metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.15D12, (C2×C8).2D6, C8⋊Dic3⋊7C2, C24⋊1C4⋊3C2, C6.7(C4○D8), C22⋊C8.5S3, C2.9(C4○D24), (C2×C4).117D12, (C2×C12).239D4, (C2×C24).2C22, C2.Dic12⋊5C2, (C22×C4).93D6, (C22×C6).50D4, C6.8(C8.C22), C12.281(C4○D4), (C2×C12).740C23, C12.48D4.8C2, C2.11(C8.D6), C22.103(C2×D12), C3⋊1(C23.20D4), C4.105(D4⋊2S3), C4⋊Dic3.269C22, (C22×C12).92C22, (C2×Dic6).12C22, C23.26D6.3C2, C6.16(C22.D4), C2.12(C23.21D6), (C2×C6).123(C2×D4), (C3×C22⋊C8).7C2, (C2×C4).685(C22×S3), SmallGroup(192,282)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.15D12
G = < a,b,c,d,e | a2=b2=c2=1, d12=e2=c, dad-1=ab=ba, ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd11 >
Subgroups: 256 in 96 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×C24, C2×Dic6, C22×C12, C23.20D4, C2.Dic12, C8⋊Dic3, C24⋊1C4, C3×C22⋊C8, C12.48D4, C23.26D6, C23.15D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C22.D4, C4○D8, C8.C22, C2×D12, D4⋊2S3, C23.20D4, C23.21D6, C4○D24, C8.D6, C23.15D12
►On 96 points
Generators in S96
(2 88)(4 90)(6 92)(8 94)(10 96)(12 74)(14 76)(16 78)(18 80)(20 82)(22 84)(24 86)(25 53)(26 38)(27 55)(28 40)(29 57)(30 42)(31 59)(32 44)(33 61)(34 46)(35 63)(36 48)(37 65)(39 67)(41 69)(43 71)(45 49)(47 51)(50 62)(52 64)(54 66)(56 68)(58 70)(60 72)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 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 61 13 49)(2 32 14 44)(3 59 15 71)(4 30 16 42)(5 57 17 69)(6 28 18 40)(7 55 19 67)(8 26 20 38)(9 53 21 65)(10 48 22 36)(11 51 23 63)(12 46 24 34)(25 95 37 83)(27 93 39 81)(29 91 41 79)(31 89 43 77)(33 87 45 75)(35 85 47 73)(50 74 62 86)(52 96 64 84)(54 94 66 82)(56 92 68 80)(58 90 70 78)(60 88 72 76)
G:=sub<Sym(96)| (2,88)(4,90)(6,92)(8,94)(10,96)(12,74)(14,76)(16,78)(18,80)(20,82)(22,84)(24,86)(25,53)(26,38)(27,55)(28,40)(29,57)(30,42)(31,59)(32,44)(33,61)(34,46)(35,63)(36,48)(37,65)(39,67)(41,69)(43,71)(45,49)(47,51)(50,62)(52,64)(54,66)(56,68)(58,70)(60,72), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,61,13,49)(2,32,14,44)(3,59,15,71)(4,30,16,42)(5,57,17,69)(6,28,18,40)(7,55,19,67)(8,26,20,38)(9,53,21,65)(10,48,22,36)(11,51,23,63)(12,46,24,34)(25,95,37,83)(27,93,39,81)(29,91,41,79)(31,89,43,77)(33,87,45,75)(35,85,47,73)(50,74,62,86)(52,96,64,84)(54,94,66,82)(56,92,68,80)(58,90,70,78)(60,88,72,76)>;
G:=Group( (2,88)(4,90)(6,92)(8,94)(10,96)(12,74)(14,76)(16,78)(18,80)(20,82)(22,84)(24,86)(25,53)(26,38)(27,55)(28,40)(29,57)(30,42)(31,59)(32,44)(33,61)(34,46)(35,63)(36,48)(37,65)(39,67)(41,69)(43,71)(45,49)(47,51)(50,62)(52,64)(54,66)(56,68)(58,70)(60,72), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,61,13,49)(2,32,14,44)(3,59,15,71)(4,30,16,42)(5,57,17,69)(6,28,18,40)(7,55,19,67)(8,26,20,38)(9,53,21,65)(10,48,22,36)(11,51,23,63)(12,46,24,34)(25,95,37,83)(27,93,39,81)(29,91,41,79)(31,89,43,77)(33,87,45,75)(35,85,47,73)(50,74,62,86)(52,96,64,84)(54,94,66,82)(56,92,68,80)(58,90,70,78)(60,88,72,76) );
G=PermutationGroup([[(2,88),(4,90),(6,92),(8,94),(10,96),(12,74),(14,76),(16,78),(18,80),(20,82),(22,84),(24,86),(25,53),(26,38),(27,55),(28,40),(29,57),(30,42),(31,59),(32,44),(33,61),(34,46),(35,63),(36,48),(37,65),(39,67),(41,69),(43,71),(45,49),(47,51),(50,62),(52,64),(54,66),(56,68),(58,70),(60,72)], [(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,73),(12,74),(13,75),(14,76),(15,77),(16,78),(17,79),(18,80),(19,81),(20,82),(21,83),(22,84),(23,85),(24,86),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(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)], [(1,61,13,49),(2,32,14,44),(3,59,15,71),(4,30,16,42),(5,57,17,69),(6,28,18,40),(7,55,19,67),(8,26,20,38),(9,53,21,65),(10,48,22,36),(11,51,23,63),(12,46,24,34),(25,95,37,83),(27,93,39,81),(29,91,41,79),(31,89,43,77),(33,87,45,75),(35,85,47,73),(50,74,62,86),(52,96,64,84),(54,94,66,82),(56,92,68,80),(58,90,70,78),(60,88,72,76)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 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 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | D12 | D12 | C4○D8 | C4○D24 | C8.C22 | D4⋊2S3 | C8.D6 |
| kernel | C23.15D12 | C2.Dic12 | C8⋊Dic3 | C24⋊1C4 | C3×C22⋊C8 | C12.48D4 | C23.26D6 | C22⋊C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C12 | C2×C4 | C23 | C6 | C2 | C6 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C23.15D12 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 14 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 4 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 52 | 3 | 0 | 0 |
| 48 | 21 | 0 | 0 |
| 0 | 0 | 56 | 0 |
| 0 | 0 | 26 | 43 |
| 56 | 65 | 0 | 0 |
| 36 | 17 | 0 | 0 |
| 0 | 0 | 26 | 60 |
| 0 | 0 | 24 | 47 |
G:=sub<GL(4,GF(73))| [1,14,0,0,0,72,0,0,0,0,1,4,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[52,48,0,0,3,21,0,0,0,0,56,26,0,0,0,43],[56,36,0,0,65,17,0,0,0,0,26,24,0,0,60,47] >;
C23.15D12 in GAP, Magma, Sage, TeX
C_2^3._{15}D_{12} % in TeX
G:=Group("C2^3.15D12"); // GroupNames label
G:=SmallGroup(192,282);
// by ID
G=gap.SmallGroup(192,282);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,254,219,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=e^2=c,d*a*d^-1=a*b=b*a,a*c=c*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^11>;
// generators/relations