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## G = C2×C42.12C4order 128 = 27

### Direct product of C2 and C42.12C4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×C42.12C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C22×C42 — C2×C42.12C4
 Lower central C1 — C2 — C2×C42.12C4
 Upper central C1 — C2×C42 — C2×C42.12C4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C42.12C4

Generators and relations for C2×C42.12C4
G = < a,b,c,d | a2=b4=c4=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c2, cd=dc >

Subgroups: 332 in 264 conjugacy classes, 196 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×16], C4 [×4], C22, C22 [×10], C22 [×12], C8 [×8], C2×C4 [×2], C2×C4 [×42], C2×C4 [×20], C23, C23 [×6], C23 [×4], C42 [×16], C2×C8 [×8], C2×C8 [×8], C22×C4 [×6], C22×C4 [×20], C22×C4 [×8], C24, C4×C8 [×8], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42 [×4], C2×C42 [×8], C22×C8 [×4], C23×C4 [×3], C2×C4×C8 [×2], C2×C22⋊C8 [×2], C2×C4⋊C8 [×2], C42.12C4 [×8], C22×C42, C2×C42.12C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], M4(2) [×4], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C22×C8 [×14], C2×M4(2) [×6], C23×C4, C2×C4○D4 [×2], C42.12C4 [×4], C2×C42⋊C2, C23×C8, C22×M4(2), C2×C42.12C4

Smallest permutation representation of C2×C42.12C4
On 64 points
Generators in S64
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 3 5 7)(2 18 6 22)(4 20 8 24)(9 11 13 15)(10 44 14 48)(12 46 16 42)(17 19 21 23)(25 27 29 31)(26 50 30 54)(28 52 32 56)(33 35 37 39)(34 58 38 62)(36 60 40 64)(41 43 45 47)(49 51 53 55)(57 59 61 63)
(1 57 19 39)(2 58 20 40)(3 59 21 33)(4 60 22 34)(5 61 23 35)(6 62 24 36)(7 63 17 37)(8 64 18 38)(9 25 45 51)(10 26 46 52)(11 27 47 53)(12 28 48 54)(13 29 41 55)(14 30 42 56)(15 31 43 49)(16 32 44 50)
(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)

G:=sub<Sym(64)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,3,5,7)(2,18,6,22)(4,20,8,24)(9,11,13,15)(10,44,14,48)(12,46,16,42)(17,19,21,23)(25,27,29,31)(26,50,30,54)(28,52,32,56)(33,35,37,39)(34,58,38,62)(36,60,40,64)(41,43,45,47)(49,51,53,55)(57,59,61,63), (1,57,19,39)(2,58,20,40)(3,59,21,33)(4,60,22,34)(5,61,23,35)(6,62,24,36)(7,63,17,37)(8,64,18,38)(9,25,45,51)(10,26,46,52)(11,27,47,53)(12,28,48,54)(13,29,41,55)(14,30,42,56)(15,31,43,49)(16,32,44,50), (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)>;

G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,3,5,7)(2,18,6,22)(4,20,8,24)(9,11,13,15)(10,44,14,48)(12,46,16,42)(17,19,21,23)(25,27,29,31)(26,50,30,54)(28,52,32,56)(33,35,37,39)(34,58,38,62)(36,60,40,64)(41,43,45,47)(49,51,53,55)(57,59,61,63), (1,57,19,39)(2,58,20,40)(3,59,21,33)(4,60,22,34)(5,61,23,35)(6,62,24,36)(7,63,17,37)(8,64,18,38)(9,25,45,51)(10,26,46,52)(11,27,47,53)(12,28,48,54)(13,29,41,55)(14,30,42,56)(15,31,43,49)(16,32,44,50), (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) );

G=PermutationGroup([(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,49),(8,50),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(1,3,5,7),(2,18,6,22),(4,20,8,24),(9,11,13,15),(10,44,14,48),(12,46,16,42),(17,19,21,23),(25,27,29,31),(26,50,30,54),(28,52,32,56),(33,35,37,39),(34,58,38,62),(36,60,40,64),(41,43,45,47),(49,51,53,55),(57,59,61,63)], [(1,57,19,39),(2,58,20,40),(3,59,21,33),(4,60,22,34),(5,61,23,35),(6,62,24,36),(7,63,17,37),(8,64,18,38),(9,25,45,51),(10,26,46,52),(11,27,47,53),(12,28,48,54),(13,29,41,55),(14,30,42,56),(15,31,43,49),(16,32,44,50)], [(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)])

80 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4X 4Y ··· 4AJ 8A ··· 8AF order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 type + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C8 M4(2) C4○D4 kernel C2×C42.12C4 C2×C4×C8 C2×C22⋊C8 C2×C4⋊C8 C42.12C4 C22×C42 C2×C42 C23×C4 C22×C4 C2×C4 C2×C4 # reps 1 2 2 2 8 1 12 4 32 8 8

Matrix representation of C2×C42.12C4 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 16 0 0 0 0 16 0 0 0 0 1
,
 13 0 0 0 0 16 0 0 0 0 13 0 0 0 0 13
,
 8 0 0 0 0 16 0 0 0 0 0 16 0 0 1 0
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1],[13,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[8,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0] >;

C2×C42.12C4 in GAP, Magma, Sage, TeX

C_2\times C_4^2._{12}C_4
% in TeX

G:=Group("C2xC4^2.12C4");
// GroupNames label

G:=SmallGroup(128,1649);
// by ID

G=gap.SmallGroup(128,1649);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^4=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^2,c*d=d*c>;
// generators/relations

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