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G = C42.470C23order 128 = 27

331st non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.470C23, C4.502+ 1+4, (C8×D4)⋊23C2, C4⋊D814C2, C87D414C2, C4⋊C4.411D4, Q86D47C2, D4⋊D412C2, C4.Q1613C2, (C4×SD16)⋊42C2, (C2×D4).240D4, C2.46(D4○D8), C4.73(C4○D8), C4.4D830C2, C4⋊C8.298C22, C4⋊C4.405C23, (C2×C4).497C24, (C2×C8).351C23, (C4×C8).275C22, Q8.20(C4○D4), C22⋊C4.109D4, (C2×D8).37C22, C23.113(C2×D4), C4⋊Q8.146C22, C2.D8.57C22, (C2×D4).227C23, (C4×D4).338C22, C4⋊D4.77C22, C23.19D47C2, C41D4.85C22, (C2×Q8).395C23, (C4×Q8).153C22, C2.133(D45D4), C4.Q8.167C22, C23.24D413C2, C22⋊C8.206C22, (C22×C8).163C22, C22.757(C22×D4), D4⋊C4.120C22, (C22×C4).1141C23, C22.49C244C2, Q8⋊C4.159C22, (C2×SD16).158C22, C42⋊C2.185C22, C4⋊C4(Q8⋊C4), C2.65(C2×C4○D8), C4.222(C2×C4○D4), (C2×C4).926(C2×D4), (C2×C4○D4).203C22, SmallGroup(128,2037)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.470C23
C1C2C4C2×C4C22×C4C2×C4○D4Q86D4 — C42.470C23
C1C2C2×C4 — C42.470C23
C1C22C4×D4 — C42.470C23
C1C2C2C2×C4 — C42.470C23

Generators and relations for C42.470C23
 G = < a,b,c,d,e | a4=b4=1, c2=a2b2, d2=e2=b2, ab=ba, cac-1=eae-1=a-1, ad=da, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2c, de=ed >

Subgroups: 440 in 204 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×15], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×15], D4 [×19], Q8 [×2], Q8 [×2], C23 [×2], C23 [×3], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×D4 [×8], C2×Q8, C2×Q8, C4○D4 [×6], C4×C8, C22⋊C8 [×2], D4⋊C4, D4⋊C4 [×6], Q8⋊C4, Q8⋊C4 [×2], C4⋊C8, C4.Q8, C2.D8 [×2], C42⋊C2 [×2], C42⋊C2, C4×D4 [×2], C4×D4, C4×Q8, C4⋊D4 [×4], C4⋊D4 [×3], C4.4D4 [×2], C41D4, C41D4, C4⋊Q8, C22×C8 [×2], C2×D8 [×2], C2×SD16, C2×C4○D4 [×2], C23.24D4 [×2], C8×D4, C4×SD16, D4⋊D4 [×2], C4⋊D8, C87D4 [×2], C4.Q16, C23.19D4 [×2], C4.4D8, Q86D4, C22.49C24, C42.470C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C4○D8 [×2], C22×D4, C2×C4○D4, 2+ 1+4, D45D4, C2×C4○D8, D4○D8, C42.470C23

Smallest permutation representation of C42.470C23
On 64 points
Generators in S64
(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 42 46 17)(2 43 47 18)(3 44 48 19)(4 41 45 20)(5 62 33 39)(6 63 34 40)(7 64 35 37)(8 61 36 38)(9 15 23 49)(10 16 24 50)(11 13 21 51)(12 14 22 52)(25 56 60 31)(26 53 57 32)(27 54 58 29)(28 55 59 30)
(1 57 48 28)(2 60 45 27)(3 59 46 26)(4 58 47 25)(5 52 35 16)(6 51 36 15)(7 50 33 14)(8 49 34 13)(9 63 21 38)(10 62 22 37)(11 61 23 40)(12 64 24 39)(17 32 44 55)(18 31 41 54)(19 30 42 53)(20 29 43 56)
(1 11 46 21)(2 12 47 22)(3 9 48 23)(4 10 45 24)(5 27 33 58)(6 28 34 59)(7 25 35 60)(8 26 36 57)(13 42 51 17)(14 43 52 18)(15 44 49 19)(16 41 50 20)(29 39 54 62)(30 40 55 63)(31 37 56 64)(32 38 53 61)
(1 10 46 24)(2 9 47 23)(3 12 48 22)(4 11 45 21)(5 30 33 55)(6 29 34 54)(7 32 35 53)(8 31 36 56)(13 20 51 41)(14 19 52 44)(15 18 49 43)(16 17 50 42)(25 38 60 61)(26 37 57 64)(27 40 58 63)(28 39 59 62)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4H4I···4M4N4O4P8A8B8C8D8E···8J
order1222222224···44···444488888···8
size1111448882···24···488822224···4

35 irreducible representations

dim1111111111112222244
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4C4○D4C4○D82+ 1+4D4○D8
kernelC42.470C23C23.24D4C8×D4C4×SD16D4⋊D4C4⋊D8C87D4C4.Q16C23.19D4C4.4D8Q86D4C22.49C24C22⋊C4C4⋊C4C2×D4Q8C4C4C2
# reps1211212121112114812

Matrix representation of C42.470C23 in GL4(𝔽17) generated by

16000
01600
0012
001616
,
11500
11600
0010
0001
,
0700
12000
00130
0044
,
4000
41300
0010
0001
,
13000
01300
0012
00016
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,16,0,0,2,16],[1,1,0,0,15,16,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,7,0,0,0,0,0,13,4,0,0,0,4],[4,4,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,13,0,0,0,0,1,0,0,0,2,16] >;

C42.470C23 in GAP, Magma, Sage, TeX

C_4^2._{470}C_2^3
% in TeX

G:=Group("C4^2.470C2^3");
// GroupNames label

G:=SmallGroup(128,2037);
// by ID

G=gap.SmallGroup(128,2037);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,352,346,248,4037,1027,124]);
// Polycyclic

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

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