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G = C42.390C23order 128 = 27

251st non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.390C23, C8⋊Q88C2, C4⋊C4.250D4, (C4×Q16)⋊30C2, C8.2D47C2, C88D4.3C2, C8.D47C2, C8⋊D4.1C2, (C4×SD16)⋊17C2, C8.29(C4○D4), C2.26(Q8○D8), C22⋊C4.90D4, C23.87(C2×D4), Q16⋊C415C2, C8.18D427C2, C4⋊C4.117C23, (C2×C4).376C24, (C2×C8).278C23, (C4×C8).183C22, C4.SD1643C2, (C4×D4).96C22, C4⋊Q8.118C22, SD16⋊C421C2, (C4×Q8).93C22, C82M4(2)⋊19C2, C4.Q8.28C22, C2.39(D4○SD16), (C2×D4).130C23, C4⋊D4.37C22, (C2×Q8).118C23, C8⋊C4.133C22, C2.D8.220C22, C22⋊Q8.37C22, (C22×C8).278C22, (C2×Q16).159C22, (C2×SD16).24C22, C4.4D4.36C22, C22.636(C22×D4), C42.C2.22C22, D4⋊C4.148C22, (C22×C4).1056C23, C22.35C244C2, Q8⋊C4.140C22, C42⋊C2.333C22, C42.78C2229C2, (C2×M4(2)).286C22, C22.36C24.2C2, C2.73(C22.26C24), C4.61(C2×C4○D4), (C2×C4).148(C2×D4), SmallGroup(128,1910)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.390C23
C1C2C4C2×C4C42C4×C8C82M4(2) — C42.390C23
C1C2C2×C4 — C42.390C23
C1C22C42⋊C2 — C42.390C23
C1C2C2C2×C4 — C42.390C23

Generators and relations for C42.390C23
 G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=b, ab=ba, ac=ca, dad=ab2, ae=ea, cbc=b-1, bd=db, be=eb, dcd=a2c, ece-1=b-1c, ede-1=b2d >

Subgroups: 308 in 175 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×10], D4 [×3], Q8 [×7], C23, C23, C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], SD16 [×4], Q16 [×4], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8 [×3], C2×Q8, C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×D4, C4×Q8 [×3], C4⋊D4, C22⋊Q8 [×3], C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2 [×2], C422C2 [×3], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C82M4(2), C4×SD16, C4×Q16, SD16⋊C4, Q16⋊C4, C88D4, C8.18D4, C8⋊D4, C8.D4, C4.SD16, C42.78C22, C8.2D4, C8⋊Q8, C22.35C24, C22.36C24, C42.390C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.26C24, D4○SD16, Q8○D8, C42.390C23

Smallest permutation representation of C42.390C23
On 64 points
Generators in S64
(1 38 31 10)(2 39 32 11)(3 40 25 12)(4 33 26 13)(5 34 27 14)(6 35 28 15)(7 36 29 16)(8 37 30 9)(17 57 50 42)(18 58 51 43)(19 59 52 44)(20 60 53 45)(21 61 54 46)(22 62 55 47)(23 63 56 48)(24 64 49 41)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 54)(18 49)(19 52)(20 55)(21 50)(22 53)(23 56)(24 51)(25 29)(26 32)(28 30)(33 39)(35 37)(36 40)(41 58)(42 61)(43 64)(44 59)(45 62)(46 57)(47 60)(48 63)
(1 44)(2 41)(3 46)(4 43)(5 48)(6 45)(7 42)(8 47)(9 51)(10 56)(11 53)(12 50)(13 55)(14 52)(15 49)(16 54)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(25 61)(26 58)(27 63)(28 60)(29 57)(30 62)(31 59)(32 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)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H4I4J···4P8A8B8C8D8E···8J
order1222224···44444···488888···8
size1111482···24448···822224···4

32 irreducible representations

dim111111111111111122244
type++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○SD16Q8○D8
kernelC42.390C23C82M4(2)C4×SD16C4×Q16SD16⋊C4Q16⋊C4C88D4C8.18D4C8⋊D4C8.D4C4.SD16C42.78C22C8.2D4C8⋊Q8C22.35C24C22.36C24C22⋊C4C4⋊C4C8C2C2
# reps111111111111111122822

Matrix representation of C42.390C23 in GL6(𝔽17)

400000
040000
000010
000001
0016000
0001600
,
100000
010000
000100
0016000
000001
0000160
,
100000
3160000
001000
0001600
000010
0000016
,
1420000
1330000
000701
00100160
0001010
0016070
,
1600000
0160000
0000314
000033
0014300
00141400

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,3,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[14,13,0,0,0,0,2,3,0,0,0,0,0,0,0,10,0,16,0,0,7,0,1,0,0,0,0,16,0,7,0,0,1,0,10,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,3,3,0,0,0,0,14,3,0,0] >;

C42.390C23 in GAP, Magma, Sage, TeX

C_4^2._{390}C_2^3
% in TeX

G:=Group("C4^2.390C2^3");
// GroupNames label

G:=SmallGroup(128,1910);
// by ID

G=gap.SmallGroup(128,1910);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,723,184,1018,80,4037,124]);
// Polycyclic

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

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