direct product, metabelian, soluble, monomial
Aliases: C2×Q8×A4, C22⋊(C6×Q8), C23⋊3(C3×Q8), (Q8×C23)⋊2C3, (C23×C4).3C6, C4.7(C22×A4), C2.4(C23×A4), (C22×Q8)⋊6C6, C24.27(C2×C6), (C4×A4).20C22, (C2×A4).13C23, C23.30(C22×C6), C22.19(C22×A4), (C22×A4).17C22, (C2×C4×A4).9C2, (C2×C4).11(C2×A4), (C22×C4).4(C2×C6), SmallGroup(192,1499)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 520 in 205 conjugacy classes, 57 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×6], C4 [×6], C22 [×2], C22 [×11], C6 [×3], C2×C4 [×3], C2×C4 [×27], Q8 [×4], Q8 [×20], C23, C23 [×2], C23 [×4], C12 [×6], A4, C2×C6, C22×C4 [×6], C22×C4 [×12], C2×Q8, C2×Q8 [×37], C24, C2×C12 [×3], C3×Q8 [×4], C2×A4, C2×A4 [×2], C23×C4 [×3], C22×Q8 [×4], C22×Q8 [×8], C4×A4 [×6], C6×Q8, C22×A4, Q8×C23, C2×C4×A4 [×3], Q8×A4 [×4], C2×Q8×A4
Quotients:
C1, C2 [×7], C3, C22 [×7], C6 [×7], Q8 [×2], C23, A4, C2×C6 [×7], C2×Q8, C3×Q8 [×2], C2×A4 [×7], C22×C6, C6×Q8, C22×A4 [×7], Q8×A4 [×2], C23×A4, C2×Q8×A4
Generators and relations
G = < a,b,c,d,e,f | a2=b4=d2=e2=f3=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >
►On 48 points
(1 11)(2 12)(3 9)(4 10)(5 27)(6 28)(7 25)(8 26)(13 31)(14 32)(15 29)(16 30)(17 21)(18 22)(19 23)(20 24)(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)
(1 41 3 43)(2 44 4 42)(5 30 7 32)(6 29 8 31)(9 35 11 33)(10 34 12 36)(13 28 15 26)(14 27 16 25)(17 37 19 39)(18 40 20 38)(21 45 23 47)(22 48 24 46)
(5 27)(6 28)(7 25)(8 26)(13 31)(14 32)(15 29)(16 30)(17 21)(18 22)(19 23)(20 24)(37 45)(38 46)(39 47)(40 48)
(1 11)(2 12)(3 9)(4 10)(17 21)(18 22)(19 23)(20 24)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 7 19)(2 8 20)(3 5 17)(4 6 18)(9 27 21)(10 28 22)(11 25 23)(12 26 24)(13 46 36)(14 47 33)(15 48 34)(16 45 35)(29 40 42)(30 37 43)(31 38 44)(32 39 41)
G:=sub<Sym(48)| (1,11)(2,12)(3,9)(4,10)(5,27)(6,28)(7,25)(8,26)(13,31)(14,32)(15,29)(16,30)(17,21)(18,22)(19,23)(20,24)(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), (1,41,3,43)(2,44,4,42)(5,30,7,32)(6,29,8,31)(9,35,11,33)(10,34,12,36)(13,28,15,26)(14,27,16,25)(17,37,19,39)(18,40,20,38)(21,45,23,47)(22,48,24,46), (5,27)(6,28)(7,25)(8,26)(13,31)(14,32)(15,29)(16,30)(17,21)(18,22)(19,23)(20,24)(37,45)(38,46)(39,47)(40,48), (1,11)(2,12)(3,9)(4,10)(17,21)(18,22)(19,23)(20,24)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,27,21)(10,28,22)(11,25,23)(12,26,24)(13,46,36)(14,47,33)(15,48,34)(16,45,35)(29,40,42)(30,37,43)(31,38,44)(32,39,41)>;
G:=Group( (1,11)(2,12)(3,9)(4,10)(5,27)(6,28)(7,25)(8,26)(13,31)(14,32)(15,29)(16,30)(17,21)(18,22)(19,23)(20,24)(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), (1,41,3,43)(2,44,4,42)(5,30,7,32)(6,29,8,31)(9,35,11,33)(10,34,12,36)(13,28,15,26)(14,27,16,25)(17,37,19,39)(18,40,20,38)(21,45,23,47)(22,48,24,46), (5,27)(6,28)(7,25)(8,26)(13,31)(14,32)(15,29)(16,30)(17,21)(18,22)(19,23)(20,24)(37,45)(38,46)(39,47)(40,48), (1,11)(2,12)(3,9)(4,10)(17,21)(18,22)(19,23)(20,24)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,27,21)(10,28,22)(11,25,23)(12,26,24)(13,46,36)(14,47,33)(15,48,34)(16,45,35)(29,40,42)(30,37,43)(31,38,44)(32,39,41) );
G=PermutationGroup([(1,11),(2,12),(3,9),(4,10),(5,27),(6,28),(7,25),(8,26),(13,31),(14,32),(15,29),(16,30),(17,21),(18,22),(19,23),(20,24),(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)], [(1,41,3,43),(2,44,4,42),(5,30,7,32),(6,29,8,31),(9,35,11,33),(10,34,12,36),(13,28,15,26),(14,27,16,25),(17,37,19,39),(18,40,20,38),(21,45,23,47),(22,48,24,46)], [(5,27),(6,28),(7,25),(8,26),(13,31),(14,32),(15,29),(16,30),(17,21),(18,22),(19,23),(20,24),(37,45),(38,46),(39,47),(40,48)], [(1,11),(2,12),(3,9),(4,10),(17,21),(18,22),(19,23),(20,24),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(1,7,19),(2,8,20),(3,5,17),(4,6,18),(9,27,21),(10,28,22),(11,25,23),(12,26,24),(13,46,36),(14,47,33),(15,48,34),(16,45,35),(29,40,42),(30,37,43),(31,38,44),(32,39,41)])
Matrix representation ►G ⊆ GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 11 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,1,0,0,0,11,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,5,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[9,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | ··· | 4F | 4G | ··· | 4L | 6A | ··· | 6F | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 8 | ··· | 8 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 6 |
| type | + | + | + | - | + | + | + | - | ||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | Q8 | C3×Q8 | A4 | C2×A4 | C2×A4 | Q8×A4 |
| kernel | C2×Q8×A4 | C2×C4×A4 | Q8×A4 | Q8×C23 | C23×C4 | C22×Q8 | C2×A4 | C23 | C2×Q8 | C2×C4 | Q8 | C2 |
| # reps | 1 | 3 | 4 | 2 | 6 | 8 | 2 | 4 | 1 | 3 | 4 | 2 |
In GAP, Magma, Sage, TeX
C_2\times Q_8\times A_4
% in TeX
G:=Group("C2xQ8xA4"); // GroupNames label
G:=SmallGroup(192,1499);
// by ID
G=gap.SmallGroup(192,1499);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,2,176,303,142,530,909]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^4=d^2=e^2=f^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations