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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 [×7], C2 [×2], C4 [×4], C4 [×8], C22 [×7], C22 [×10], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×16], D4 [×6], C23, C23 [×8], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, C2.C42, D4⋊C4 [×4], D4⋊C4 [×2], C4.Q8 [×2], C2×C42, C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C22×C8 [×2], C22×D4, C22.7C42, C22.4Q16, C4×C4⋊C4, C24.3C22, C2×D4⋊C4 [×2], C2×C4.Q8, C4.67(C4×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, SD16 [×2], C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C2×SD16, C4○D8, C8⋊C22 [×2], 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 47 19)(2 29 48 22)(3 32 41 17)(4 27 42 20)(5 30 43 23)(6 25 44 18)(7 28 45 21)(8 31 46 24)(9 51 64 38)(10 54 57 33)(11 49 58 36)(12 52 59 39)(13 55 60 34)(14 50 61 37)(15 53 62 40)(16 56 63 35)
(1 44 35 53)(2 52 36 43)(3 42 37 51)(4 50 38 41)(5 48 39 49)(6 56 40 47)(7 46 33 55)(8 54 34 45)(9 17 27 61)(10 60 28 24)(11 23 29 59)(12 58 30 22)(13 21 31 57)(14 64 32 20)(15 19 25 63)(16 62 26 18)
(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), (1,26,47,19)(2,29,48,22)(3,32,41,17)(4,27,42,20)(5,30,43,23)(6,25,44,18)(7,28,45,21)(8,31,46,24)(9,51,64,38)(10,54,57,33)(11,49,58,36)(12,52,59,39)(13,55,60,34)(14,50,61,37)(15,53,62,40)(16,56,63,35), (1,44,35,53)(2,52,36,43)(3,42,37,51)(4,50,38,41)(5,48,39,49)(6,56,40,47)(7,46,33,55)(8,54,34,45)(9,17,27,61)(10,60,28,24)(11,23,29,59)(12,58,30,22)(13,21,31,57)(14,64,32,20)(15,19,25,63)(16,62,26,18), (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), (1,26,47,19)(2,29,48,22)(3,32,41,17)(4,27,42,20)(5,30,43,23)(6,25,44,18)(7,28,45,21)(8,31,46,24)(9,51,64,38)(10,54,57,33)(11,49,58,36)(12,52,59,39)(13,55,60,34)(14,50,61,37)(15,53,62,40)(16,56,63,35), (1,44,35,53)(2,52,36,43)(3,42,37,51)(4,50,38,41)(5,48,39,49)(6,56,40,47)(7,46,33,55)(8,54,34,45)(9,17,27,61)(10,60,28,24)(11,23,29,59)(12,58,30,22)(13,21,31,57)(14,64,32,20)(15,19,25,63)(16,62,26,18), (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)], [(1,26,47,19),(2,29,48,22),(3,32,41,17),(4,27,42,20),(5,30,43,23),(6,25,44,18),(7,28,45,21),(8,31,46,24),(9,51,64,38),(10,54,57,33),(11,49,58,36),(12,52,59,39),(13,55,60,34),(14,50,61,37),(15,53,62,40),(16,56,63,35)], [(1,44,35,53),(2,52,36,43),(3,42,37,51),(4,50,38,41),(5,48,39,49),(6,56,40,47),(7,46,33,55),(8,54,34,45),(9,17,27,61),(10,60,28,24),(11,23,29,59),(12,58,30,22),(13,21,31,57),(14,64,32,20),(15,19,25,63),(16,62,26,18)], [(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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