metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.234D6, C12.45(C4⋊C4), C12.65(C2×Q8), (C2×C12).22Q8, C6.84(C4○D8), (C2×C12).495D4, C6.Q16⋊44C2, C4.30(C2×Dic6), (C2×C4).32Dic6, (C22×C6).73D4, C42⋊C2.6S3, C12.63(C22×C4), C12.Q8⋊43C2, (C22×C4).353D6, C4.32(Dic3⋊C4), (C2×C12).327C23, C2.1(Q8.13D6), C23.44(C3⋊D4), C3⋊3(C23.25D4), C4⋊Dic3.325C22, C22.9(Dic3⋊C4), (C22×C12).148C22, C23.26D6.13C2, (C2×C3⋊C8)⋊8C4, C4.88(S3×C2×C4), C6.40(C2×C4⋊C4), C3⋊C8.19(C2×C4), (C22×C3⋊C8).7C2, (C2×C6).12(C4⋊C4), (C2×C4).155(C4×S3), (C2×C12).89(C2×C4), (C2×C6).456(C2×D4), (C2×C3⋊C8).242C22, C2.14(C2×Dic3⋊C4), C22.71(C2×C3⋊D4), (C2×C4).273(C3⋊D4), (C3×C4⋊C4).265C22, (C2×C4).427(C22×S3), (C3×C42⋊C2).7C2, SmallGroup(192,557)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.234D6
G = < a,b,c,d | a4=b4=c6=1, d2=ab2, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, dbd-1=a-1b-1, dcd-1=c-1 >
Subgroups: 232 in 114 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C2×C12, C22×C6, C4.Q8, C2.D8, C42⋊C2, C42⋊C2, C22×C8, C2×C3⋊C8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C23.25D4, C6.Q16, C12.Q8, C22×C3⋊C8, C23.26D6, C3×C42⋊C2, C4⋊C4.234D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C3⋊D4, C22×S3, C2×C4⋊C4, C4○D8, Dic3⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C23.25D4, C2×Dic3⋊C4, Q8.13D6, C4⋊C4.234D6
►On 96 points
(1 88 5 84)(2 81 6 85)(3 82 7 86)(4 83 8 87)(9 78 13 74)(10 79 14 75)(11 80 15 76)(12 73 16 77)(17 94 21 90)(18 95 22 91)(19 96 23 92)(20 89 24 93)(25 58 29 62)(26 59 30 63)(27 60 31 64)(28 61 32 57)(33 50 37 54)(34 51 38 55)(35 52 39 56)(36 53 40 49)(41 71 45 67)(42 72 46 68)(43 65 47 69)(44 66 48 70)
(1 76 86 9)(2 79 87 12)(3 74 88 15)(4 77 81 10)(5 80 82 13)(6 75 83 16)(7 78 84 11)(8 73 85 14)(17 61 92 30)(18 64 93 25)(19 59 94 28)(20 62 95 31)(21 57 96 26)(22 60 89 29)(23 63 90 32)(24 58 91 27)(33 71 56 43)(34 66 49 46)(35 69 50 41)(36 72 51 44)(37 67 52 47)(38 70 53 42)(39 65 54 45)(40 68 55 48)
(1 24 65)(2 66 17)(3 18 67)(4 68 19)(5 20 69)(6 70 21)(7 22 71)(8 72 23)(9 31 39 13 27 35)(10 36 28 14 40 32)(11 25 33 15 29 37)(12 38 30 16 34 26)(41 82 95)(42 96 83)(43 84 89)(44 90 85)(45 86 91)(46 92 87)(47 88 93)(48 94 81)(49 57 79 53 61 75)(50 76 62 54 80 58)(51 59 73 55 63 77)(52 78 64 56 74 60)
(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)
G:=sub<Sym(96)| (1,88,5,84)(2,81,6,85)(3,82,7,86)(4,83,8,87)(9,78,13,74)(10,79,14,75)(11,80,15,76)(12,73,16,77)(17,94,21,90)(18,95,22,91)(19,96,23,92)(20,89,24,93)(25,58,29,62)(26,59,30,63)(27,60,31,64)(28,61,32,57)(33,50,37,54)(34,51,38,55)(35,52,39,56)(36,53,40,49)(41,71,45,67)(42,72,46,68)(43,65,47,69)(44,66,48,70), (1,76,86,9)(2,79,87,12)(3,74,88,15)(4,77,81,10)(5,80,82,13)(6,75,83,16)(7,78,84,11)(8,73,85,14)(17,61,92,30)(18,64,93,25)(19,59,94,28)(20,62,95,31)(21,57,96,26)(22,60,89,29)(23,63,90,32)(24,58,91,27)(33,71,56,43)(34,66,49,46)(35,69,50,41)(36,72,51,44)(37,67,52,47)(38,70,53,42)(39,65,54,45)(40,68,55,48), (1,24,65)(2,66,17)(3,18,67)(4,68,19)(5,20,69)(6,70,21)(7,22,71)(8,72,23)(9,31,39,13,27,35)(10,36,28,14,40,32)(11,25,33,15,29,37)(12,38,30,16,34,26)(41,82,95)(42,96,83)(43,84,89)(44,90,85)(45,86,91)(46,92,87)(47,88,93)(48,94,81)(49,57,79,53,61,75)(50,76,62,54,80,58)(51,59,73,55,63,77)(52,78,64,56,74,60), (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)>;
G:=Group( (1,88,5,84)(2,81,6,85)(3,82,7,86)(4,83,8,87)(9,78,13,74)(10,79,14,75)(11,80,15,76)(12,73,16,77)(17,94,21,90)(18,95,22,91)(19,96,23,92)(20,89,24,93)(25,58,29,62)(26,59,30,63)(27,60,31,64)(28,61,32,57)(33,50,37,54)(34,51,38,55)(35,52,39,56)(36,53,40,49)(41,71,45,67)(42,72,46,68)(43,65,47,69)(44,66,48,70), (1,76,86,9)(2,79,87,12)(3,74,88,15)(4,77,81,10)(5,80,82,13)(6,75,83,16)(7,78,84,11)(8,73,85,14)(17,61,92,30)(18,64,93,25)(19,59,94,28)(20,62,95,31)(21,57,96,26)(22,60,89,29)(23,63,90,32)(24,58,91,27)(33,71,56,43)(34,66,49,46)(35,69,50,41)(36,72,51,44)(37,67,52,47)(38,70,53,42)(39,65,54,45)(40,68,55,48), (1,24,65)(2,66,17)(3,18,67)(4,68,19)(5,20,69)(6,70,21)(7,22,71)(8,72,23)(9,31,39,13,27,35)(10,36,28,14,40,32)(11,25,33,15,29,37)(12,38,30,16,34,26)(41,82,95)(42,96,83)(43,84,89)(44,90,85)(45,86,91)(46,92,87)(47,88,93)(48,94,81)(49,57,79,53,61,75)(50,76,62,54,80,58)(51,59,73,55,63,77)(52,78,64,56,74,60), (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) );
G=PermutationGroup([[(1,88,5,84),(2,81,6,85),(3,82,7,86),(4,83,8,87),(9,78,13,74),(10,79,14,75),(11,80,15,76),(12,73,16,77),(17,94,21,90),(18,95,22,91),(19,96,23,92),(20,89,24,93),(25,58,29,62),(26,59,30,63),(27,60,31,64),(28,61,32,57),(33,50,37,54),(34,51,38,55),(35,52,39,56),(36,53,40,49),(41,71,45,67),(42,72,46,68),(43,65,47,69),(44,66,48,70)], [(1,76,86,9),(2,79,87,12),(3,74,88,15),(4,77,81,10),(5,80,82,13),(6,75,83,16),(7,78,84,11),(8,73,85,14),(17,61,92,30),(18,64,93,25),(19,59,94,28),(20,62,95,31),(21,57,96,26),(22,60,89,29),(23,63,90,32),(24,58,91,27),(33,71,56,43),(34,66,49,46),(35,69,50,41),(36,72,51,44),(37,67,52,47),(38,70,53,42),(39,65,54,45),(40,68,55,48)], [(1,24,65),(2,66,17),(3,18,67),(4,68,19),(5,20,69),(6,70,21),(7,22,71),(8,72,23),(9,31,39,13,27,35),(10,36,28,14,40,32),(11,25,33,15,29,37),(12,38,30,16,34,26),(41,82,95),(42,96,83),(43,84,89),(44,90,85),(45,86,91),(46,92,87),(47,88,93),(48,94,81),(49,57,79,53,61,75),(50,76,62,54,80,58),(51,59,73,55,63,77),(52,78,64,56,74,60)], [(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)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D4 | D6 | D6 | Dic6 | C4×S3 | C3⋊D4 | C3⋊D4 | C4○D8 | Q8.13D6 |
| kernel | C4⋊C4.234D6 | C6.Q16 | C12.Q8 | C22×C3⋊C8 | C23.26D6 | C3×C42⋊C2 | C2×C3⋊C8 | C42⋊C2 | C2×C12 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 8 | 4 |
Matrix representation of C4⋊C4.234D6 ►in GL6(𝔽73)
| 27 | 0 | 0 | 0 | 0 | 0 |
| 55 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 25 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 70 | 0 | 0 | 0 | 0 |
| 25 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 70 | 0 | 0 |
| 0 | 0 | 48 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 48 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 18 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 22 | 0 | 0 | 0 | 0 | 0 |
| 38 | 63 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 0 | 0 | 0 |
| 0 | 0 | 4 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
G:=sub<GL(6,GF(73))| [27,55,0,0,0,0,0,46,0,0,0,0,0,0,46,25,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,25,0,0,0,0,70,1,0,0,0,0,0,0,27,48,0,0,0,0,70,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,48,0,0,0,0,0,72,0,0,0,0,0,0,1,18,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[22,38,0,0,0,0,0,63,0,0,0,0,0,0,51,4,0,0,0,0,0,10,0,0,0,0,0,0,1,1,0,0,0,0,0,72] >;
C4⋊C4.234D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{234}D_6 % in TeX
G:=Group("C4:C4.234D6"); // GroupNames label
G:=SmallGroup(192,557);
// by ID
G=gap.SmallGroup(192,557);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,422,387,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a*b^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations