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## G = C2×C42.6C22order 128 = 27

### Direct product of C2 and C42.6C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C42.6C22
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C2×C42⋊C2 — C2×C42.6C22
 Lower central C1 — C22 — C2×C42.6C22
 Upper central C1 — C22×C4 — C2×C42.6C22
 Jennings C1 — C2 — C2 — C2×C4 — C2×C42.6C22

Generators and relations for C2×C42.6C22
G = < a,b,c,d,e | a2=b4=c4=1, d2=c, e2=b2c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1c2, ebe-1=bc2, cd=dc, ce=ec, ede-1=b2c2d >

Subgroups: 332 in 256 conjugacy classes, 180 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×10], C22 [×12], C8 [×8], C2×C4 [×2], C2×C4 [×34], C2×C4 [×8], C23, C23 [×6], C23 [×4], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C2×C8 [×8], C2×C8 [×16], M4(2) [×8], C22×C4 [×2], C22×C4 [×16], C24, C4⋊C8 [×16], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C22×C8 [×8], C22×C8 [×4], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4, C2×C4⋊C8 [×4], C42.6C22 [×8], C2×C42⋊C2, C23×C8, C22×M4(2), C2×C42.6C22
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C8○D4 [×4], C23×C4, C22×D4, C22×Q8, C42.6C22 [×4], C22×C4⋊C4, C2×C8○D4 [×2], C2×C42.6C22

Smallest permutation representation of C2×C42.6C22
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 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 59 19 39)(2 36 20 64)(3 61 21 33)(4 38 22 58)(5 63 23 35)(6 40 24 60)(7 57 17 37)(8 34 18 62)(9 29 45 49)(10 54 46 26)(11 31 47 51)(12 56 48 28)(13 25 41 53)(14 50 42 30)(15 27 43 55)(16 52 44 32)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(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 63 23 39)(2 40 24 64)(3 57 17 33)(4 34 18 58)(5 59 19 35)(6 36 20 60)(7 61 21 37)(8 38 22 62)(9 29 41 53)(10 54 42 30)(11 31 43 55)(12 56 44 32)(13 25 45 49)(14 50 46 26)(15 27 47 51)(16 52 48 28)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I 4J 4K 4L 4M ··· 4T 8A ··· 8P 8Q ··· 8X order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 Q8 C8○D4 kernel C2×C42.6C22 C2×C4⋊C8 C42.6C22 C2×C42⋊C2 C23×C8 C22×M4(2) C2×C22⋊C4 C2×C4⋊C4 C42⋊C2 C22×C4 C22×C4 C22 # reps 1 4 8 1 1 1 4 4 8 4 4 16

Matrix representation of C2×C42.6C22 in GL5(𝔽17)

 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 16 0
,
 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 16 0 0 0 0 0 2 0 0 0 0 0 15 0 0 0 0 0 15 0 0 0 0 0 15
,
 1 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,2,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,15],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0] >;

C2×C42.6C22 in GAP, Magma, Sage, TeX

C_2\times C_4^2._6C_2^2
% in TeX

G:=Group("C2xC4^2.6C2^2");
// GroupNames label

G:=SmallGroup(128,1636);
// by ID

G=gap.SmallGroup(128,1636);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,124]);
// Polycyclic

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

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