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G = C2×C4×Q8order 64 = 26

Direct product of C2×C4 and Q8

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, C43(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

C1C2 — C2×C4×Q8
C1C2C22C23C22×C4C2×C42 — C2×C4×Q8
C1C2 — C2×C4×Q8
C1C22×C4 — C2×C4×Q8
C1C22 — C2×C4×Q8

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 [×3], C2 [×4], C4 [×16], C4 [×6], C22, C22 [×6], C2×C4 [×30], C2×C4 [×6], Q8 [×16], C23, C42 [×12], C4⋊C4 [×12], C22×C4, C22×C4 [×6], C2×Q8 [×12], C2×C42 [×3], C2×C4⋊C4 [×3], C4×Q8 [×8], C22×Q8, C2×C4×Q8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, C2×C4×Q8

Smallest permutation representation of C2×C4×Q8
Regular action on 64 points
Generators in S64
(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)  Q85M4(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  Q84C42  C4214Q8  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  C425Q8  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
C4214Q8  C23.211C24  C42.33Q8  C424Q8  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···2G4A···4H4I···4AF
order12···24···44···4
size11···11···12···2

40 irreducible representations

dim11111122
type+++++-
imageC1C2C2C2C2C4Q8C4○D4
kernelC2×C4×Q8C2×C42C2×C4⋊C4C4×Q8C22×Q8C2×Q8C2×C4C22
# reps133811644

Matrix representation of C2×C4×Q8 in GL4(𝔽5) generated by

4000
0400
0040
0004
,
1000
0200
0010
0001
,
1000
0400
0001
0040
,
4000
0400
0003
0030
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

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