direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C23.C8, C23.C24, C24.104D4, M5(2)⋊3C6, C12.34M4(2), (C2×C4).C24, (C2×C8).3C12, (C2×C24).6C4, (C2×C12).2C8, C8.24(C3×D4), (C22×C6).1C8, C22.4(C2×C24), (C22×C4).5C12, (C22×C12).9C4, C4.7(C3×M4(2)), C6.26(C22⋊C8), (C3×M5(2))⋊11C2, (C2×C24).309C22, (C6×M4(2)).22C2, (C2×M4(2)).10C6, C12.112(C22⋊C4), (C2×C6).22(C2×C8), (C2×C8).46(C2×C6), C2.7(C3×C22⋊C8), (C2×C4).67(C2×C12), C4.29(C3×C22⋊C4), (C2×C12).328(C2×C4), SmallGroup(192,155)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.C8
G = < a,b,c,d,e | a3=b2=c2=d2=1, e8=d, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >
Subgroups: 90 in 58 conjugacy classes, 34 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C16, C2×C8, M4(2), C22×C4, C24, C24, C2×C12, C2×C12, C22×C6, M5(2), C2×M4(2), C48, C2×C24, C3×M4(2), C22×C12, C23.C8, C3×M5(2), C6×M4(2), C3×C23.C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C8, M4(2), C24, C2×C12, C3×D4, C22⋊C8, C3×C22⋊C4, C2×C24, C3×M4(2), C23.C8, C3×C22⋊C8, C3×C23.C8
►On 48 points
(1 28 45)(2 29 46)(3 30 47)(4 31 48)(5 32 33)(6 17 34)(7 18 35)(8 19 36)(9 20 37)(10 21 38)(11 22 39)(12 23 40)(13 24 41)(14 25 42)(15 26 43)(16 27 44)
(2 10)(3 11)(6 14)(7 15)(17 25)(18 26)(21 29)(22 30)(34 42)(35 43)(38 46)(39 47)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(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)
G:=sub<Sym(48)| (1,28,45)(2,29,46)(3,30,47)(4,31,48)(5,32,33)(6,17,34)(7,18,35)(8,19,36)(9,20,37)(10,21,38)(11,22,39)(12,23,40)(13,24,41)(14,25,42)(15,26,43)(16,27,44), (2,10)(3,11)(6,14)(7,15)(17,25)(18,26)(21,29)(22,30)(34,42)(35,43)(38,46)(39,47), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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)>;
G:=Group( (1,28,45)(2,29,46)(3,30,47)(4,31,48)(5,32,33)(6,17,34)(7,18,35)(8,19,36)(9,20,37)(10,21,38)(11,22,39)(12,23,40)(13,24,41)(14,25,42)(15,26,43)(16,27,44), (2,10)(3,11)(6,14)(7,15)(17,25)(18,26)(21,29)(22,30)(34,42)(35,43)(38,46)(39,47), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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) );
G=PermutationGroup([[(1,28,45),(2,29,46),(3,30,47),(4,31,48),(5,32,33),(6,17,34),(7,18,35),(8,19,36),(9,20,37),(10,21,38),(11,22,39),(12,23,40),(13,24,41),(14,25,42),(15,26,43),(16,27,44)], [(2,10),(3,11),(6,14),(7,15),(17,25),(18,26),(21,29),(22,30),(34,42),(35,43),(38,46),(39,47)], [(2,10),(4,12),(6,14),(8,16),(17,25),(19,27),(21,29),(23,31),(34,42),(36,44),(38,46),(40,48)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(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)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 16A | ··· | 16H | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C8 | C12 | C12 | C24 | C24 | D4 | M4(2) | C3×D4 | C3×M4(2) | C23.C8 | C3×C23.C8 |
| kernel | C3×C23.C8 | C3×M5(2) | C6×M4(2) | C23.C8 | C2×C24 | C22×C12 | M5(2) | C2×M4(2) | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C4 | C23 | C24 | C12 | C8 | C4 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C3×C23.C8 ►in GL4(𝔽97) generated by
| 61 | 0 | 0 | 0 |
| 0 | 61 | 0 | 0 |
| 0 | 0 | 61 | 0 |
| 0 | 0 | 0 | 61 |
| 1 | 0 | 0 | 7 |
| 0 | 96 | 7 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 96 |
| 1 | 0 | 90 | 7 |
| 0 | 1 | 90 | 90 |
| 0 | 0 | 96 | 0 |
| 0 | 0 | 0 | 96 |
| 96 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 96 | 0 |
| 0 | 0 | 0 | 96 |
| 57 | 90 | 67 | 30 |
| 40 | 90 | 30 | 67 |
| 0 | 95 | 7 | 7 |
| 53 | 0 | 57 | 40 |
G:=sub<GL(4,GF(97))| [61,0,0,0,0,61,0,0,0,0,61,0,0,0,0,61],[1,0,0,0,0,96,0,0,0,7,1,0,7,0,0,96],[1,0,0,0,0,1,0,0,90,90,96,0,7,90,0,96],[96,0,0,0,0,96,0,0,0,0,96,0,0,0,0,96],[57,40,0,53,90,90,95,0,67,30,7,57,30,67,7,40] >;
C3×C23.C8 in GAP, Magma, Sage, TeX
C_3\times C_2^3.C_8
% in TeX
G:=Group("C3xC2^3.C8"); // GroupNames label
G:=SmallGroup(192,155);
// by ID
G=gap.SmallGroup(192,155);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,3027,2111,102,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^2=1,e^8=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations