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## G = C2×C42.6C4order 128 = 27

### Direct product of C2 and C42.6C4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C42.6C4
G = < a,b,c,d | a2=b4=c4=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c2, dcd-1=b2c >

Subgroups: 332 in 244 conjugacy classes, 156 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C22×C8, C23×C4, C2×C8⋊C4, C2×C22⋊C8, C2×C4⋊C8, C42.6C4, C22×C42, C2×C42.6C4
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C4○D4, C24, C42⋊C2, C2×M4(2), C23×C4, C2×C4○D4, C42.6C4, C2×C42⋊C2, C22×M4(2), C2×C42.6C4

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4AB 8A ··· 8P 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 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 M4(2) C4○D4 M4(2) kernel C2×C42.6C4 C2×C8⋊C4 C2×C22⋊C8 C2×C4⋊C8 C42.6C4 C22×C42 C2×C42 C23×C4 C2×C4 C2×C4 C23 # reps 1 2 2 2 8 1 12 4 8 8 8

Matrix representation of C2×C42.6C4 in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 16 0 0 0 0 0 16 0 0 0 0 13 1 0 0 0 0 0 4 0 0 0 0 0 4
,
 16 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 13
,
 16 0 0 0 0 0 4 15 0 0 0 0 13 0 0 0 0 0 0 16 0 0 0 13 0

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,13,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,13],[16,0,0,0,0,0,4,0,0,0,0,15,13,0,0,0,0,0,0,13,0,0,0,16,0] >;

C2×C42.6C4 in GAP, Magma, Sage, TeX

C_2\times C_4^2._6C_4
% in TeX

G:=Group("C2xC4^2.6C4");
// GroupNames label

G:=SmallGroup(128,1650);
// by ID

G=gap.SmallGroup(128,1650);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,100,124]);
// 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,d*c*d^-1=b^2*c>;
// generators/relations

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