direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12⋊7D4, C23⋊5D12, C24.82D6, (C2×C12)⋊37D4, C12⋊15(C2×D4), (C23×C4)⋊9S3, C6⋊3(C4⋊D4), (C23×C12)⋊8C2, C22⋊3(C2×D12), (C22×C6)⋊15D4, (C22×C4)⋊47D6, D6⋊C4⋊42C22, (C22×D12)⋊12C2, (C2×D12)⋊50C22, (C2×C6).288C24, C4⋊Dic3⋊64C22, C6.134(C22×D4), C2.33(C22×D12), (C2×C12).705C23, (C22×C12)⋊60C22, C22.83(C4○D12), (S3×C23).75C22, (C23×C6).110C22, C22.303(S3×C23), C23.244(C22×S3), (C22×C6).417C23, (C22×S3).126C23, (C2×Dic3).150C23, (C22×Dic3).162C22, C4⋊4(C2×C3⋊D4), C3⋊4(C2×C4⋊D4), (C2×C6)⋊11(C2×D4), (C2×D6⋊C4)⋊14C2, C6.63(C2×C4○D4), (C2×C4)⋊16(C3⋊D4), (C2×C4⋊Dic3)⋊29C2, C2.71(C2×C4○D12), C2.7(C22×C3⋊D4), (C2×C3⋊D4)⋊42C22, (C22×C3⋊D4)⋊11C2, (C2×C6).114(C4○D4), (C2×C4).658(C22×S3), C22.104(C2×C3⋊D4), SmallGroup(192,1349)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C12⋊7D4
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1176 in 426 conjugacy classes, 143 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, C4⋊Dic3, D6⋊C4, C2×D12, C2×D12, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C22×C12, C22×C12, S3×C23, C23×C6, C2×C4⋊D4, C2×C4⋊Dic3, C2×D6⋊C4, C12⋊7D4, C22×D12, C22×C3⋊D4, C23×C12, C2×C12⋊7D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, D12, C3⋊D4, C22×S3, C4⋊D4, C22×D4, C2×C4○D4, C2×D12, C4○D12, C2×C3⋊D4, S3×C23, C2×C4⋊D4, C12⋊7D4, C22×D12, C2×C4○D12, C22×C3⋊D4, C2×C12⋊7D4
►On 96 points
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(37 50)(38 51)(39 52)(40 53)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 49)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 85)(84 86)
(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 95 58 26)(2 94 59 25)(3 93 60 36)(4 92 49 35)(5 91 50 34)(6 90 51 33)(7 89 52 32)(8 88 53 31)(9 87 54 30)(10 86 55 29)(11 85 56 28)(12 96 57 27)(13 68 74 40)(14 67 75 39)(15 66 76 38)(16 65 77 37)(17 64 78 48)(18 63 79 47)(19 62 80 46)(20 61 81 45)(21 72 82 44)(22 71 83 43)(23 70 84 42)(24 69 73 41)
(1 39)(2 38)(3 37)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(49 64)(50 63)(51 62)(52 61)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(73 85)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)
G:=sub<Sym(96)| (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,49)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,85)(84,86), (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), (1,95,58,26)(2,94,59,25)(3,93,60,36)(4,92,49,35)(5,91,50,34)(6,90,51,33)(7,89,52,32)(8,88,53,31)(9,87,54,30)(10,86,55,29)(11,85,56,28)(12,96,57,27)(13,68,74,40)(14,67,75,39)(15,66,76,38)(16,65,77,37)(17,64,78,48)(18,63,79,47)(19,62,80,46)(20,61,81,45)(21,72,82,44)(22,71,83,43)(23,70,84,42)(24,69,73,41), (1,39)(2,38)(3,37)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(49,64)(50,63)(51,62)(52,61)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(73,85)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86)>;
G:=Group( (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,49)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,85)(84,86), (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), (1,95,58,26)(2,94,59,25)(3,93,60,36)(4,92,49,35)(5,91,50,34)(6,90,51,33)(7,89,52,32)(8,88,53,31)(9,87,54,30)(10,86,55,29)(11,85,56,28)(12,96,57,27)(13,68,74,40)(14,67,75,39)(15,66,76,38)(16,65,77,37)(17,64,78,48)(18,63,79,47)(19,62,80,46)(20,61,81,45)(21,72,82,44)(22,71,83,43)(23,70,84,42)(24,69,73,41), (1,39)(2,38)(3,37)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(49,64)(50,63)(51,62)(52,61)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(73,85)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,88)(83,87)(84,86) );
G=PermutationGroup([[(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(37,50),(38,51),(39,52),(40,53),(41,54),(42,55),(43,56),(44,57),(45,58),(46,59),(47,60),(48,49),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,85),(84,86)], [(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)], [(1,95,58,26),(2,94,59,25),(3,93,60,36),(4,92,49,35),(5,91,50,34),(6,90,51,33),(7,89,52,32),(8,88,53,31),(9,87,54,30),(10,86,55,29),(11,85,56,28),(12,96,57,27),(13,68,74,40),(14,67,75,39),(15,66,76,38),(16,65,77,37),(17,64,78,48),(18,63,79,47),(19,62,80,46),(20,61,81,45),(21,72,82,44),(22,71,83,43),(23,70,84,42),(24,69,73,41)], [(1,39),(2,38),(3,37),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,42),(11,41),(12,40),(13,27),(14,26),(15,25),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,28),(49,64),(50,63),(51,62),(52,61),(53,72),(54,71),(55,70),(56,69),(57,68),(58,67),(59,66),(60,65),(73,85),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(81,89),(82,88),(83,87),(84,86)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6O | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | D12 | C4○D12 |
| kernel | C2×C12⋊7D4 | C2×C4⋊Dic3 | C2×D6⋊C4 | C12⋊7D4 | C22×D12 | C22×C3⋊D4 | C23×C12 | C23×C4 | C2×C12 | C22×C6 | C22×C4 | C24 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 2 | 8 | 1 | 2 | 1 | 1 | 4 | 4 | 6 | 1 | 4 | 8 | 8 | 8 |
Matrix representation of C2×C12⋊7D4 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 10 |
| 0 | 0 | 0 | 3 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,0,8,0,0,0,8,0,0,0,0,0,0,6,3,0,0,0,10,3],[1,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0] >;
C2×C12⋊7D4 in GAP, Magma, Sage, TeX
C_2\times C_{12}\rtimes_7D_4 % in TeX
G:=Group("C2xC12:7D4"); // GroupNames label
G:=SmallGroup(192,1349);
// by ID
G=gap.SmallGroup(192,1349);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations