metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊C4⋊12C2, C6.7(C8○D4), (C2×C8).193D6, C22⋊C8.8S3, Dic3⋊C8⋊22C2, (C8×Dic3)⋊14C2, C2.9(C8○D12), C4⋊Dic3.10C4, C23.16(C4×S3), (C22×C4).90D6, C2.9(D12.C4), C6.D4.3C4, C12.296(C4○D4), (C2×C12).818C23, (C2×C24).212C22, C12.55D4.2C2, C4.122(D4⋊2S3), C6.22(C42⋊C2), (C22×C12).91C22, C3⋊3(C42.7C22), C23.26D6.2C2, (C4×Dic3).270C22, C2.10(C23.16D6), (C2×C4).30(C4×S3), (C2×C12).38(C2×C4), C22.102(S3×C2×C4), (C3×C22⋊C8).14C2, (C2×C3⋊C8).299C22, (C22×C6).36(C2×C4), (C2×C6).73(C22×C4), (C2×C4).760(C22×S3), (C2×Dic3).16(C2×C4), SmallGroup(192,279)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊C4⋊C2
G = < a,b,c | a24=b4=c2=1, bab-1=a5, cac=ab2, cbc=a12b >
Subgroups: 192 in 96 conjugacy classes, 47 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C4×C8, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C42⋊C2, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C22×C12, C42.7C22, C8×Dic3, Dic3⋊C8, C24⋊C4, C12.55D4, C3×C22⋊C8, C23.26D6, C24⋊C4⋊C2
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, C8○D4, S3×C2×C4, D4⋊2S3, C42.7C22, C23.16D6, C8○D12, D12.C4, C24⋊C4⋊C2
►On 96 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)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 54 82 29)(2 59 83 34)(3 64 84 39)(4 69 85 44)(5 50 86 25)(6 55 87 30)(7 60 88 35)(8 65 89 40)(9 70 90 45)(10 51 91 26)(11 56 92 31)(12 61 93 36)(13 66 94 41)(14 71 95 46)(15 52 96 27)(16 57 73 32)(17 62 74 37)(18 67 75 42)(19 72 76 47)(20 53 77 28)(21 58 78 33)(22 63 79 38)(23 68 80 43)(24 49 81 48)
(2 83)(4 85)(6 87)(8 89)(10 91)(12 93)(14 95)(16 73)(18 75)(20 77)(22 79)(24 81)(25 37)(26 63)(27 39)(28 65)(29 41)(30 67)(31 43)(32 69)(33 45)(34 71)(35 47)(36 49)(38 51)(40 53)(42 55)(44 57)(46 59)(48 61)(50 62)(52 64)(54 66)(56 68)(58 70)(60 72)
G:=sub<Sym(96)| (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,54,82,29)(2,59,83,34)(3,64,84,39)(4,69,85,44)(5,50,86,25)(6,55,87,30)(7,60,88,35)(8,65,89,40)(9,70,90,45)(10,51,91,26)(11,56,92,31)(12,61,93,36)(13,66,94,41)(14,71,95,46)(15,52,96,27)(16,57,73,32)(17,62,74,37)(18,67,75,42)(19,72,76,47)(20,53,77,28)(21,58,78,33)(22,63,79,38)(23,68,80,43)(24,49,81,48), (2,83)(4,85)(6,87)(8,89)(10,91)(12,93)(14,95)(16,73)(18,75)(20,77)(22,79)(24,81)(25,37)(26,63)(27,39)(28,65)(29,41)(30,67)(31,43)(32,69)(33,45)(34,71)(35,47)(36,49)(38,51)(40,53)(42,55)(44,57)(46,59)(48,61)(50,62)(52,64)(54,66)(56,68)(58,70)(60,72)>;
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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,54,82,29)(2,59,83,34)(3,64,84,39)(4,69,85,44)(5,50,86,25)(6,55,87,30)(7,60,88,35)(8,65,89,40)(9,70,90,45)(10,51,91,26)(11,56,92,31)(12,61,93,36)(13,66,94,41)(14,71,95,46)(15,52,96,27)(16,57,73,32)(17,62,74,37)(18,67,75,42)(19,72,76,47)(20,53,77,28)(21,58,78,33)(22,63,79,38)(23,68,80,43)(24,49,81,48), (2,83)(4,85)(6,87)(8,89)(10,91)(12,93)(14,95)(16,73)(18,75)(20,77)(22,79)(24,81)(25,37)(26,63)(27,39)(28,65)(29,41)(30,67)(31,43)(32,69)(33,45)(34,71)(35,47)(36,49)(38,51)(40,53)(42,55)(44,57)(46,59)(48,61)(50,62)(52,64)(54,66)(56,68)(58,70)(60,72) );
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),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,54,82,29),(2,59,83,34),(3,64,84,39),(4,69,85,44),(5,50,86,25),(6,55,87,30),(7,60,88,35),(8,65,89,40),(9,70,90,45),(10,51,91,26),(11,56,92,31),(12,61,93,36),(13,66,94,41),(14,71,95,46),(15,52,96,27),(16,57,73,32),(17,62,74,37),(18,67,75,42),(19,72,76,47),(20,53,77,28),(21,58,78,33),(22,63,79,38),(23,68,80,43),(24,49,81,48)], [(2,83),(4,85),(6,87),(8,89),(10,91),(12,93),(14,95),(16,73),(18,75),(20,77),(22,79),(24,81),(25,37),(26,63),(27,39),(28,65),(29,41),(30,67),(31,43),(32,69),(33,45),(34,71),(35,47),(36,49),(38,51),(40,53),(42,55),(44,57),(46,59),(48,61),(50,62),(52,64),(54,66),(56,68),(58,70),(60,72)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D6 | D6 | C4○D4 | C4×S3 | C4×S3 | C8○D4 | C8○D12 | D4⋊2S3 | D12.C4 |
| kernel | C24⋊C4⋊C2 | C8×Dic3 | Dic3⋊C8 | C24⋊C4 | C12.55D4 | C3×C22⋊C8 | C23.26D6 | C4⋊Dic3 | C6.D4 | C22⋊C8 | C2×C8 | C22×C4 | C12 | C2×C4 | C23 | C6 | C2 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 2 | 1 | 4 | 2 | 2 | 8 | 8 | 2 | 2 |
Matrix representation of C24⋊C4⋊C2 ►in GL4(𝔽73) generated by
| 21 | 0 | 0 | 0 |
| 1 | 43 | 0 | 0 |
| 0 | 0 | 20 | 71 |
| 0 | 0 | 17 | 53 |
| 46 | 63 | 0 | 0 |
| 29 | 27 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 1 | 0 | 0 | 0 |
| 53 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 20 | 72 |
G:=sub<GL(4,GF(73))| [21,1,0,0,0,43,0,0,0,0,20,17,0,0,71,53],[46,29,0,0,63,27,0,0,0,0,46,0,0,0,0,46],[1,53,0,0,0,72,0,0,0,0,1,20,0,0,0,72] >;
C24⋊C4⋊C2 in GAP, Magma, Sage, TeX
C_{24}\rtimes C_4\rtimes C_2 % in TeX
G:=Group("C24:C4:C2"); // GroupNames label
G:=SmallGroup(192,279);
// by ID
G=gap.SmallGroup(192,279);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a*b^2,c*b*c=a^12*b>;
// generators/relations