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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×C4×Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4×Q8
 Lower central C1 — C2 — C2×C4×Q8
 Upper central C1 — C22×C4 — C2×C4×Q8
 Jennings C1 — C22 — 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)])

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

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