metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊26D6, (C4×S3)⋊13D4, (C2×Q8)⋊18D6, D6⋊4(C4○D4), D6.42(C2×D4), C4.186(S3×D4), C22⋊Q8⋊26S3, D6⋊D4⋊15C2, C12⋊D4⋊23C2, (C6×Q8)⋊6C22, C12⋊7D4⋊35C2, D6⋊C4⋊19C22, D6⋊3Q8⋊13C2, C12.231(C2×D4), C22⋊C4.56D6, Dic3⋊5D4⋊24C2, C6.73(C22×D4), D6.D4⋊15C2, (C2×D12)⋊24C22, (C2×C6).171C24, C4⋊Dic3⋊35C22, Dic3.64(C2×D4), (C22×C4).388D6, Dic3⋊4D4⋊14C2, (C2×C12).598C23, Dic3⋊C4⋊17C22, C3⋊5(C22.19C24), C22⋊2(Q8⋊3S3), (C4×Dic3)⋊27C22, (C22×C6).199C23, C22.192(S3×C23), C23.198(C22×S3), (C2×Dic3).86C23, (S3×C23).109C22, (C22×S3).193C23, (C22×C12).251C22, (C22×Dic3).225C22, C2.46(C2×S3×D4), (S3×C22×C4)⋊5C2, (C2×C6)⋊6(C4○D4), (S3×C2×C4)⋊17C22, C4⋊C4⋊7S3⋊24C2, C2.48(S3×C4○D4), (C3×C22⋊Q8)⋊7C2, (C3×C4⋊C4)⋊18C22, (C2×Q8⋊3S3)⋊5C2, C6.160(C2×C4○D4), (C2×C4).46(C22×S3), C2.16(C2×Q8⋊3S3), (C2×C3⋊D4).38C22, (C3×C22⋊C4).26C22, SmallGroup(192,1186)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊26D6
G = < a,b,c,d | a4=b4=c6=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b-1, dbd=a2b, dcd=c-1 >
Subgroups: 896 in 330 conjugacy classes, 109 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C22×S3, C22×C6, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, Q8⋊3S3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×Q8, S3×C23, C22.19C24, Dic3⋊4D4, D6⋊D4, C4⋊C4⋊7S3, Dic3⋊5D4, D6.D4, C12⋊D4, C12⋊7D4, D6⋊3Q8, C3×C22⋊Q8, S3×C22×C4, C2×Q8⋊3S3, C4⋊C4⋊26D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, S3×D4, Q8⋊3S3, S3×C23, C22.19C24, C2×S3×D4, C2×Q8⋊3S3, S3×C4○D4, C4⋊C4⋊26D6
►On 48 points
(1 31 7 25)(2 32 8 26)(3 33 9 27)(4 34 10 28)(5 35 11 29)(6 36 12 30)(13 46 16 43)(14 47 17 44)(15 48 18 45)(19 40 22 37)(20 41 23 38)(21 42 24 39)
(1 19 4 16)(2 14 5 23)(3 21 6 18)(7 22 10 13)(8 17 11 20)(9 24 12 15)(25 40 28 43)(26 47 29 38)(27 42 30 45)(31 37 34 46)(32 44 35 41)(33 39 36 48)
(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)
(1 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 21)(14 20)(15 19)(16 24)(17 23)(18 22)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 45)(38 44)(39 43)(40 48)(41 47)(42 46)
G:=sub<Sym(48)| (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,46,16,43)(14,47,17,44)(15,48,18,45)(19,40,22,37)(20,41,23,38)(21,42,24,39), (1,19,4,16)(2,14,5,23)(3,21,6,18)(7,22,10,13)(8,17,11,20)(9,24,12,15)(25,40,28,43)(26,47,29,38)(27,42,30,45)(31,37,34,46)(32,44,35,41)(33,39,36,48), (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), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22)(25,30)(26,29)(27,28)(31,36)(32,35)(33,34)(37,45)(38,44)(39,43)(40,48)(41,47)(42,46)>;
G:=Group( (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,46,16,43)(14,47,17,44)(15,48,18,45)(19,40,22,37)(20,41,23,38)(21,42,24,39), (1,19,4,16)(2,14,5,23)(3,21,6,18)(7,22,10,13)(8,17,11,20)(9,24,12,15)(25,40,28,43)(26,47,29,38)(27,42,30,45)(31,37,34,46)(32,44,35,41)(33,39,36,48), (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), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22)(25,30)(26,29)(27,28)(31,36)(32,35)(33,34)(37,45)(38,44)(39,43)(40,48)(41,47)(42,46) );
G=PermutationGroup([[(1,31,7,25),(2,32,8,26),(3,33,9,27),(4,34,10,28),(5,35,11,29),(6,36,12,30),(13,46,16,43),(14,47,17,44),(15,48,18,45),(19,40,22,37),(20,41,23,38),(21,42,24,39)], [(1,19,4,16),(2,14,5,23),(3,21,6,18),(7,22,10,13),(8,17,11,20),(9,24,12,15),(25,40,28,43),(26,47,29,38),(27,42,30,45),(31,37,34,46),(32,44,35,41),(33,39,36,48)], [(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)], [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,21),(14,20),(15,19),(16,24),(17,23),(18,22),(25,30),(26,29),(27,28),(31,36),(32,35),(33,34),(37,45),(38,44),(39,43),(40,48),(41,47),(42,46)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | S3×D4 | Q8⋊3S3 | S3×C4○D4 |
| kernel | C4⋊C4⋊26D6 | Dic3⋊4D4 | D6⋊D4 | C4⋊C4⋊7S3 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | C12⋊7D4 | D6⋊3Q8 | C3×C22⋊Q8 | S3×C22×C4 | C2×Q8⋊3S3 | C22⋊Q8 | C4×S3 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | D6 | C2×C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 1 | 1 | 4 | 4 | 2 | 2 | 2 |
Matrix representation of C4⋊C4⋊26D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 5 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,11,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,12,12] >;
C4⋊C4⋊26D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_{26}D_6 % in TeX
G:=Group("C4:C4:26D6"); // GroupNames label
G:=SmallGroup(192,1186);
// by ID
G=gap.SmallGroup(192,1186);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,1123,794,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations