metabelian, supersoluble, monomial
Aliases: C62⋊5C4, C62.25C22, (C2×C6)⋊4Dic3, (C3×C6).38D4, (C2×C6).34D6, (C22×C6).9S3, (C2×C62).3C2, C6.26(C3⋊D4), C23.2(C3⋊S3), C32⋊8(C22⋊C4), C6.16(C2×Dic3), C3⋊2(C6.D4), C22⋊2(C3⋊Dic3), C2.3(C32⋊7D4), (C3×C6).34(C2×C4), (C2×C3⋊Dic3)⋊3C2, C22.7(C2×C3⋊S3), C2.5(C2×C3⋊Dic3), SmallGroup(144,100)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62⋊5C4
G = < a,b,c | a6=b6=c4=1, ab=ba, cac-1=a-1b3, cbc-1=b-1 >
Subgroups: 250 in 102 conjugacy classes, 51 normal (9 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C23, C32, Dic3, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C22×C6, C3⋊Dic3, C62, C62, C62, C6.D4, C2×C3⋊Dic3, C2×C62, C62⋊5C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C3⋊S3, C2×Dic3, C3⋊D4, C3⋊Dic3, C2×C3⋊S3, C6.D4, C2×C3⋊Dic3, C32⋊7D4, C62⋊5C4
►On 72 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 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 65 66)(67 68 69 70 71 72)
(1 17 10 23 13 19)(2 18 11 24 14 20)(3 16 12 22 15 21)(4 31 34 30 8 25)(5 32 35 28 9 26)(6 33 36 29 7 27)(37 54 48 40 51 45)(38 49 43 41 52 46)(39 50 44 42 53 47)(55 67 62 58 70 65)(56 68 63 59 71 66)(57 69 64 60 72 61)
(1 55 32 37)(2 57 33 39)(3 59 31 41)(4 52 21 71)(5 54 19 67)(6 50 20 69)(7 42 24 60)(8 38 22 56)(9 40 23 58)(10 70 28 51)(11 72 29 53)(12 68 30 49)(13 62 26 48)(14 64 27 44)(15 66 25 46)(16 63 34 43)(17 65 35 45)(18 61 36 47)
G:=sub<Sym(72)| (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)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,17,10,23,13,19)(2,18,11,24,14,20)(3,16,12,22,15,21)(4,31,34,30,8,25)(5,32,35,28,9,26)(6,33,36,29,7,27)(37,54,48,40,51,45)(38,49,43,41,52,46)(39,50,44,42,53,47)(55,67,62,58,70,65)(56,68,63,59,71,66)(57,69,64,60,72,61), (1,55,32,37)(2,57,33,39)(3,59,31,41)(4,52,21,71)(5,54,19,67)(6,50,20,69)(7,42,24,60)(8,38,22,56)(9,40,23,58)(10,70,28,51)(11,72,29,53)(12,68,30,49)(13,62,26,48)(14,64,27,44)(15,66,25,46)(16,63,34,43)(17,65,35,45)(18,61,36,47)>;
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)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,17,10,23,13,19)(2,18,11,24,14,20)(3,16,12,22,15,21)(4,31,34,30,8,25)(5,32,35,28,9,26)(6,33,36,29,7,27)(37,54,48,40,51,45)(38,49,43,41,52,46)(39,50,44,42,53,47)(55,67,62,58,70,65)(56,68,63,59,71,66)(57,69,64,60,72,61), (1,55,32,37)(2,57,33,39)(3,59,31,41)(4,52,21,71)(5,54,19,67)(6,50,20,69)(7,42,24,60)(8,38,22,56)(9,40,23,58)(10,70,28,51)(11,72,29,53)(12,68,30,49)(13,62,26,48)(14,64,27,44)(15,66,25,46)(16,63,34,43)(17,65,35,45)(18,61,36,47) );
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),(61,62,63,64,65,66),(67,68,69,70,71,72)], [(1,17,10,23,13,19),(2,18,11,24,14,20),(3,16,12,22,15,21),(4,31,34,30,8,25),(5,32,35,28,9,26),(6,33,36,29,7,27),(37,54,48,40,51,45),(38,49,43,41,52,46),(39,50,44,42,53,47),(55,67,62,58,70,65),(56,68,63,59,71,66),(57,69,64,60,72,61)], [(1,55,32,37),(2,57,33,39),(3,59,31,41),(4,52,21,71),(5,54,19,67),(6,50,20,69),(7,42,24,60),(8,38,22,56),(9,40,23,58),(10,70,28,51),(11,72,29,53),(12,68,30,49),(13,62,26,48),(14,64,27,44),(15,66,25,46),(16,63,34,43),(17,65,35,45),(18,61,36,47)]])
C62⋊5C4 is a maximal subgroup of
C62.31D4 C62.110D4 C62.38D4 C62.97C23 C62.98C23 C62.101C23 C62.56D4 C62⋊3Q8 S3×C6.D4 C62.111C23 C62.112C23 Dic3×C3⋊D4 C62⋊4D4 C62.221C23 C62⋊6Q8 C62.223C23 C22⋊C4×C3⋊S3 C62.227C23 C62.229C23 C62⋊10Q8 C62.247C23 C4×C32⋊7D4 C62.129D4 D4×C3⋊Dic3 C62.72D4 C62.254C23 C62⋊13D4 C62.256C23 C62⋊14D4 C62⋊24D4 C62⋊3C12 C62.127D6 C62.10Dic3 C62.77D6 C63.C2 C62⋊10Dic3
C62⋊5C4 is a maximal quotient of
C62⋊7C8 C62.15Q8 C62.116D4 (C6×D4).S3 C62.38D4 C62.117D4 (C6×C12).C4 C62.39D4 C62.127D6 C62⋊4Dic3 C62.77D6 C63.C2
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C4 | S3 | D4 | Dic3 | D6 | C3⋊D4 |
| kernel | C62⋊5C4 | C2×C3⋊Dic3 | C2×C62 | C62 | C22×C6 | C3×C6 | C2×C6 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 4 | 4 | 2 | 8 | 4 | 16 |
Matrix representation of C62⋊5C4 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 8 | 9 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 2 | 8 |
| 0 | 0 | 1 | 11 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,10,0,0,0,0,3,8,0,0,0,9],[0,12,0,0,1,0,0,0,0,0,2,1,0,0,8,11] >;
C62⋊5C4 in GAP, Magma, Sage, TeX
C_6^2\rtimes_5C_4
% in TeX
G:=Group("C6^2:5C4"); // GroupNames label
G:=SmallGroup(144,100);
// by ID
G=gap.SmallGroup(144,100);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,964,3461]);
// Polycyclic
G:=Group<a,b,c|a^6=b^6=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b^3,c*b*c^-1=b^-1>;
// generators/relations