metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.236D6, (C4×S3)⋊3Q8, C12⋊Q8⋊34C2, D6.3(C2×Q8), C4.38(S3×Q8), C4⋊C4.204D6, C12.49(C2×Q8), (S3×C42).8C2, D6⋊Q8.1C2, C42.C2⋊15S3, C6.41(C22×Q8), (C2×C12).86C23, (C2×C6).232C24, D6⋊C4.38C22, C12.6Q8⋊22C2, Dic3.16(C2×Q8), Dic6⋊C4⋊34C2, (C4×C12).192C22, Dic3.12(C4○D4), Dic3⋊C4.50C22, C4⋊Dic3.239C22, C22.253(S3×C23), (C22×S3).219C23, C3⋊5(C23.37C23), (C2×Dic3).311C23, (C4×Dic3).139C22, (C2×Dic6).178C22, C2.24(C2×S3×Q8), C2.84(S3×C4○D4), C6.195(C2×C4○D4), C4⋊C4⋊7S3.11C2, (C3×C42.C2)⋊5C2, (S3×C2×C4).249C22, (C2×C4).77(C22×S3), (C3×C4⋊C4).187C22, SmallGroup(192,1247)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 480 in 222 conjugacy classes, 107 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×16], C22, C22 [×4], S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×15], Q8 [×8], C23, Dic3 [×6], Dic3 [×4], C12 [×2], C12 [×6], D6 [×2], D6 [×2], C2×C6, C42, C42 [×7], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×3], C2×Q8 [×4], Dic6 [×8], C4×S3 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×6], C2×C12, C2×C12 [×6], C22×S3, C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2, C42.C2, C4⋊Q8 [×2], C4×Dic3, C4×Dic3 [×6], Dic3⋊C4 [×8], C4⋊Dic3 [×2], D6⋊C4 [×4], C4×C12, C3×C4⋊C4 [×6], C2×Dic6 [×4], S3×C2×C4, S3×C2×C4 [×2], C23.37C23, C12.6Q8, S3×C42, Dic6⋊C4 [×4], C12⋊Q8 [×2], C4⋊C4⋊7S3 [×2], D6⋊Q8 [×4], C3×C42.C2, C42.236D6
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×4], C24, C22×S3 [×7], C22×Q8, C2×C4○D4 [×2], S3×Q8 [×2], S3×C23, C23.37C23, C2×S3×Q8, S3×C4○D4 [×2], C42.236D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=a2b-1, dcd-1=a2c5 >
►On 96 points
(1 19 64 33)(2 34 65 20)(3 21 66 35)(4 36 67 22)(5 23 68 25)(6 26 69 24)(7 13 70 27)(8 28 71 14)(9 15 72 29)(10 30 61 16)(11 17 62 31)(12 32 63 18)(37 95 73 49)(38 50 74 96)(39 85 75 51)(40 52 76 86)(41 87 77 53)(42 54 78 88)(43 89 79 55)(44 56 80 90)(45 91 81 57)(46 58 82 92)(47 93 83 59)(48 60 84 94)
(1 75 7 81)(2 46 8 40)(3 77 9 83)(4 48 10 42)(5 79 11 73)(6 38 12 44)(13 57 19 51)(14 86 20 92)(15 59 21 53)(16 88 22 94)(17 49 23 55)(18 90 24 96)(25 89 31 95)(26 50 32 56)(27 91 33 85)(28 52 34 58)(29 93 35 87)(30 54 36 60)(37 68 43 62)(39 70 45 64)(41 72 47 66)(61 78 67 84)(63 80 69 74)(65 82 71 76)
(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 6 70 63)(2 62 71 5)(3 4 72 61)(7 12 64 69)(8 68 65 11)(9 10 66 67)(13 18 33 26)(14 25 34 17)(15 16 35 36)(19 24 27 32)(20 31 28 23)(21 22 29 30)(37 46 79 76)(38 75 80 45)(39 44 81 74)(40 73 82 43)(41 42 83 84)(47 48 77 78)(49 58 89 86)(50 85 90 57)(51 56 91 96)(52 95 92 55)(53 54 93 94)(59 60 87 88)
G:=sub<Sym(96)| (1,19,64,33)(2,34,65,20)(3,21,66,35)(4,36,67,22)(5,23,68,25)(6,26,69,24)(7,13,70,27)(8,28,71,14)(9,15,72,29)(10,30,61,16)(11,17,62,31)(12,32,63,18)(37,95,73,49)(38,50,74,96)(39,85,75,51)(40,52,76,86)(41,87,77,53)(42,54,78,88)(43,89,79,55)(44,56,80,90)(45,91,81,57)(46,58,82,92)(47,93,83,59)(48,60,84,94), (1,75,7,81)(2,46,8,40)(3,77,9,83)(4,48,10,42)(5,79,11,73)(6,38,12,44)(13,57,19,51)(14,86,20,92)(15,59,21,53)(16,88,22,94)(17,49,23,55)(18,90,24,96)(25,89,31,95)(26,50,32,56)(27,91,33,85)(28,52,34,58)(29,93,35,87)(30,54,36,60)(37,68,43,62)(39,70,45,64)(41,72,47,66)(61,78,67,84)(63,80,69,74)(65,82,71,76), (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), (1,6,70,63)(2,62,71,5)(3,4,72,61)(7,12,64,69)(8,68,65,11)(9,10,66,67)(13,18,33,26)(14,25,34,17)(15,16,35,36)(19,24,27,32)(20,31,28,23)(21,22,29,30)(37,46,79,76)(38,75,80,45)(39,44,81,74)(40,73,82,43)(41,42,83,84)(47,48,77,78)(49,58,89,86)(50,85,90,57)(51,56,91,96)(52,95,92,55)(53,54,93,94)(59,60,87,88)>;
G:=Group( (1,19,64,33)(2,34,65,20)(3,21,66,35)(4,36,67,22)(5,23,68,25)(6,26,69,24)(7,13,70,27)(8,28,71,14)(9,15,72,29)(10,30,61,16)(11,17,62,31)(12,32,63,18)(37,95,73,49)(38,50,74,96)(39,85,75,51)(40,52,76,86)(41,87,77,53)(42,54,78,88)(43,89,79,55)(44,56,80,90)(45,91,81,57)(46,58,82,92)(47,93,83,59)(48,60,84,94), (1,75,7,81)(2,46,8,40)(3,77,9,83)(4,48,10,42)(5,79,11,73)(6,38,12,44)(13,57,19,51)(14,86,20,92)(15,59,21,53)(16,88,22,94)(17,49,23,55)(18,90,24,96)(25,89,31,95)(26,50,32,56)(27,91,33,85)(28,52,34,58)(29,93,35,87)(30,54,36,60)(37,68,43,62)(39,70,45,64)(41,72,47,66)(61,78,67,84)(63,80,69,74)(65,82,71,76), (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), (1,6,70,63)(2,62,71,5)(3,4,72,61)(7,12,64,69)(8,68,65,11)(9,10,66,67)(13,18,33,26)(14,25,34,17)(15,16,35,36)(19,24,27,32)(20,31,28,23)(21,22,29,30)(37,46,79,76)(38,75,80,45)(39,44,81,74)(40,73,82,43)(41,42,83,84)(47,48,77,78)(49,58,89,86)(50,85,90,57)(51,56,91,96)(52,95,92,55)(53,54,93,94)(59,60,87,88) );
G=PermutationGroup([(1,19,64,33),(2,34,65,20),(3,21,66,35),(4,36,67,22),(5,23,68,25),(6,26,69,24),(7,13,70,27),(8,28,71,14),(9,15,72,29),(10,30,61,16),(11,17,62,31),(12,32,63,18),(37,95,73,49),(38,50,74,96),(39,85,75,51),(40,52,76,86),(41,87,77,53),(42,54,78,88),(43,89,79,55),(44,56,80,90),(45,91,81,57),(46,58,82,92),(47,93,83,59),(48,60,84,94)], [(1,75,7,81),(2,46,8,40),(3,77,9,83),(4,48,10,42),(5,79,11,73),(6,38,12,44),(13,57,19,51),(14,86,20,92),(15,59,21,53),(16,88,22,94),(17,49,23,55),(18,90,24,96),(25,89,31,95),(26,50,32,56),(27,91,33,85),(28,52,34,58),(29,93,35,87),(30,54,36,60),(37,68,43,62),(39,70,45,64),(41,72,47,66),(61,78,67,84),(63,80,69,74),(65,82,71,76)], [(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)], [(1,6,70,63),(2,62,71,5),(3,4,72,61),(7,12,64,69),(8,68,65,11),(9,10,66,67),(13,18,33,26),(14,25,34,17),(15,16,35,36),(19,24,27,32),(20,31,28,23),(21,22,29,30),(37,46,79,76),(38,75,80,45),(39,44,81,74),(40,73,82,43),(41,42,83,84),(47,48,77,78),(49,58,89,86),(50,85,90,57),(51,56,91,96),(52,95,92,55),(53,54,93,94),(59,60,87,88)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 9 |
| 0 | 0 | 0 | 0 | 9 | 10 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,9,0,0,0,0,9,10],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 4U | 4V | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | C4○D4 | S3×Q8 | S3×C4○D4 |
| kernel | C42.236D6 | C12.6Q8 | S3×C42 | Dic6⋊C4 | C12⋊Q8 | C4⋊C4⋊7S3 | D6⋊Q8 | C3×C42.C2 | C42.C2 | C4×S3 | C42 | C4⋊C4 | Dic3 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 1 | 1 | 4 | 1 | 6 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{236}D_6 % in TeX
G:=Group("C4^2.236D6"); // GroupNames label
G:=SmallGroup(192,1247);
// by ID
G=gap.SmallGroup(192,1247);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,1123,570,409,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=a^2*c^5>;
// generators/relations