direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C3⋊D4, C12⋊8D4, C23.27D6, C3⋊4(C4×D4), D6⋊4(C2×C4), C4○3(D6⋊C4), D6⋊C4⋊18C2, (C22×C4)⋊4S3, C22⋊3(C4×S3), C6.41(C2×D4), (C22×C12)⋊9C2, Dic3⋊2(C2×C4), (C2×C4).103D6, C4○3(Dic3⋊C4), Dic3⋊C4⋊18C2, (C4×Dic3)⋊16C2, C6.17(C4○D4), C2.5(C4○D12), (C2×C6).46C23, C6.19(C22×C4), C4○2(C6.D4), C6.D4⋊14C2, (C2×C12).77C22, (C22×C6).38C22, C22.24(C22×S3), (C22×S3).24C22, (C2×Dic3).36C22, (S3×C2×C4)⋊14C2, (C2×C6)⋊5(C2×C4), C2.20(S3×C2×C4), C2.3(C2×C3⋊D4), (C2×C3⋊D4).7C2, (C2×C4)○(C6.D4), SmallGroup(96,135)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C3⋊D4
G = < a,b,c,d | a4=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 194 in 94 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4×D4, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C4×C3⋊D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C3⋊D4, C22×S3, C4×D4, S3×C2×C4, C4○D12, C2×C3⋊D4, C4×C3⋊D4
►On 48 points
(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 37 23)(2 38 24)(3 39 21)(4 40 22)(5 36 46)(6 33 47)(7 34 48)(8 35 45)(9 29 27)(10 30 28)(11 31 25)(12 32 26)(13 43 17)(14 44 18)(15 41 19)(16 42 20)
(1 47 27 14)(2 48 28 15)(3 45 25 16)(4 46 26 13)(5 32 43 22)(6 29 44 23)(7 30 41 24)(8 31 42 21)(9 18 37 33)(10 19 38 34)(11 20 39 35)(12 17 40 36)
(1 3)(2 4)(5 19)(6 20)(7 17)(8 18)(9 31)(10 32)(11 29)(12 30)(13 48)(14 45)(15 46)(16 47)(21 37)(22 38)(23 39)(24 40)(25 27)(26 28)(33 42)(34 43)(35 44)(36 41)
G:=sub<Sym(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), (1,37,23)(2,38,24)(3,39,21)(4,40,22)(5,36,46)(6,33,47)(7,34,48)(8,35,45)(9,29,27)(10,30,28)(11,31,25)(12,32,26)(13,43,17)(14,44,18)(15,41,19)(16,42,20), (1,47,27,14)(2,48,28,15)(3,45,25,16)(4,46,26,13)(5,32,43,22)(6,29,44,23)(7,30,41,24)(8,31,42,21)(9,18,37,33)(10,19,38,34)(11,20,39,35)(12,17,40,36), (1,3)(2,4)(5,19)(6,20)(7,17)(8,18)(9,31)(10,32)(11,29)(12,30)(13,48)(14,45)(15,46)(16,47)(21,37)(22,38)(23,39)(24,40)(25,27)(26,28)(33,42)(34,43)(35,44)(36,41)>;
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), (1,37,23)(2,38,24)(3,39,21)(4,40,22)(5,36,46)(6,33,47)(7,34,48)(8,35,45)(9,29,27)(10,30,28)(11,31,25)(12,32,26)(13,43,17)(14,44,18)(15,41,19)(16,42,20), (1,47,27,14)(2,48,28,15)(3,45,25,16)(4,46,26,13)(5,32,43,22)(6,29,44,23)(7,30,41,24)(8,31,42,21)(9,18,37,33)(10,19,38,34)(11,20,39,35)(12,17,40,36), (1,3)(2,4)(5,19)(6,20)(7,17)(8,18)(9,31)(10,32)(11,29)(12,30)(13,48)(14,45)(15,46)(16,47)(21,37)(22,38)(23,39)(24,40)(25,27)(26,28)(33,42)(34,43)(35,44)(36,41) );
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)], [(1,37,23),(2,38,24),(3,39,21),(4,40,22),(5,36,46),(6,33,47),(7,34,48),(8,35,45),(9,29,27),(10,30,28),(11,31,25),(12,32,26),(13,43,17),(14,44,18),(15,41,19),(16,42,20)], [(1,47,27,14),(2,48,28,15),(3,45,25,16),(4,46,26,13),(5,32,43,22),(6,29,44,23),(7,30,41,24),(8,31,42,21),(9,18,37,33),(10,19,38,34),(11,20,39,35),(12,17,40,36)], [(1,3),(2,4),(5,19),(6,20),(7,17),(8,18),(9,31),(10,32),(11,29),(12,30),(13,48),(14,45),(15,46),(16,47),(21,37),(22,38),(23,39),(24,40),(25,27),(26,28),(33,42),(34,43),(35,44),(36,41)]])
C4×C3⋊D4 is a maximal subgroup of
C3⋊D4⋊C8 D6⋊2M4(2) Dic3⋊M4(2) C3⋊C8⋊26D4 C24⋊33D4 C24⋊D4 C24⋊21D4 C42.277D6 C24.35D6 C24.38D6 C24.41D6 C24.42D6 C6.82+ 1+4 C6.2- 1+4 C6.102+ 1+4 C6.52- 1+4 C6.112+ 1+4 C6.62- 1+4 C42.93D6 C42.94D6 C42.95D6 C42.97D6 C42.98D6 C42.102D6 C42.104D6 C4×S3×D4 C42⋊13D6 C42.108D6 C42⋊14D6 C42.228D6 C42⋊18D6 C42.229D6 C42.113D6 C42.114D6 C42⋊19D6 C42.118D6 Dic6⋊19D4 Dic6⋊20D4 C6.342+ 1+4 D12⋊19D4 C6.402+ 1+4 C6.732- 1+4 D12⋊20D4 C6.422+ 1+4 C6.432+ 1+4 C6.442+ 1+4 C6.452+ 1+4 C6.1152+ 1+4 D12⋊21D4 D12⋊22D4 Dic6⋊21D4 Dic6⋊22D4 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 C6.612+ 1+4 C6.622+ 1+4 C6.632+ 1+4 C6.642+ 1+4 C6.652+ 1+4 C6.662+ 1+4 C6.672+ 1+4 C24.83D6 C24.53D6 C6.452- 1+4 C6.1042- 1+4 (C2×D4)⋊43D6 C6.1452+ 1+4 C6.1072- 1+4 (C2×C12)⋊17D4 C6.1482+ 1+4 C62.49C23 C62.74C23 C62.94C23 D6⋊(C4×D5) C15⋊20(C4×D4) C15⋊26(C4×D4)
C4×C3⋊D4 is a maximal quotient of
(C2×C42).6S3 (C2×C42)⋊3S3 C24.14D6 C24.15D6 C24.23D6 C24.24D6 Dic3⋊(C4⋊C4) C6.67(C4×D4) D6⋊C4⋊6C4 D6⋊C4⋊7C4 C42.48D6 C42.51D6 C42.56D6 C42.59D6 C24⋊33D4 C24⋊D4 C24⋊21D4 C24.100D4 C24.54D4 C24.73D6 C24.76D6 C62.49C23 C62.74C23 C62.94C23 D6⋊(C4×D5) C15⋊20(C4×D4) C15⋊26(C4×D4)
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 6A | ··· | 6G | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C4×S3 | C4○D12 |
| kernel | C4×C3⋊D4 | C4×Dic3 | Dic3⋊C4 | D6⋊C4 | C6.D4 | S3×C2×C4 | C2×C3⋊D4 | C22×C12 | C3⋊D4 | C22×C4 | C12 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 4 |
Matrix representation of C4×C3⋊D4 ►in GL3(𝔽13) generated by
| 8 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 1 | 0 | 0 |
| 0 | 12 | 12 |
| 0 | 1 | 0 |
| 12 | 0 | 0 |
| 0 | 11 | 9 |
| 0 | 11 | 2 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 12 | 12 |
G:=sub<GL(3,GF(13))| [8,0,0,0,12,0,0,0,12],[1,0,0,0,12,1,0,12,0],[12,0,0,0,11,11,0,9,2],[1,0,0,0,1,12,0,0,12] >;
C4×C3⋊D4 in GAP, Magma, Sage, TeX
C_4\times C_3\rtimes D_4
% in TeX
G:=Group("C4xC3:D4"); // GroupNames label
G:=SmallGroup(96,135);
// by ID
G=gap.SmallGroup(96,135);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,50,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations