direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C24⋊C2, C24⋊10C23, C12.54C24, Dic6⋊4C23, C23.65D12, D12.20C23, (C2×C8)⋊35D6, C8⋊9(C22×S3), (C2×C6)⋊9SD16, C6⋊1(C2×SD16), (C22×C8)⋊14S3, C4.44(C2×D12), (C2×C4).99D12, C3⋊1(C22×SD16), (C2×C24)⋊46C22, (C22×C24)⋊14C2, (C2×C12).390D4, C12.289(C2×D4), C4.51(S3×C23), C6.21(C22×D4), (C22×D12).9C2, C2.23(C22×D12), C22.69(C2×D12), (C22×C4).458D6, (C22×C6).144D4, (C2×C12).786C23, (C22×Dic6)⋊11C2, (C2×Dic6)⋊56C22, (C2×D12).228C22, (C22×C12).525C22, (C2×C6).177(C2×D4), (C2×C4).735(C22×S3), SmallGroup(192,1298)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 920 in 298 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4, C4 [×3], C4 [×4], C22 [×7], C22 [×16], S3 [×4], C6, C6 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×6], D4 [×10], Q8 [×10], C23, C23 [×10], Dic3 [×4], C12, C12 [×3], D6 [×16], C2×C6 [×7], C2×C8 [×6], SD16 [×16], C22×C4, C22×C4, C2×D4 [×9], C2×Q8 [×9], C24, C24 [×4], Dic6 [×4], Dic6 [×6], D12 [×4], D12 [×6], C2×Dic3 [×6], C2×C12 [×6], C22×S3 [×10], C22×C6, C22×C8, C2×SD16 [×12], C22×D4, C22×Q8, C24⋊C2 [×16], C2×C24 [×6], C2×Dic6 [×6], C2×Dic6 [×3], C2×D12 [×6], C2×D12 [×3], C22×Dic3, C22×C12, S3×C23, C22×SD16, C2×C24⋊C2 [×12], C22×C24, C22×Dic6, C22×D12, C22×C24⋊C2
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], SD16 [×4], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C2×SD16 [×6], C22×D4, C24⋊C2 [×4], C2×D12 [×6], S3×C23, C22×SD16, C2×C24⋊C2 [×6], C22×D12, C22×C24⋊C2
Generators and relations
G = < a,b,c,d | a2=b2=c24=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c11 >
►On 96 points
(1 44)(2 45)(3 46)(4 47)(5 48)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 56)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 72)(42 49)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)
(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)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 27)(26 38)(28 36)(29 47)(30 34)(31 45)(33 43)(35 41)(37 39)(40 48)(42 46)(49 53)(50 64)(52 62)(54 60)(55 71)(56 58)(57 69)(59 67)(61 65)(66 72)(68 70)(73 89)(74 76)(75 87)(77 85)(78 96)(79 83)(80 94)(82 92)(84 90)(86 88)(91 95)
G:=sub<Sym(96)| (1,44)(2,45)(3,46)(4,47)(5,48)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78), (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55), (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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,27)(26,38)(28,36)(29,47)(30,34)(31,45)(33,43)(35,41)(37,39)(40,48)(42,46)(49,53)(50,64)(52,62)(54,60)(55,71)(56,58)(57,69)(59,67)(61,65)(66,72)(68,70)(73,89)(74,76)(75,87)(77,85)(78,96)(79,83)(80,94)(82,92)(84,90)(86,88)(91,95)>;
G:=Group( (1,44)(2,45)(3,46)(4,47)(5,48)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78), (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55), (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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,27)(26,38)(28,36)(29,47)(30,34)(31,45)(33,43)(35,41)(37,39)(40,48)(42,46)(49,53)(50,64)(52,62)(54,60)(55,71)(56,58)(57,69)(59,67)(61,65)(66,72)(68,70)(73,89)(74,76)(75,87)(77,85)(78,96)(79,83)(80,94)(82,92)(84,90)(86,88)(91,95) );
G=PermutationGroup([(1,44),(2,45),(3,46),(4,47),(5,48),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,40),(22,41),(23,42),(24,43),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,73),(68,74),(69,75),(70,76),(71,77),(72,78)], [(1,81),(2,82),(3,83),(4,84),(5,85),(6,86),(7,87),(8,88),(9,89),(10,90),(11,91),(12,92),(13,93),(14,94),(15,95),(16,96),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,56),(26,57),(27,58),(28,59),(29,60),(30,61),(31,62),(32,63),(33,64),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71),(41,72),(42,49),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55)], [(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)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,27),(26,38),(28,36),(29,47),(30,34),(31,45),(33,43),(35,41),(37,39),(40,48),(42,46),(49,53),(50,64),(52,62),(54,60),(55,71),(56,58),(57,69),(59,67),(61,65),(66,72),(68,70),(73,89),(74,76),(75,87),(77,85),(78,96),(79,83),(80,94),(82,92),(84,90),(86,88),(91,95)])
Matrix representation ►G ⊆ GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 11 | 48 |
| 0 | 0 | 25 | 36 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,72,0,0,0,0,11,25,0,0,48,36],[72,0,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 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 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | SD16 | D12 | D12 | C24⋊C2 |
| kernel | C22×C24⋊C2 | C2×C24⋊C2 | C22×C24 | C22×Dic6 | C22×D12 | C22×C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 1 | 1 | 1 | 3 | 1 | 6 | 1 | 8 | 6 | 2 | 16 |
In GAP, Magma, Sage, TeX
C_2^2\times C_{24}\rtimes C_2 % in TeX
G:=Group("C2^2xC24:C2"); // GroupNames label
G:=SmallGroup(192,1298);
// by ID
G=gap.SmallGroup(192,1298);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,80,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^24=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^11>;
// generators/relations