direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3xC22:Q8, C12.62D4, C4:C4:3C6, (C2xC6):2Q8, (C6xQ8):8C2, (C2xQ8):3C6, C2.6(C6xD4), C2.3(C6xQ8), C4.13(C3xD4), C6.69(C2xD4), C6.20(C2xQ8), C22:2(C3xQ8), C22:C4.1C6, (C22xC4).7C6, C6.42(C4oD4), (C2xC6).77C23, C23.12(C2xC6), (C22xC12).15C2, (C2xC12).124C22, (C22xC6).28C22, C22.12(C22xC6), (C3xC4:C4):12C2, (C2xC4).4(C2xC6), C2.5(C3xC4oD4), (C3xC22:C4).4C2, SmallGroup(96,169)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xC22:Q8
G = < a,b,c,d,e | a3=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 100 in 74 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2xC4, C2xC4, C2xC4, Q8, C23, C12, C12, C2xC6, C2xC6, C2xC6, C22:C4, C4:C4, C4:C4, C22xC4, C2xQ8, C2xC12, C2xC12, C2xC12, C3xQ8, C22xC6, C22:Q8, C3xC22:C4, C3xC4:C4, C3xC4:C4, C22xC12, C6xQ8, C3xC22:Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2xC6, C2xD4, C2xQ8, C4oD4, C3xD4, C3xQ8, C22xC6, C22:Q8, C6xD4, C6xQ8, C3xC4oD4, C3xC22:Q8
►On 48 points
(1 5 19)(2 6 20)(3 7 17)(4 8 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 43)(30 37 44)(31 38 41)(32 39 42)
(13 37)(14 38)(15 39)(16 40)(29 35)(30 36)(31 33)(32 34)(41 47)(42 48)(43 45)(44 46)
(1 11)(2 12)(3 9)(4 10)(5 25)(6 26)(7 27)(8 28)(13 37)(14 38)(15 39)(16 40)(17 21)(18 22)(19 23)(20 24)(29 35)(30 36)(31 33)(32 34)(41 47)(42 48)(43 45)(44 46)
(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 31 3 29)(2 30 4 32)(5 38 7 40)(6 37 8 39)(9 35 11 33)(10 34 12 36)(13 28 15 26)(14 27 16 25)(17 43 19 41)(18 42 20 44)(21 45 23 47)(22 48 24 46)
G:=sub<Sym(48)| (1,5,19)(2,6,20)(3,7,17)(4,8,18)(9,27,21)(10,28,22)(11,25,23)(12,26,24)(13,46,36)(14,47,33)(15,48,34)(16,45,35)(29,40,43)(30,37,44)(31,38,41)(32,39,42), (13,37)(14,38)(15,39)(16,40)(29,35)(30,36)(31,33)(32,34)(41,47)(42,48)(43,45)(44,46), (1,11)(2,12)(3,9)(4,10)(5,25)(6,26)(7,27)(8,28)(13,37)(14,38)(15,39)(16,40)(17,21)(18,22)(19,23)(20,24)(29,35)(30,36)(31,33)(32,34)(41,47)(42,48)(43,45)(44,46), (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), (1,31,3,29)(2,30,4,32)(5,38,7,40)(6,37,8,39)(9,35,11,33)(10,34,12,36)(13,28,15,26)(14,27,16,25)(17,43,19,41)(18,42,20,44)(21,45,23,47)(22,48,24,46)>;
G:=Group( (1,5,19)(2,6,20)(3,7,17)(4,8,18)(9,27,21)(10,28,22)(11,25,23)(12,26,24)(13,46,36)(14,47,33)(15,48,34)(16,45,35)(29,40,43)(30,37,44)(31,38,41)(32,39,42), (13,37)(14,38)(15,39)(16,40)(29,35)(30,36)(31,33)(32,34)(41,47)(42,48)(43,45)(44,46), (1,11)(2,12)(3,9)(4,10)(5,25)(6,26)(7,27)(8,28)(13,37)(14,38)(15,39)(16,40)(17,21)(18,22)(19,23)(20,24)(29,35)(30,36)(31,33)(32,34)(41,47)(42,48)(43,45)(44,46), (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), (1,31,3,29)(2,30,4,32)(5,38,7,40)(6,37,8,39)(9,35,11,33)(10,34,12,36)(13,28,15,26)(14,27,16,25)(17,43,19,41)(18,42,20,44)(21,45,23,47)(22,48,24,46) );
G=PermutationGroup([[(1,5,19),(2,6,20),(3,7,17),(4,8,18),(9,27,21),(10,28,22),(11,25,23),(12,26,24),(13,46,36),(14,47,33),(15,48,34),(16,45,35),(29,40,43),(30,37,44),(31,38,41),(32,39,42)], [(13,37),(14,38),(15,39),(16,40),(29,35),(30,36),(31,33),(32,34),(41,47),(42,48),(43,45),(44,46)], [(1,11),(2,12),(3,9),(4,10),(5,25),(6,26),(7,27),(8,28),(13,37),(14,38),(15,39),(16,40),(17,21),(18,22),(19,23),(20,24),(29,35),(30,36),(31,33),(32,34),(41,47),(42,48),(43,45),(44,46)], [(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)], [(1,31,3,29),(2,30,4,32),(5,38,7,40),(6,37,8,39),(9,35,11,33),(10,34,12,36),(13,28,15,26),(14,27,16,25),(17,43,19,41),(18,42,20,44),(21,45,23,47),(22,48,24,46)]])
C3xC22:Q8 is a maximal subgroup of
(C6xQ8):C4 (C2xQ8).49D6 (C2xC6).Q16 (C2xQ8).51D6 D12.36D4 D12.37D4 C3:C8:24D4 C3:C8:6D4 Dic6.37D4 C3:C8.29D4 C3:C8.6D4 (Q8xDic3):C2 C6.752- 1+4 C4:C4.187D6 C6.152- 1+4 C4:C4:26D6 C6.162- 1+4 C6.172- 1+4 D12:21D4 D12:22D4 Dic6:21D4 Dic6:22D4 C6.512+ 1+4 C6.1182+ 1+4 C6.522+ 1+4 C6.532+ 1+4 C6.202- 1+4 C6.212- 1+4 C6.222- 1+4 C6.232- 1+4 C6.772- 1+4 C6.242- 1+4 C6.562+ 1+4 C6.782- 1+4 C6.252- 1+4 C6.592+ 1+4 C3xD4xQ8
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 12A | ··· | 12H | 12I | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | Q8 | C4oD4 | C3xD4 | C3xQ8 | C3xC4oD4 |
| kernel | C3xC22:Q8 | C3xC22:C4 | C3xC4:C4 | C22xC12 | C6xQ8 | C22:Q8 | C22:C4 | C4:C4 | C22xC4 | C2xQ8 | C12 | C2xC6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 3 | 1 | 1 | 2 | 4 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C3xC22:Q8 ►in GL4(F13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 9 | 10 | 0 | 0 |
| 10 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[0,12,0,0,1,0,0,0,0,0,12,0,0,0,0,12],[9,10,0,0,10,4,0,0,0,0,0,1,0,0,1,0] >;
C3xC22:Q8 in GAP, Magma, Sage, TeX
C_3\times C_2^2\rtimes Q_8
% in TeX
G:=Group("C3xC2^2:Q8"); // GroupNames label
G:=SmallGroup(96,169);
// by ID
G=gap.SmallGroup(96,169);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,144,313,151,938]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations