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## G = C2×C4.C42order 128 = 27

### Direct product of C2 and C4.C42

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C4.C42
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C23×C8 — C2×C4.C42
 Lower central C1 — C2 — C4 — C2×C4.C42
 Upper central C1 — C22×C4 — C23×C4 — C2×C4.C42
 Jennings C1 — C2 — C2 — C22×C4 — C2×C4.C42

Generators and relations for C2×C4.C42
G = < a,b,c,d | a2=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c >

Subgroups: 276 in 196 conjugacy classes, 116 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C23, C23, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C4.C42, C23×C8, C22×M4(2), C2×C4.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2.C42, C8.C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.C42, C2×C2.C42, C2×C8.C4, C2×C4.C42

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

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

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

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

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 1 2 2 2 2 type + + + + + - - image C1 C2 C2 C2 C4 C4 D4 Q8 Q8 C8.C4 kernel C2×C4.C42 C4.C42 C23×C8 C22×M4(2) C22×C8 C2×M4(2) C22×C4 C22×C4 C24 C22 # reps 1 4 1 2 8 16 6 1 1 16

Matrix representation of C2×C4.C42 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 1 0 0 0 0 4 0 0 0 0 13
,
 1 0 0 0 0 1 0 0 0 0 0 16 0 0 13 0
,
 4 0 0 0 0 16 0 0 0 0 2 0 0 0 0 8
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,0,13,0,0,16,0],[4,0,0,0,0,16,0,0,0,0,2,0,0,0,0,8] >;

C2×C4.C42 in GAP, Magma, Sage, TeX

C_2\times C_4.C_4^2
% in TeX

G:=Group("C2xC4.C4^2");
// GroupNames label

G:=SmallGroup(128,469);
// by ID

G=gap.SmallGroup(128,469);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,2019,248,172,4037,124]);
// Polycyclic

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

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