metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.28D12, C4○D12⋊2C4, (C22×C8)⋊9S3, (C22×C24)⋊4C2, (C2×C8).295D6, C4.55(D6⋊C4), D12.23(C2×C4), C2.5(C4○D24), C6.16(C4○D8), C12.411(C2×D4), (C2×C12).403D4, C2.D24⋊44C2, (C2×C4).172D12, C22.6(D6⋊C4), C2.Dic12⋊44C2, Dic6.24(C2×C4), C22.54(C2×D12), (C22×C6).139D4, (C22×C4).445D6, C12.66(C22⋊C4), (C2×C24).356C22, C12.113(C22×C4), (C2×C12).767C23, C23.26D6⋊2C2, C3⋊4(C23.24D4), (C2×D12).199C22, C4⋊Dic3.282C22, (C22×C12).542C22, (C2×Dic6).219C22, C4.71(S3×C2×C4), C2.25(C2×D6⋊C4), (C2×C4).117(C4×S3), (C2×C4○D12).5C2, (C2×C6).157(C2×D4), C4.104(C2×C3⋊D4), C6.53(C2×C22⋊C4), (C2×C12).235(C2×C4), (C2×C4).255(C3⋊D4), (C2×C6).64(C22⋊C4), (C2×C4).715(C22×S3), SmallGroup(192,672)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.28D12
G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd11 >
Subgroups: 440 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, C23.24D4, C2.Dic12, C2.D24, C23.26D6, C22×C24, C2×C4○D12, C23.28D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C4○D8, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C23.24D4, C4○D24, C2×D6⋊C4, C23.28D12
►On 96 points
Generators in S96
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(49 94)(50 95)(51 96)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 49)(23 50)(24 51)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)
(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 63 64 24)(2 23 65 62)(3 61 66 22)(4 21 67 60)(5 59 68 20)(6 19 69 58)(7 57 70 18)(8 17 71 56)(9 55 72 16)(10 15 49 54)(11 53 50 14)(12 13 51 52)(25 46 92 89)(26 88 93 45)(27 44 94 87)(28 86 95 43)(29 42 96 85)(30 84 73 41)(31 40 74 83)(32 82 75 39)(33 38 76 81)(34 80 77 37)(35 36 78 79)(47 48 90 91)
G:=sub<Sym(96)| (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(49,94)(50,95)(51,96)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,49)(23,50)(24,51)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79), (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), (1,63,64,24)(2,23,65,62)(3,61,66,22)(4,21,67,60)(5,59,68,20)(6,19,69,58)(7,57,70,18)(8,17,71,56)(9,55,72,16)(10,15,49,54)(11,53,50,14)(12,13,51,52)(25,46,92,89)(26,88,93,45)(27,44,94,87)(28,86,95,43)(29,42,96,85)(30,84,73,41)(31,40,74,83)(32,82,75,39)(33,38,76,81)(34,80,77,37)(35,36,78,79)(47,48,90,91)>;
G:=Group( (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(49,94)(50,95)(51,96)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,49)(23,50)(24,51)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79), (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), (1,63,64,24)(2,23,65,62)(3,61,66,22)(4,21,67,60)(5,59,68,20)(6,19,69,58)(7,57,70,18)(8,17,71,56)(9,55,72,16)(10,15,49,54)(11,53,50,14)(12,13,51,52)(25,46,92,89)(26,88,93,45)(27,44,94,87)(28,86,95,43)(29,42,96,85)(30,84,73,41)(31,40,74,83)(32,82,75,39)(33,38,76,81)(34,80,77,37)(35,36,78,79)(47,48,90,91) );
G=PermutationGroup([[(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,41),(49,94),(50,95),(51,96),(52,73),(53,74),(54,75),(55,76),(56,77),(57,78),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,49),(23,50),(24,51),(25,80),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79)], [(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)], [(1,63,64,24),(2,23,65,62),(3,61,66,22),(4,21,67,60),(5,59,68,20),(6,19,69,58),(7,57,70,18),(8,17,71,56),(9,55,72,16),(10,15,49,54),(11,53,50,14),(12,13,51,52),(25,46,92,89),(26,88,93,45),(27,44,94,87),(28,86,95,43),(29,42,96,85),(30,84,73,41),(31,40,74,83),(32,82,75,39),(33,38,76,81),(34,80,77,37),(35,36,78,79),(47,48,90,91)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | D12 | C4○D8 | C4○D24 |
| kernel | C23.28D12 | C2.Dic12 | C2.D24 | C23.26D6 | C22×C24 | C2×C4○D12 | C4○D12 | C22×C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 3 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 8 | 16 |
Matrix representation of C23.28D12 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 43 | 13 |
| 0 | 60 | 30 |
| 72 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 1 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 46 | 0 | 0 |
| 0 | 50 | 55 |
| 0 | 18 | 68 |
| 27 | 0 | 0 |
| 0 | 50 | 55 |
| 0 | 5 | 23 |
G:=sub<GL(3,GF(73))| [72,0,0,0,43,60,0,13,30],[72,0,0,0,72,0,0,0,72],[1,0,0,0,72,0,0,0,72],[46,0,0,0,50,18,0,55,68],[27,0,0,0,50,5,0,55,23] >;
C23.28D12 in GAP, Magma, Sage, TeX
C_2^3._{28}D_{12} % in TeX
G:=Group("C2^3.28D12"); // GroupNames label
G:=SmallGroup(192,672);
// by ID
G=gap.SmallGroup(192,672);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,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