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## G = C22×C8.C4order 128 = 27

### Direct product of C22 and C8.C4

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C22×C8.C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C23×C8 — C22×C8.C4
 Lower central C1 — C2 — C4 — C22×C8.C4
 Upper central C1 — C22×C4 — C23×C4 — C22×C8.C4
 Jennings C1 — C2 — C2 — C2×C4 — C22×C8.C4

Generators and relations for C22×C8.C4
G = < a,b,c,d | a2=b2=c8=1, d4=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 300 in 240 conjugacy classes, 180 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×10], C22 [×12], C8 [×8], C8 [×8], C2×C4, C2×C4 [×27], C23, C23 [×6], C23 [×4], C2×C8 [×28], C2×C8 [×12], M4(2) [×8], M4(2) [×12], C22×C4 [×2], C22×C4 [×12], C24, C8.C4 [×16], C22×C8 [×2], C22×C8 [×12], C22×C8 [×2], C2×M4(2) [×12], C2×M4(2) [×6], C23×C4, C2×C8.C4 [×12], C23×C8, C22×M4(2) [×2], C22×C8.C4
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, C8.C4 [×4], C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, C2×C8.C4 [×6], C22×C4⋊C4, C22×C8.C4

Smallest permutation representation of C22×C8.C4
On 64 points
Generators in S64
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 49)(32 50)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 64)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(32 63)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)
(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 38 45 57 5 34 41 61)(2 37 46 64 6 33 42 60)(3 36 47 63 7 40 43 59)(4 35 48 62 8 39 44 58)(9 52 19 28 13 56 23 32)(10 51 20 27 14 55 24 31)(11 50 21 26 15 54 17 30)(12 49 22 25 16 53 18 29)

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + + - - image C1 C2 C2 C2 C4 D4 Q8 Q8 C8.C4 kernel C22×C8.C4 C2×C8.C4 C23×C8 C22×M4(2) C22×C8 C22×C4 C22×C4 C24 C22 # reps 1 12 1 2 16 4 3 1 16

Matrix representation of C22×C8.C4 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 16 0 0 0 0 8 0 0 0 0 15
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 13 0
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,8,0,0,0,0,15],[1,0,0,0,0,1,0,0,0,0,0,13,0,0,1,0] >;

C22×C8.C4 in GAP, Magma, Sage, TeX

C_2^2\times C_8.C_4
% in TeX

G:=Group("C2^2xC8.C4");
// GroupNames label

G:=SmallGroup(128,1646);
// by ID

G=gap.SmallGroup(128,1646);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,2804,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^8=1,d^4=c^4,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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