metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.5C42, C23.32D6, C22.11D12, C22.3Dic6, (C2×C12)⋊3C4, C6.9(C4⋊C4), (C2×C6).4Q8, (C2×C4)⋊2Dic3, (C2×C6).32D4, C2.2(D6⋊C4), (C2×Dic3)⋊2C4, C3⋊(C2.C42), (C22×C4).4S3, C2.5(C4×Dic3), C22.12(C4×S3), (C22×C12).1C2, C2.2(C4⋊Dic3), C6.11(C22⋊C4), C2.2(Dic3⋊C4), C2.2(C6.D4), C22.16(C3⋊D4), (C22×C6).31C22, (C22×Dic3).1C2, C22.10(C2×Dic3), (C2×C6).13(C2×C4), SmallGroup(96,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.32D6
G = < a,b,c,d,e | a2=b2=c2=1, d6=b, e2=abc, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd5 >
Subgroups: 146 in 76 conjugacy classes, 45 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C22 [×3], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×10], C23, Dic3 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×4], C22×C4, C22×C4 [×2], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×2], C22×C6, C2.C42, C22×Dic3 [×2], C22×C12, C23.32D6
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C2×C4 [×3], D4 [×3], Q8, Dic3 [×2], D6, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic6, C4×S3 [×2], D12, C2×Dic3, C3⋊D4 [×2], C2.C42, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], C6.D4, C23.32D6
►Regular action on 96 points
Generators in S96
(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 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 85)(34 86)(35 87)(36 88)(37 57)(38 58)(39 59)(40 60)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(61 77)(62 78)(63 79)(64 80)(65 81)(66 82)(67 83)(68 84)(69 73)(70 74)(71 75)(72 76)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 61)(35 62)(36 63)(73 94)(74 95)(75 96)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(82 91)(83 92)(84 93)
(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 36 57 73)(2 68 58 87)(3 34 59 83)(4 66 60 85)(5 32 49 81)(6 64 50 95)(7 30 51 79)(8 62 52 93)(9 28 53 77)(10 72 54 91)(11 26 55 75)(12 70 56 89)(13 88 37 69)(14 84 38 35)(15 86 39 67)(16 82 40 33)(17 96 41 65)(18 80 42 31)(19 94 43 63)(20 78 44 29)(21 92 45 61)(22 76 46 27)(23 90 47 71)(24 74 48 25)
G:=sub<Sym(96)| (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,85)(34,86)(35,87)(36,88)(37,57)(38,58)(39,59)(40,60)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(69,73)(70,74)(71,75)(72,76), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,61)(35,62)(36,63)(73,94)(74,95)(75,96)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93), (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), (1,36,57,73)(2,68,58,87)(3,34,59,83)(4,66,60,85)(5,32,49,81)(6,64,50,95)(7,30,51,79)(8,62,52,93)(9,28,53,77)(10,72,54,91)(11,26,55,75)(12,70,56,89)(13,88,37,69)(14,84,38,35)(15,86,39,67)(16,82,40,33)(17,96,41,65)(18,80,42,31)(19,94,43,63)(20,78,44,29)(21,92,45,61)(22,76,46,27)(23,90,47,71)(24,74,48,25)>;
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,85)(34,86)(35,87)(36,88)(37,57)(38,58)(39,59)(40,60)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(69,73)(70,74)(71,75)(72,76), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,61)(35,62)(36,63)(73,94)(74,95)(75,96)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93), (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), (1,36,57,73)(2,68,58,87)(3,34,59,83)(4,66,60,85)(5,32,49,81)(6,64,50,95)(7,30,51,79)(8,62,52,93)(9,28,53,77)(10,72,54,91)(11,26,55,75)(12,70,56,89)(13,88,37,69)(14,84,38,35)(15,86,39,67)(16,82,40,33)(17,96,41,65)(18,80,42,31)(19,94,43,63)(20,78,44,29)(21,92,45,61)(22,76,46,27)(23,90,47,71)(24,74,48,25) );
G=PermutationGroup([(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(33,85),(34,86),(35,87),(36,88),(37,57),(38,58),(39,59),(40,60),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56),(61,77),(62,78),(63,79),(64,80),(65,81),(66,82),(67,83),(68,84),(69,73),(70,74),(71,75),(72,76)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,49),(24,50),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,70),(32,71),(33,72),(34,61),(35,62),(36,63),(73,94),(74,95),(75,96),(76,85),(77,86),(78,87),(79,88),(80,89),(81,90),(82,91),(83,92),(84,93)], [(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)], [(1,36,57,73),(2,68,58,87),(3,34,59,83),(4,66,60,85),(5,32,49,81),(6,64,50,95),(7,30,51,79),(8,62,52,93),(9,28,53,77),(10,72,54,91),(11,26,55,75),(12,70,56,89),(13,88,37,69),(14,84,38,35),(15,86,39,67),(16,82,40,33),(17,96,41,65),(18,80,42,31),(19,94,43,63),(20,78,44,29),(21,92,45,61),(22,76,46,27),(23,90,47,71),(24,74,48,25)])
36 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6G | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | - | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | Dic3 | D6 | Dic6 | C4×S3 | D12 | C3⋊D4 |
| kernel | C23.32D6 | C22×Dic3 | C22×C12 | C2×Dic3 | C2×C12 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C23 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 8 | 4 | 1 | 3 | 1 | 2 | 1 | 2 | 4 | 2 | 4 |
Matrix representation of C23.32D6 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 10 | 3 |
| 0 | 0 | 10 | 7 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 10 | 3 |
| 0 | 0 | 6 | 3 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,5,0,0,0,0,10,10,0,0,3,7],[8,0,0,0,0,8,0,0,0,0,10,6,0,0,3,3] >;
C23.32D6 in GAP, Magma, Sage, TeX
C_2^3._{32}D_6 % in TeX
G:=Group("C2^3.32D6"); // GroupNames label
G:=SmallGroup(96,38);
// by ID
G=gap.SmallGroup(96,38);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,217,55,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=b,e^2=a*b*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^5>;
// generators/relations