metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C4.Dic9, C36.1C4, C9⋊2M4(2), C12.53D6, C4.15D18, C22.Dic9, C12.1Dic3, C36.15C22, C9⋊C8⋊5C2, (C2×C4).2D9, (C2×C18).3C4, (C2×C36).4C2, C18.7(C2×C4), (C2×C12).10S3, C6.7(C2×Dic3), (C2×C6).4Dic3, C2.3(C2×Dic9), C3.(C4.Dic3), SmallGroup(144,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.Dic9
G = < a,b,c | a4=1, b18=a2, c2=b9, ab=ba, cac-1=a-1, cbc-1=b17 >
Smallest permutation representation of C4.Dic9
►On 72 points
(1 10 19 28)(2 11 20 29)(3 12 21 30)(4 13 22 31)(5 14 23 32)(6 15 24 33)(7 16 25 34)(8 17 26 35)(9 18 27 36)(37 64 55 46)(38 65 56 47)(39 66 57 48)(40 67 58 49)(41 68 59 50)(42 69 60 51)(43 70 61 52)(44 71 62 53)(45 72 63 54)
(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 50 10 59 19 68 28 41)(2 67 11 40 20 49 29 58)(3 48 12 57 21 66 30 39)(4 65 13 38 22 47 31 56)(5 46 14 55 23 64 32 37)(6 63 15 72 24 45 33 54)(7 44 16 53 25 62 34 71)(8 61 17 70 26 43 35 52)(9 42 18 51 27 60 36 69)
G:=sub<Sym(72)| (1,10,19,28)(2,11,20,29)(3,12,21,30)(4,13,22,31)(5,14,23,32)(6,15,24,33)(7,16,25,34)(8,17,26,35)(9,18,27,36)(37,64,55,46)(38,65,56,47)(39,66,57,48)(40,67,58,49)(41,68,59,50)(42,69,60,51)(43,70,61,52)(44,71,62,53)(45,72,63,54), (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), (1,50,10,59,19,68,28,41)(2,67,11,40,20,49,29,58)(3,48,12,57,21,66,30,39)(4,65,13,38,22,47,31,56)(5,46,14,55,23,64,32,37)(6,63,15,72,24,45,33,54)(7,44,16,53,25,62,34,71)(8,61,17,70,26,43,35,52)(9,42,18,51,27,60,36,69)>;
G:=Group( (1,10,19,28)(2,11,20,29)(3,12,21,30)(4,13,22,31)(5,14,23,32)(6,15,24,33)(7,16,25,34)(8,17,26,35)(9,18,27,36)(37,64,55,46)(38,65,56,47)(39,66,57,48)(40,67,58,49)(41,68,59,50)(42,69,60,51)(43,70,61,52)(44,71,62,53)(45,72,63,54), (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), (1,50,10,59,19,68,28,41)(2,67,11,40,20,49,29,58)(3,48,12,57,21,66,30,39)(4,65,13,38,22,47,31,56)(5,46,14,55,23,64,32,37)(6,63,15,72,24,45,33,54)(7,44,16,53,25,62,34,71)(8,61,17,70,26,43,35,52)(9,42,18,51,27,60,36,69) );
G=PermutationGroup([[(1,10,19,28),(2,11,20,29),(3,12,21,30),(4,13,22,31),(5,14,23,32),(6,15,24,33),(7,16,25,34),(8,17,26,35),(9,18,27,36),(37,64,55,46),(38,65,56,47),(39,66,57,48),(40,67,58,49),(41,68,59,50),(42,69,60,51),(43,70,61,52),(44,71,62,53),(45,72,63,54)], [(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)], [(1,50,10,59,19,68,28,41),(2,67,11,40,20,49,29,58),(3,48,12,57,21,66,30,39),(4,65,13,38,22,47,31,56),(5,46,14,55,23,64,32,37),(6,63,15,72,24,45,33,54),(7,44,16,53,25,62,34,71),(8,61,17,70,26,43,35,52),(9,42,18,51,27,60,36,69)]])
C4.Dic9 is a maximal subgroup of
C42⋊4D9 C72.C4 C36.53D4 C4.D36 C36.48D4 C36.D4 C36.9D4 Q8⋊3Dic9 D36.2C4 M4(2)×D9 D36⋊6C22 C36.C23 D4.Dic9 D4.D18 D4⋊D18 C4.Dic27 D6.Dic9 C36.C12 C36.69D6
C4.Dic9 is a maximal quotient of
C42.D9 C36⋊C8 C36.55D4 C4.Dic27 D6.Dic9 C36.69D6
42 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | - | |||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | Dic3 | M4(2) | D9 | Dic9 | D18 | Dic9 | C4.Dic3 | C4.Dic9 |
| kernel | C4.Dic9 | C9⋊C8 | C2×C36 | C36 | C2×C18 | C2×C12 | C12 | C12 | C2×C6 | C9 | C2×C4 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 12 |
Matrix representation of C4.Dic9 ►in GL2(𝔽37) generated by
| 31 | 0 |
| 0 | 6 |
| 5 | 0 |
| 0 | 22 |
| 0 | 6 |
| 1 | 0 |
G:=sub<GL(2,GF(37))| [31,0,0,6],[5,0,0,22],[0,1,6,0] >;
C4.Dic9 in GAP, Magma, Sage, TeX
C_4.{\rm Dic}_9 % in TeX
G:=Group("C4.Dic9"); // GroupNames label
G:=SmallGroup(144,10);
// by ID
G=gap.SmallGroup(144,10);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,50,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c|a^4=1,b^18=a^2,c^2=b^9,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^17>;
// generators/relations
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