direct product, metabelian, nilpotent (class 2), monomial
Aliases: M4(2)×He3, C62.2C12, C12.41C62, C4.(C4×He3), (C3×C24)⋊7C6, C8⋊3(C2×He3), C24.14(C3×C6), (C6×C12).10C6, (C3×C12).5C12, C12.8(C3×C12), C22.(C4×He3), C6.26(C6×C12), (C8×He3)⋊11C2, (C4×He3).10C4, (C32×M4(2))⋊C3, C4.6(C22×He3), C32⋊7(C3×M4(2)), (C22×He3).3C4, (C4×He3).55C22, C3.2(C32×M4(2)), (C3×M4(2)).2C32, C2.5(C2×C4×He3), (C2×C4×He3).14C2, (C2×C4).2(C2×He3), (C2×C12).14(C3×C6), (C3×C12).68(C2×C6), (C2×C6).10(C3×C12), (C3×C6).34(C2×C12), (C2×He3).39(C2×C4), SmallGroup(432,213)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×He3
G = < a,b,c,d,e | a8=b2=c3=d3=e3=1, bab=a5, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 209 in 110 conjugacy classes, 63 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C32, C12, C12, C2×C6, C2×C6, M4(2), C3×C6, C3×C6, C24, C24, C2×C12, C2×C12, He3, C3×C12, C62, C3×M4(2), C3×M4(2), C2×He3, C2×He3, C3×C24, C6×C12, C4×He3, C22×He3, C32×M4(2), C8×He3, C2×C4×He3, M4(2)×He3
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, M4(2), C3×C6, C2×C12, He3, C3×C12, C62, C3×M4(2), C2×He3, C6×C12, C4×He3, C22×He3, C32×M4(2), C2×C4×He3, M4(2)×He3
►On 72 points
Generators in S72
(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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(41 45)(43 47)(49 53)(51 55)(58 62)(60 64)(66 70)(68 72)
(1 50 67)(2 51 68)(3 52 69)(4 53 70)(5 54 71)(6 55 72)(7 56 65)(8 49 66)(9 61 35)(10 62 36)(11 63 37)(12 64 38)(13 57 39)(14 58 40)(15 59 33)(16 60 34)(17 27 41)(18 28 42)(19 29 43)(20 30 44)(21 31 45)(22 32 46)(23 25 47)(24 26 48)
(1 32 61)(2 25 62)(3 26 63)(4 27 64)(5 28 57)(6 29 58)(7 30 59)(8 31 60)(9 67 22)(10 68 23)(11 69 24)(12 70 17)(13 71 18)(14 72 19)(15 65 20)(16 66 21)(33 56 44)(34 49 45)(35 50 46)(36 51 47)(37 52 48)(38 53 41)(39 54 42)(40 55 43)
(1 67 35)(2 68 36)(3 69 37)(4 70 38)(5 71 39)(6 72 40)(7 65 33)(8 66 34)(9 46 61)(10 47 62)(11 48 63)(12 41 64)(13 42 57)(14 43 58)(15 44 59)(16 45 60)(17 53 27)(18 54 28)(19 55 29)(20 56 30)(21 49 31)(22 50 32)(23 51 25)(24 52 26)
G:=sub<Sym(72)| (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)(49,53)(51,55)(58,62)(60,64)(66,70)(68,72), (1,50,67)(2,51,68)(3,52,69)(4,53,70)(5,54,71)(6,55,72)(7,56,65)(8,49,66)(9,61,35)(10,62,36)(11,63,37)(12,64,38)(13,57,39)(14,58,40)(15,59,33)(16,60,34)(17,27,41)(18,28,42)(19,29,43)(20,30,44)(21,31,45)(22,32,46)(23,25,47)(24,26,48), (1,32,61)(2,25,62)(3,26,63)(4,27,64)(5,28,57)(6,29,58)(7,30,59)(8,31,60)(9,67,22)(10,68,23)(11,69,24)(12,70,17)(13,71,18)(14,72,19)(15,65,20)(16,66,21)(33,56,44)(34,49,45)(35,50,46)(36,51,47)(37,52,48)(38,53,41)(39,54,42)(40,55,43), (1,67,35)(2,68,36)(3,69,37)(4,70,38)(5,71,39)(6,72,40)(7,65,33)(8,66,34)(9,46,61)(10,47,62)(11,48,63)(12,41,64)(13,42,57)(14,43,58)(15,44,59)(16,45,60)(17,53,27)(18,54,28)(19,55,29)(20,56,30)(21,49,31)(22,50,32)(23,51,25)(24,52,26)>;
G:=Group( (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)(49,53)(51,55)(58,62)(60,64)(66,70)(68,72), (1,50,67)(2,51,68)(3,52,69)(4,53,70)(5,54,71)(6,55,72)(7,56,65)(8,49,66)(9,61,35)(10,62,36)(11,63,37)(12,64,38)(13,57,39)(14,58,40)(15,59,33)(16,60,34)(17,27,41)(18,28,42)(19,29,43)(20,30,44)(21,31,45)(22,32,46)(23,25,47)(24,26,48), (1,32,61)(2,25,62)(3,26,63)(4,27,64)(5,28,57)(6,29,58)(7,30,59)(8,31,60)(9,67,22)(10,68,23)(11,69,24)(12,70,17)(13,71,18)(14,72,19)(15,65,20)(16,66,21)(33,56,44)(34,49,45)(35,50,46)(36,51,47)(37,52,48)(38,53,41)(39,54,42)(40,55,43), (1,67,35)(2,68,36)(3,69,37)(4,70,38)(5,71,39)(6,72,40)(7,65,33)(8,66,34)(9,46,61)(10,47,62)(11,48,63)(12,41,64)(13,42,57)(14,43,58)(15,44,59)(16,45,60)(17,53,27)(18,54,28)(19,55,29)(20,56,30)(21,49,31)(22,50,32)(23,51,25)(24,52,26) );
G=PermutationGroup([[(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)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40),(41,45),(43,47),(49,53),(51,55),(58,62),(60,64),(66,70),(68,72)], [(1,50,67),(2,51,68),(3,52,69),(4,53,70),(5,54,71),(6,55,72),(7,56,65),(8,49,66),(9,61,35),(10,62,36),(11,63,37),(12,64,38),(13,57,39),(14,58,40),(15,59,33),(16,60,34),(17,27,41),(18,28,42),(19,29,43),(20,30,44),(21,31,45),(22,32,46),(23,25,47),(24,26,48)], [(1,32,61),(2,25,62),(3,26,63),(4,27,64),(5,28,57),(6,29,58),(7,30,59),(8,31,60),(9,67,22),(10,68,23),(11,69,24),(12,70,17),(13,71,18),(14,72,19),(15,65,20),(16,66,21),(33,56,44),(34,49,45),(35,50,46),(36,51,47),(37,52,48),(38,53,41),(39,54,42),(40,55,43)], [(1,67,35),(2,68,36),(3,69,37),(4,70,38),(5,71,39),(6,72,40),(7,65,33),(8,66,34),(9,46,61),(10,47,62),(11,48,63),(12,41,64),(13,42,57),(14,43,58),(15,44,59),(16,45,60),(17,53,27),(18,54,28),(19,55,29),(20,56,30),(21,49,31),(22,50,32),(23,51,25),(24,52,26)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | ··· | 3J | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 6M | ··· | 6T | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | ··· | 12V | 12W | ··· | 12AD | 24A | ··· | 24H | 24I | ··· | 24AN |
| order | 1 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 1 | 1 | 3 | ··· | 3 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | ··· | 2 | 6 | ··· | 6 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | M4(2) | C3×M4(2) | He3 | C2×He3 | C2×He3 | C4×He3 | C4×He3 | M4(2)×He3 |
| kernel | M4(2)×He3 | C8×He3 | C2×C4×He3 | C32×M4(2) | C4×He3 | C22×He3 | C3×C24 | C6×C12 | C3×C12 | C62 | He3 | C32 | M4(2) | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 16 | 8 | 16 | 16 | 2 | 16 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of M4(2)×He3 ►in GL5(𝔽73)
| 18 | 54 | 0 | 0 | 0 |
| 30 | 55 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 |
| 48 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 9 | 65 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 64 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 64 | 16 | 18 |
| 0 | 0 | 40 | 9 | 65 |
| 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(5,GF(73))| [18,30,0,0,0,54,55,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[1,48,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,8,0,40,0,0,0,0,9,0,0,0,1,65],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[8,0,0,0,0,0,8,0,0,0,0,0,64,40,0,0,0,16,9,8,0,0,18,65,0] >;
M4(2)×He3 in GAP, Magma, Sage, TeX
M_4(2)\times {\rm He}_3 % in TeX
G:=Group("M4(2)xHe3"); // GroupNames label
G:=SmallGroup(432,213);
// by ID
G=gap.SmallGroup(432,213);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,252,3053,605,242]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=b^2=c^3=d^3=e^3=1,b*a*b=a^5,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,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations