(4 33)(5 10)(6 11)(7 12)(8 9)(13 27)(14 28)(15 25)(16 26)(17 41)(18 42)(19 43)(20 44)(21 30)(22 31)(23 32)(24 29)(37 47)(38 48)(39 45)(40 46)(49 59)(50 60)(51 57)(52 58)(53 63)(54 64)(55 61)(56 62)
(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 53 3 55)(2 54 4 56)(5 23 7 21)(6 24 8 22)(9 31 11 29)(10 32 12 30)(13 52 15 50)(14 49 16 51)(17 47 19 45)(18 48 20 46)(25 60 27 58)(26 57 28 59)(33 62 35 64)(34 63 36 61)(37 43 39 41)(38 44 40 42)
(1 45 3 47)(2 46 4 48)(5 26 7 28)(6 27 8 25)(9 15 11 13)(10 16 12 14)(17 53 19 55)(18 54 20 56)(21 57 23 59)(22 58 24 60)(29 50 31 52)(30 51 32 49)(33 38 35 40)(34 39 36 37)(41 63 43 61)(42 64 44 62)
(5 23 59)(6 24 60)(7 21 57)(8 22 58)(9 31 52)(10 32 49)(11 29 50)(12 30 51)(17 53 45)(18 54 46)(19 55 47)(20 56 48)(37 43 61)(38 44 62)(39 41 63)(40 42 64)
G:=sub<Sym(64)| (1,15)(2,16)(3,13)(4,14)(5,38)(6,39)(7,40)(8,37)(9,47)(10,48)(11,45)(12,46)(17,29)(18,30)(19,31)(20,32)(21,42)(22,43)(23,44)(24,41)(25,34)(26,35)(27,36)(28,33)(49,56)(50,53)(51,54)(52,55)(57,64)(58,61)(59,62)(60,63), (1,34)(2,35)(3,36)(4,33)(5,10)(6,11)(7,12)(8,9)(13,27)(14,28)(15,25)(16,26)(17,41)(18,42)(19,43)(20,44)(21,30)(22,31)(23,32)(24,29)(37,47)(38,48)(39,45)(40,46)(49,59)(50,60)(51,57)(52,58)(53,63)(54,64)(55,61)(56,62), (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), (1,53,3,55)(2,54,4,56)(5,23,7,21)(6,24,8,22)(9,31,11,29)(10,32,12,30)(13,52,15,50)(14,49,16,51)(17,47,19,45)(18,48,20,46)(25,60,27,58)(26,57,28,59)(33,62,35,64)(34,63,36,61)(37,43,39,41)(38,44,40,42), (1,45,3,47)(2,46,4,48)(5,26,7,28)(6,27,8,25)(9,15,11,13)(10,16,12,14)(17,53,19,55)(18,54,20,56)(21,57,23,59)(22,58,24,60)(29,50,31,52)(30,51,32,49)(33,38,35,40)(34,39,36,37)(41,63,43,61)(42,64,44,62), (5,23,59)(6,24,60)(7,21,57)(8,22,58)(9,31,52)(10,32,49)(11,29,50)(12,30,51)(17,53,45)(18,54,46)(19,55,47)(20,56,48)(37,43,61)(38,44,62)(39,41,63)(40,42,64)>;
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,38)(6,39)(7,40)(8,37)(9,47)(10,48)(11,45)(12,46)(17,29)(18,30)(19,31)(20,32)(21,42)(22,43)(23,44)(24,41)(25,34)(26,35)(27,36)(28,33)(49,56)(50,53)(51,54)(52,55)(57,64)(58,61)(59,62)(60,63), (1,34)(2,35)(3,36)(4,33)(5,10)(6,11)(7,12)(8,9)(13,27)(14,28)(15,25)(16,26)(17,41)(18,42)(19,43)(20,44)(21,30)(22,31)(23,32)(24,29)(37,47)(38,48)(39,45)(40,46)(49,59)(50,60)(51,57)(52,58)(53,63)(54,64)(55,61)(56,62), (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), (1,53,3,55)(2,54,4,56)(5,23,7,21)(6,24,8,22)(9,31,11,29)(10,32,12,30)(13,52,15,50)(14,49,16,51)(17,47,19,45)(18,48,20,46)(25,60,27,58)(26,57,28,59)(33,62,35,64)(34,63,36,61)(37,43,39,41)(38,44,40,42), (1,45,3,47)(2,46,4,48)(5,26,7,28)(6,27,8,25)(9,15,11,13)(10,16,12,14)(17,53,19,55)(18,54,20,56)(21,57,23,59)(22,58,24,60)(29,50,31,52)(30,51,32,49)(33,38,35,40)(34,39,36,37)(41,63,43,61)(42,64,44,62), (5,23,59)(6,24,60)(7,21,57)(8,22,58)(9,31,52)(10,32,49)(11,29,50)(12,30,51)(17,53,45)(18,54,46)(19,55,47)(20,56,48)(37,43,61)(38,44,62)(39,41,63)(40,42,64) );
G=PermutationGroup([(1,15),(2,16),(3,13),(4,14),(5,38),(6,39),(7,40),(8,37),(9,47),(10,48),(11,45),(12,46),(17,29),(18,30),(19,31),(20,32),(21,42),(22,43),(23,44),(24,41),(25,34),(26,35),(27,36),(28,33),(49,56),(50,53),(51,54),(52,55),(57,64),(58,61),(59,62),(60,63)], [(1,34),(2,35),(3,36),(4,33),(5,10),(6,11),(7,12),(8,9),(13,27),(14,28),(15,25),(16,26),(17,41),(18,42),(19,43),(20,44),(21,30),(22,31),(23,32),(24,29),(37,47),(38,48),(39,45),(40,46),(49,59),(50,60),(51,57),(52,58),(53,63),(54,64),(55,61),(56,62)], [(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)], [(1,53,3,55),(2,54,4,56),(5,23,7,21),(6,24,8,22),(9,31,11,29),(10,32,12,30),(13,52,15,50),(14,49,16,51),(17,47,19,45),(18,48,20,46),(25,60,27,58),(26,57,28,59),(33,62,35,64),(34,63,36,61),(37,43,39,41),(38,44,40,42)], [(1,45,3,47),(2,46,4,48),(5,26,7,28),(6,27,8,25),(9,15,11,13),(10,16,12,14),(17,53,19,55),(18,54,20,56),(21,57,23,59),(22,58,24,60),(29,50,31,52),(30,51,32,49),(33,38,35,40),(34,39,36,37),(41,63,43,61),(42,64,44,62)], [(5,23,59),(6,24,60),(7,21,57),(8,22,58),(9,31,52),(10,32,49),(11,29,50),(12,30,51),(17,53,45),(18,54,46),(19,55,47),(20,56,48),(37,43,61),(38,44,62),(39,41,63),(40,42,64)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 2 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 3 |
| 0 | 0 | 5 | 6 |
| 3 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,12,2,0,0,12,1],[1,0,0,0,0,1,0,0,0,0,7,5,0,0,3,6],[3,0,0,0,0,9,0,0,0,0,1,0,0,0,4,9] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6N | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 4 | 4 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 |
| type | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C4.A4 | A4 | C2×A4 | C2×A4 |
| kernel | C22×C4.A4 | C22×SL2(𝔽3) | C2×C4.A4 | C22×C4○D4 | C22×Q8 | C2×C4○D4 | C22 | C22×C4 | C2×C4 | C23 |
| # reps | 1 | 1 | 6 | 2 | 2 | 12 | 24 | 1 | 6 | 1 |
In GAP, Magma, Sage, TeX
C_2^2\times C_4.A_4
% in TeX
G:=Group("C2^2xC4.A4"); // GroupNames label
G:=SmallGroup(192,1500);
// by ID
G=gap.SmallGroup(192,1500);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,520,235,172,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=f^3=1,d^2=e^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,e*d*e^-1=c^2*d,f*d*f^-1=c^2*d*e,f*e*f^-1=d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀