direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C4×Q8, C22.9C24, C42.86C22, C23.70C23, C4○(C4×Q8), C2.5(C23×C4), C2.2(C22×Q8), C4⋊C4.78C22, C4.17(C22×C4), (C2×C42).17C2, C22.17(C2×Q8), (C2×C4).127C23, (C22×Q8).10C2, (C2×Q8).70C22, C22.28(C4○D4), C22.26(C22×C4), (C22×C4).120C22, C4○3(C2×C4⋊C4), (C2×C4)○(C4×Q8), (C2×C4)○4(C4⋊C4), C2.3(C2×C4○D4), (C2×C4⋊C4).22C2, (C2×C4).49(C2×C4), (C2×C4)○3(C2×C4⋊C4), SmallGroup(64,197)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4×Q8
G = < a,b,c,d | a2=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 161 in 149 conjugacy classes, 137 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×C42, C2×C4⋊C4, C4×Q8, C22×Q8, C2×C4×Q8
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, C24, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, C2×C4×Q8
►Regular action on 64 points
(1 9)(2 10)(3 11)(4 12)(5 24)(6 21)(7 22)(8 23)(13 63)(14 64)(15 61)(16 62)(17 25)(18 26)(19 27)(20 28)(29 39)(30 40)(31 37)(32 38)(33 41)(34 42)(35 43)(36 44)(45 55)(46 56)(47 53)(48 54)(49 57)(50 58)(51 59)(52 60)
(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)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 29 21 18)(2 30 22 19)(3 31 23 20)(4 32 24 17)(5 25 12 38)(6 26 9 39)(7 27 10 40)(8 28 11 37)(13 47 60 36)(14 48 57 33)(15 45 58 34)(16 46 59 35)(41 64 54 49)(42 61 55 50)(43 62 56 51)(44 63 53 52)
(1 48 21 33)(2 45 22 34)(3 46 23 35)(4 47 24 36)(5 44 12 53)(6 41 9 54)(7 42 10 55)(8 43 11 56)(13 17 60 32)(14 18 57 29)(15 19 58 30)(16 20 59 31)(25 52 38 63)(26 49 39 64)(27 50 40 61)(28 51 37 62)
G:=sub<Sym(64)| (1,9)(2,10)(3,11)(4,12)(5,24)(6,21)(7,22)(8,23)(13,63)(14,64)(15,61)(16,62)(17,25)(18,26)(19,27)(20,28)(29,39)(30,40)(31,37)(32,38)(33,41)(34,42)(35,43)(36,44)(45,55)(46,56)(47,53)(48,54)(49,57)(50,58)(51,59)(52,60), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,29,21,18)(2,30,22,19)(3,31,23,20)(4,32,24,17)(5,25,12,38)(6,26,9,39)(7,27,10,40)(8,28,11,37)(13,47,60,36)(14,48,57,33)(15,45,58,34)(16,46,59,35)(41,64,54,49)(42,61,55,50)(43,62,56,51)(44,63,53,52), (1,48,21,33)(2,45,22,34)(3,46,23,35)(4,47,24,36)(5,44,12,53)(6,41,9,54)(7,42,10,55)(8,43,11,56)(13,17,60,32)(14,18,57,29)(15,19,58,30)(16,20,59,31)(25,52,38,63)(26,49,39,64)(27,50,40,61)(28,51,37,62)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,24)(6,21)(7,22)(8,23)(13,63)(14,64)(15,61)(16,62)(17,25)(18,26)(19,27)(20,28)(29,39)(30,40)(31,37)(32,38)(33,41)(34,42)(35,43)(36,44)(45,55)(46,56)(47,53)(48,54)(49,57)(50,58)(51,59)(52,60), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,29,21,18)(2,30,22,19)(3,31,23,20)(4,32,24,17)(5,25,12,38)(6,26,9,39)(7,27,10,40)(8,28,11,37)(13,47,60,36)(14,48,57,33)(15,45,58,34)(16,46,59,35)(41,64,54,49)(42,61,55,50)(43,62,56,51)(44,63,53,52), (1,48,21,33)(2,45,22,34)(3,46,23,35)(4,47,24,36)(5,44,12,53)(6,41,9,54)(7,42,10,55)(8,43,11,56)(13,17,60,32)(14,18,57,29)(15,19,58,30)(16,20,59,31)(25,52,38,63)(26,49,39,64)(27,50,40,61)(28,51,37,62) );
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,24),(6,21),(7,22),(8,23),(13,63),(14,64),(15,61),(16,62),(17,25),(18,26),(19,27),(20,28),(29,39),(30,40),(31,37),(32,38),(33,41),(34,42),(35,43),(36,44),(45,55),(46,56),(47,53),(48,54),(49,57),(50,58),(51,59),(52,60)], [(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),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,29,21,18),(2,30,22,19),(3,31,23,20),(4,32,24,17),(5,25,12,38),(6,26,9,39),(7,27,10,40),(8,28,11,37),(13,47,60,36),(14,48,57,33),(15,45,58,34),(16,46,59,35),(41,64,54,49),(42,61,55,50),(43,62,56,51),(44,63,53,52)], [(1,48,21,33),(2,45,22,34),(3,46,23,35),(4,47,24,36),(5,44,12,53),(6,41,9,54),(7,42,10,55),(8,43,11,56),(13,17,60,32),(14,18,57,29),(15,19,58,30),(16,20,59,31),(25,52,38,63),(26,49,39,64),(27,50,40,61),(28,51,37,62)]])
C2×C4×Q8 is a maximal subgroup of
C42.394D4 C42.44D4 C42.399D4 Q8⋊M4(2) Q8⋊5M4(2) Q8⋊C42 C42.97D4 C42.99D4 C42.101D4 Q8⋊(C4⋊C4) Q8⋊C4⋊C4 (C2×C4)⋊9Q16 (C2×SD16)⋊15C4 C42.327D4 C42.120D4 Q8⋊4C42 C42⋊14Q8 C23.202C24 C42.159D4 C42.160D4 C42.161D4 C23.223C24 C23.233C24 C23.237C24 C23.238C24 C24.558C23 C23.244C24 C23.247C24 C24.220C23 C42.162D4 C42.163D4 C42⋊5Q8 C23.321C24 C23.323C24 C24.259C23 C23.329C24 C23.346C24 C23.348C24 C23.351C24 C23.353C24 C24.279C23 C23.362C24 C24.285C23 C23.369C24 C42.165D4 C42.166D4 C42.168D4 C42.169D4 C42.171D4 C42.174D4 C42.176D4 C42.177D4 C42.178D4 C42.179D4 C42.180D4 C42.181D4 C42.182D4 C42.183D4 C42.184D4 C42.189D4 C42.191D4 C42.192D4 C42.695C23 C42.302C23 Q8.4M4(2) C42.212D4 C42.220D4 C42.223D4 C42.224D4 C42.226D4 C42.230D4 C42.231D4 C42.235D4 C22.50C25 C22.69C25 C22.71C25 C4⋊2- 1+4 C22.91C25 C22.96C25 C22.105C25 C22.111C25 C23.146C24
C2×C4×Q8 is a maximal quotient of
C42⋊14Q8 C23.211C24 C42.33Q8 C42⋊4Q8 C23.227C24 C23.237C24 C24.558C23 C23.247C24 C23.250C24 C23.251C24 C23.252C24 C42.286C23 C42.287C23 M4(2)⋊9Q8
40 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | Q8 | C4○D4 |
| kernel | C2×C4×Q8 | C2×C42 | C2×C4⋊C4 | C4×Q8 | C22×Q8 | C2×Q8 | C2×C4 | C22 |
| # reps | 1 | 3 | 3 | 8 | 1 | 16 | 4 | 4 |
Matrix representation of C2×C4×Q8 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,4,0,0,0,0,0,4,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,0,3,0,0,3,0] >;
C2×C4×Q8 in GAP, Magma, Sage, TeX
C_2\times C_4\times Q_8
% in TeX
G:=Group("C2xC4xQ8"); // GroupNames label
G:=SmallGroup(64,197);
// by ID
G=gap.SmallGroup(64,197);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,192,217,103,230]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations