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## G = C4.67(C4×D4)  order 128 = 27

### 18th non-split extension by C4 of C4×D4 acting via C4×D4/C22⋊C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4.67(C4×D4)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4⋊C4 — C4×C4⋊C4 — C4.67(C4×D4)
 Lower central C1 — C2 — C2×C4 — C4.67(C4×D4)
 Upper central C1 — C23 — C2×C42 — C4.67(C4×D4)
 Jennings C1 — C2 — C2 — C22×C4 — C4.67(C4×D4)

Generators and relations for C4.67(C4×D4)
G = < a,b,c,d | a4=b4=c4=1, d2=a, bab-1=cac-1=a-1, ad=da, bc=cb, dbd-1=a-1b, dcd-1=ac-1 >

Subgroups: 340 in 145 conjugacy classes, 58 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, D4⋊C4, D4⋊C4, C4.Q8, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C22.7C42, C22.4Q16, C4×C4⋊C4, C24.3C22, C2×D4⋊C4, C2×C4.Q8, C4.67(C4×D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, SD16, C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C422C2, C2×SD16, C4○D8, C8⋊C22, C24.C22, C4×SD16, D8⋊C4, C4⋊SD16, D4.2D4, C23.46D4, C23.19D4, C4.67(C4×D4)

Smallest permutation representation of C4.67(C4×D4)
On 64 points
Generators in S64
(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 26 51 18)(2 29 52 21)(3 32 53 24)(4 27 54 19)(5 30 55 22)(6 25 56 17)(7 28 49 20)(8 31 50 23)(9 47 57 33)(10 42 58 36)(11 45 59 39)(12 48 60 34)(13 43 61 37)(14 46 62 40)(15 41 63 35)(16 44 64 38)
(1 56 35 46)(2 45 36 55)(3 54 37 44)(4 43 38 53)(5 52 39 42)(6 41 40 51)(7 50 33 48)(8 47 34 49)(9 60 28 23)(10 22 29 59)(11 58 30 21)(12 20 31 57)(13 64 32 19)(14 18 25 63)(15 62 26 17)(16 24 27 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)

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4H 4I ··· 4R 4S 4T 8A ··· 8H order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 4 4 8 ··· 8 size 1 1 ··· 1 8 8 2 ··· 2 4 ··· 4 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 D4 D4 SD16 C4○D4 C4○D8 C8⋊C22 kernel C4.67(C4×D4) C22.7C42 C22.4Q16 C4×C4⋊C4 C24.3C22 C2×D4⋊C4 C2×C4.Q8 D4⋊C4 C4⋊C4 C22×C4 C2×C4 C2×C4 C22 C22 # reps 1 1 1 1 1 2 1 8 2 2 4 8 4 2

Matrix representation of C4.67(C4×D4) in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 0 4 0 0 0 0 13 0 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 12 12 0 0 0 0 12 5
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 13 15 0 0 0 0 0 4 0 0 0 0 0 0 5 5 0 0 0 0 5 12
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 13 15 0 0 0 0 16 4 0 0 0 0 0 0 5 12 0 0 0 0 5 5

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,12,12,0,0,0,0,12,5],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,15,4,0,0,0,0,0,0,5,5,0,0,0,0,5,12],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,13,16,0,0,0,0,15,4,0,0,0,0,0,0,5,5,0,0,0,0,12,5] >;

C4.67(C4×D4) in GAP, Magma, Sage, TeX

C_4._{67}(C_4\times D_4)
% in TeX

G:=Group("C4.67(C4xD4)");
// GroupNames label

G:=SmallGroup(128,658);
// by ID

G=gap.SmallGroup(128,658);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,394,2804,718,172]);
// Polycyclic

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

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