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G = C42.385C23order 128 = 27

246th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.385C23, C83Q86C2, C88D445C2, C4⋊C4.245D4, C8.8(C4○D4), C8.D46C2, C82D4.4C2, D8⋊C413C2, (C4×SD16)⋊53C2, C22⋊C4.85D4, C23.82(C2×D4), Q16⋊C413C2, C8.12D418C2, C4⋊C4.112C23, (C4×C8).290C22, (C2×C8).600C23, (C2×C4).371C24, (C4×D4).92C22, (C2×D8).63C22, C4⋊Q8.114C22, (C4×Q8).89C22, C82M4(2)⋊15C2, C2.37(D4○SD16), (C2×D4).126C23, C4⋊D4.33C22, (C2×Q8).114C23, (C2×Q16).64C22, C4.Q8.163C22, C8⋊C4.128C22, C22⋊Q8.33C22, (C22×C8).359C22, C4.4D4.34C22, C22.631(C22×D4), D4⋊C4.204C22, (C22×C4).1051C23, C22.36C244C2, Q8⋊C4.206C22, (C2×SD16).151C22, C42.28C2233C2, C42⋊C2.328C22, (C2×M4(2)).281C22, C2.68(C22.26C24), C4.56(C2×C4○D4), (C2×C4).143(C2×D4), SmallGroup(128,1905)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.385C23
C1C2C4C2×C4C42C4×C8C82M4(2) — C42.385C23
C1C2C2×C4 — C42.385C23
C1C22C42⋊C2 — C42.385C23
C1C2C2C2×C4 — C42.385C23

Generators and relations for C42.385C23
 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, de=ed >

Subgroups: 348 in 181 conjugacy classes, 88 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×6], Q8 [×6], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8 [×2], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C4.Q8 [×2], C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4.4D4 [×2], C422C2 [×2], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×D8, C2×SD16 [×2], C2×Q16, C82M4(2), C4×SD16 [×2], Q16⋊C4, D8⋊C4, C88D4 [×2], C82D4, C8.D4, C42.28C22 [×2], C8.12D4, C83Q8, C22.36C24 [×2], C42.385C23
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 [×2], C42.385C23

Smallest permutation representation of C42.385C23
On 64 points
Generators in S64
(1 39 31 11)(2 40 32 12)(3 33 25 13)(4 34 26 14)(5 35 27 15)(6 36 28 16)(7 37 29 9)(8 38 30 10)(17 57 56 45)(18 58 49 46)(19 59 50 47)(20 60 51 48)(21 61 52 41)(22 62 53 42)(23 63 54 43)(24 64 55 44)
(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 49)(2 52)(3 55)(4 50)(5 53)(6 56)(7 51)(8 54)(9 60)(10 63)(11 58)(12 61)(13 64)(14 59)(15 62)(16 57)(17 28)(18 31)(19 26)(20 29)(21 32)(22 27)(23 30)(24 25)(33 44)(34 47)(35 42)(36 45)(37 48)(38 43)(39 46)(40 41)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 56)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(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)| (1,39,31,11)(2,40,32,12)(3,33,25,13)(4,34,26,14)(5,35,27,15)(6,36,28,16)(7,37,29,9)(8,38,30,10)(17,57,56,45)(18,58,49,46)(19,59,50,47)(20,60,51,48)(21,61,52,41)(22,62,53,42)(23,63,54,43)(24,64,55,44), (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), (1,49)(2,52)(3,55)(4,50)(5,53)(6,56)(7,51)(8,54)(9,60)(10,63)(11,58)(12,61)(13,64)(14,59)(15,62)(16,57)(17,28)(18,31)(19,26)(20,29)(21,32)(22,27)(23,30)(24,25)(33,44)(34,47)(35,42)(36,45)(37,48)(38,43)(39,46)(40,41), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,56)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(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( (1,39,31,11)(2,40,32,12)(3,33,25,13)(4,34,26,14)(5,35,27,15)(6,36,28,16)(7,37,29,9)(8,38,30,10)(17,57,56,45)(18,58,49,46)(19,59,50,47)(20,60,51,48)(21,61,52,41)(22,62,53,42)(23,63,54,43)(24,64,55,44), (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), (1,49)(2,52)(3,55)(4,50)(5,53)(6,56)(7,51)(8,54)(9,60)(10,63)(11,58)(12,61)(13,64)(14,59)(15,62)(16,57)(17,28)(18,31)(19,26)(20,29)(21,32)(22,27)(23,30)(24,25)(33,44)(34,47)(35,42)(36,45)(37,48)(38,43)(39,46)(40,41), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,56)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(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([(1,39,31,11),(2,40,32,12),(3,33,25,13),(4,34,26,14),(5,35,27,15),(6,36,28,16),(7,37,29,9),(8,38,30,10),(17,57,56,45),(18,58,49,46),(19,59,50,47),(20,60,51,48),(21,61,52,41),(22,62,53,42),(23,63,54,43),(24,64,55,44)], [(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)], [(1,49),(2,52),(3,55),(4,50),(5,53),(6,56),(7,51),(8,54),(9,60),(10,63),(11,58),(12,61),(13,64),(14,59),(15,62),(16,57),(17,28),(18,31),(19,26),(20,29),(21,32),(22,27),(23,30),(24,25),(33,44),(34,47),(35,42),(36,45),(37,48),(38,43),(39,46),(40,41)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,41),(8,42),(9,56),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,55),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(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 2A2B2C2D2E2F4A···4F4G4H4I4J···4O8A8B8C8D8E···8J
order12222224···44444···488888···8
size11114882···24448···822224···4

32 irreducible representations

dim1111111111112224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○SD16
kernelC42.385C23C82M4(2)C4×SD16Q16⋊C4D8⋊C4C88D4C82D4C8.D4C42.28C22C8.12D4C83Q8C22.36C24C22⋊C4C4⋊C4C8C2
# reps1121121121122284

Matrix representation of C42.385C23 in GL6(𝔽17)

1300000
0130000
000010
00413115
0016000
000004
,
1600000
0160000
000100
0016000
00413115
0040116
,
100000
0160000
0001604
0000134
00134161
00130161
,
040000
1300000
001131210
0001450
0001230
0057146
,
010000
1600000
0012500
00121200
0031407
003057

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,4,16,0,0,0,0,13,0,0,0,0,1,1,0,0,0,0,0,15,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,4,4,0,0,1,0,13,0,0,0,0,0,1,1,0,0,0,0,15,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,13,0,0,16,0,4,0,0,0,0,13,16,16,0,0,4,4,1,1],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,11,0,0,5,0,0,3,14,12,7,0,0,12,5,3,14,0,0,10,0,0,6],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,12,12,3,3,0,0,5,12,14,0,0,0,0,0,0,5,0,0,0,0,7,7] >;

C42.385C23 in GAP, Magma, Sage, TeX

C_4^2._{385}C_2^3
% in TeX

G:=Group("C4^2.385C2^3");
// GroupNames label

G:=SmallGroup(128,1905);
// by ID

G=gap.SmallGroup(128,1905);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,184,1018,80,4037,124]);
// 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,d*e=e*d>;
// generators/relations

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