direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C22⋊C8, C22⋊2C24, C12.65D4, C23.3C12, C6.8M4(2), (C2×C8)⋊1C6, (C2×C6)⋊1C8, (C2×C24)⋊3C2, C2.1(C2×C24), C6.11(C2×C8), (C2×C12).6C4, (C2×C4).3C12, C4.16(C3×D4), (C22×C4).4C6, (C22×C6).3C4, C22.9(C2×C12), (C22×C12).3C2, C2.2(C3×M4(2)), C6.20(C22⋊C4), (C2×C12).135C22, (C2×C4).31(C2×C6), (C2×C6).38(C2×C4), C2.2(C3×C22⋊C4), SmallGroup(96,48)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C22⋊C8
G = < a,b,c,d | a3=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Smallest permutation representation of C3×C22⋊C8
►On 48 points
(1 14 39)(2 15 40)(3 16 33)(4 9 34)(5 10 35)(6 11 36)(7 12 37)(8 13 38)(17 29 41)(18 30 42)(19 31 43)(20 32 44)(21 25 45)(22 26 46)(23 27 47)(24 28 48)
(2 24)(4 18)(6 20)(8 22)(9 30)(11 32)(13 26)(15 28)(34 42)(36 44)(38 46)(40 48)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(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)| (1,14,39)(2,15,40)(3,16,33)(4,9,34)(5,10,35)(6,11,36)(7,12,37)(8,13,38)(17,29,41)(18,30,42)(19,31,43)(20,32,44)(21,25,45)(22,26,46)(23,27,47)(24,28,48), (2,24)(4,18)(6,20)(8,22)(9,30)(11,32)(13,26)(15,28)(34,42)(36,44)(38,46)(40,48), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(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( (1,14,39)(2,15,40)(3,16,33)(4,9,34)(5,10,35)(6,11,36)(7,12,37)(8,13,38)(17,29,41)(18,30,42)(19,31,43)(20,32,44)(21,25,45)(22,26,46)(23,27,47)(24,28,48), (2,24)(4,18)(6,20)(8,22)(9,30)(11,32)(13,26)(15,28)(34,42)(36,44)(38,46)(40,48), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(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([[(1,14,39),(2,15,40),(3,16,33),(4,9,34),(5,10,35),(6,11,36),(7,12,37),(8,13,38),(17,29,41),(18,30,42),(19,31,43),(20,32,44),(21,25,45),(22,26,46),(23,27,47),(24,28,48)], [(2,24),(4,18),(6,20),(8,22),(9,30),(11,32),(13,26),(15,28),(34,42),(36,44),(38,46),(40,48)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29),(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)]])
C3×C22⋊C8 is a maximal subgroup of
C23.35D12 (C22×S3)⋊C8 (C2×Dic3)⋊C8 C22.2D24 Dic3.5M4(2) Dic3.M4(2) C24⋊C4⋊C2 C23.39D12 C23.40D12 C23.15D12 C3⋊D4⋊C8 D6⋊M4(2) D6⋊C8⋊C2 D6⋊2M4(2) Dic3⋊M4(2) C3⋊C8⋊26D4 D12.31D4 D12⋊13D4 D12.32D4 D12⋊14D4 C23.43D12 C22.D24 C23.18D12 Dic6⋊14D4 Dic6.32D4 D4×C24
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | D4 | M4(2) | C3×D4 | C3×M4(2) |
| kernel | C3×C22⋊C8 | C2×C24 | C22×C12 | C22⋊C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 | C12 | C6 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C3×C22⋊C8 ►in GL3(𝔽73) generated by
| 8 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 46 | 72 |
| 1 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 63 | 0 | 0 |
| 0 | 46 | 71 |
| 0 | 0 | 27 |
G:=sub<GL(3,GF(73))| [8,0,0,0,1,0,0,0,1],[72,0,0,0,1,46,0,0,72],[1,0,0,0,72,0,0,0,72],[63,0,0,0,46,0,0,71,27] >;
C3×C22⋊C8 in GAP, Magma, Sage, TeX
C_3\times C_2^2\rtimes C_8
% in TeX
G:=Group("C3xC2^2:C8"); // GroupNames label
G:=SmallGroup(96,48);
// by ID
G=gap.SmallGroup(96,48);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,88]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
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