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## G = C42.287C23order 128 = 27

### 148th non-split extension by C42 of C23 acting via C23/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.287C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — C42.287C23
 Lower central C1 — C22 — C42.287C23
 Upper central C1 — C2×C4 — C42.287C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.287C23

Generators and relations for C42.287C23
G = < a,b,c,d,e | a4=b4=1, c2=b, d2=a2b, e2=a2, ab=ba, cac-1=a-1b2, dad-1=ab2, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, ede-1=b2d >

Subgroups: 220 in 174 conjugacy classes, 140 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×16], C2×C4 [×2], Q8 [×4], C23, C42 [×2], C42 [×6], C22⋊C4 [×4], C4⋊C4 [×16], C2×C8 [×12], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C2×Q8 [×4], C4×C8 [×4], C8⋊C4 [×8], C4⋊C8 [×12], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×C8⋊C4, C82M4(2) [×2], C4⋊M4(2), C42.6C22 [×2], C84Q8 [×8], C23.37C23, C42.287C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, C2×C4×Q8, Q8○M4(2) [×2], C42.287C23

Smallest permutation representation of C42.287C23
On 64 points
Generators in S64
(1 63 55 10)(2 15 56 60)(3 57 49 12)(4 9 50 62)(5 59 51 14)(6 11 52 64)(7 61 53 16)(8 13 54 58)(17 35 31 46)(18 43 32 40)(19 37 25 48)(20 45 26 34)(21 39 27 42)(22 47 28 36)(23 33 29 44)(24 41 30 38)
(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 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 25 49 21 5 29 53 17)(2 26 50 22 6 30 54 18)(3 27 51 23 7 31 55 19)(4 28 52 24 8 32 56 20)(9 40 64 45 13 36 60 41)(10 33 57 46 14 37 61 42)(11 34 58 47 15 38 62 43)(12 35 59 48 16 39 63 44)
(1 63 55 10)(2 11 56 64)(3 57 49 12)(4 13 50 58)(5 59 51 14)(6 15 52 60)(7 61 53 16)(8 9 54 62)(17 35 31 46)(18 47 32 36)(19 37 25 48)(20 41 26 38)(21 39 27 42)(22 43 28 40)(23 33 29 44)(24 45 30 34)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G ··· 4R 8A ··· 8H 8I ··· 8T order 1 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 1 1 1 1 2 2 4 ··· 4 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 Q8 C4○D4 Q8○M4(2) kernel C42.287C23 C2×C8⋊C4 C8○2M4(2) C4⋊M4(2) C42.6C22 C8⋊4Q8 C23.37C23 C22⋊Q8 C42.C2 C4⋊Q8 C2×C8 C2×C4 C2 # reps 1 1 2 1 2 8 1 8 4 4 4 4 4

Matrix representation of C42.287C23 in GL6(𝔽17)

 14 15 0 0 0 0 5 3 0 0 0 0 0 0 11 15 0 0 0 0 9 6 0 0 0 0 0 0 6 2 0 0 0 0 8 11
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 14 7 0 0 0 0 1 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 13 0 0 0 0 0 0 13 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 13 0 0 0 0 0 0 0 0 1 0 0 0 0 13 0
,
 3 2 0 0 0 0 12 14 0 0 0 0 0 0 11 15 0 0 0 0 9 6 0 0 0 0 0 0 11 15 0 0 0 0 9 6

G:=sub<GL(6,GF(17))| [14,5,0,0,0,0,15,3,0,0,0,0,0,0,11,9,0,0,0,0,15,6,0,0,0,0,0,0,6,8,0,0,0,0,2,11],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[14,1,0,0,0,0,7,3,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,13,0,0,0,0,1,0],[3,12,0,0,0,0,2,14,0,0,0,0,0,0,11,9,0,0,0,0,15,6,0,0,0,0,0,0,11,9,0,0,0,0,15,6] >;

C42.287C23 in GAP, Magma, Sage, TeX

C_4^2._{287}C_2^3
% in TeX

G:=Group("C4^2.287C2^3");
// GroupNames label

G:=SmallGroup(128,1693);
// by ID

G=gap.SmallGroup(128,1693);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,268,2019,124]);
// Polycyclic

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

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