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G = C42.361C23order 128 = 27

222nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.361C23, (C4×Q16)⋊6C2, C4⋊C4.354D4, Q8.Q822C2, C42Q1625C2, C4⋊C8.60C22, C4⋊C4.80C23, (C4×C8).72C22, (C2×C8).54C23, C2.20(Q8○D8), Q16⋊C412C2, (C2×C4).325C24, Q8.15(C4○D4), C22⋊C4.155D4, C23.264(C2×D4), C4⋊Q8.109C22, C22⋊Q16.4C2, (C4×Q8).83C22, C8⋊C4.17C22, C4.Q8.23C22, C22⋊C8.38C22, (C2×Q8).384C23, C2.D8.176C22, C22⋊Q8.32C22, (C22×C4).298C23, Q8⋊C4.40C22, (C2×Q16).124C22, C23.20D4.2C2, C22.585(C22×D4), C42.C2.15C22, C42.7C22.1C2, (C22×Q8).296C22, C42⋊C2.136C22, C23.32C23.6C2, C22.35C24.1C2, C2.126(C22.19C24), C4.210(C2×C4○D4), (C2×C4).509(C2×D4), SmallGroup(128,1859)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.361C23
C1C2C4C2×C4C42C4×Q8C23.32C23 — C42.361C23
C1C2C2×C4 — C42.361C23
C1C22C42⋊C2 — C42.361C23
C1C2C2C2×C4 — C42.361C23

Generators and relations for C42.361C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=e2=b2, ab=ba, ac=ca, dad=ab2, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd=a2b2c, ece-1=bc, de=ed >

Subgroups: 284 in 179 conjugacy classes, 88 normal (20 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×15], C22, C22 [×3], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], Q8 [×4], Q8 [×10], C23, C42 [×2], C42 [×7], C22⋊C4 [×2], C22⋊C4 [×3], C4⋊C4 [×6], C4⋊C4 [×12], C2×C8 [×4], Q16 [×8], C22×C4, C22×C4, C2×Q8 [×4], C2×Q8 [×5], C4×C8, C8⋊C4, C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C42⋊C2 [×2], C4×Q8 [×6], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2, C42.C2 [×2], C422C2 [×2], C4⋊Q8, C2×Q16 [×4], C22×Q8, C42.7C22, C4×Q16 [×2], Q16⋊C4 [×2], C22⋊Q16 [×2], C42Q16 [×2], Q8.Q8 [×2], C23.20D4 [×2], C23.32C23, C22.35C24, C42.361C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.19C24, Q8○D8 [×2], C42.361C23

Smallest permutation representation of C42.361C23
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 28 20 23)(2 25 17 24)(3 26 18 21)(4 27 19 22)(5 12 15 62)(6 9 16 63)(7 10 13 64)(8 11 14 61)(29 40 33 41)(30 37 34 42)(31 38 35 43)(32 39 36 44)(45 56 49 57)(46 53 50 58)(47 54 51 59)(48 55 52 60)
(1 52 20 48)(2 49 17 45)(3 50 18 46)(4 51 19 47)(5 42 15 37)(6 43 16 38)(7 44 13 39)(8 41 14 40)(9 35 63 31)(10 36 64 32)(11 33 61 29)(12 34 62 30)(21 58 26 53)(22 59 27 54)(23 60 28 55)(24 57 25 56)
(2 17)(4 19)(5 13)(6 8)(7 15)(9 11)(10 62)(12 64)(14 16)(22 27)(24 25)(29 33)(31 35)(38 43)(40 41)(45 47)(46 52)(48 50)(49 51)(53 60)(54 56)(55 58)(57 59)(61 63)
(1 36 20 32)(2 33 17 29)(3 34 18 30)(4 35 19 31)(5 50 15 46)(6 51 16 47)(7 52 13 48)(8 49 14 45)(9 54 63 59)(10 55 64 60)(11 56 61 57)(12 53 62 58)(21 42 26 37)(22 43 27 38)(23 44 28 39)(24 41 25 40)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D4A···4F4G···4Q4R4S4T4U8A8B8C8D8E8F
order122224···44···44444888888
size111142···24···48888444488

32 irreducible representations

dim11111111112224
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D4Q8○D8
kernelC42.361C23C42.7C22C4×Q16Q16⋊C4C22⋊Q16C42Q16Q8.Q8C23.20D4C23.32C23C22.35C24C22⋊C4C4⋊C4Q8C2
# reps11222222112284

Matrix representation of C42.361C23 in GL6(𝔽17)

400000
040000
000010
000001
0016000
0001600
,
100000
010000
0001600
001000
0000016
000010
,
640000
4110000
000400
004000
000004
000040
,
100000
14160000
001000
000100
0000160
0000016
,
1600000
0160000
005500
0051200
000055
0000512

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[6,4,0,0,0,0,4,11,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[1,14,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,5,5,0,0,0,0,5,12,0,0,0,0,0,0,5,5,0,0,0,0,5,12] >;

C42.361C23 in GAP, Magma, Sage, TeX

C_4^2._{361}C_2^3
% in TeX

G:=Group("C4^2.361C2^3");
// GroupNames label

G:=SmallGroup(128,1859);
// by ID

G=gap.SmallGroup(128,1859);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,352,1018,304,4037,1027,124]);
// Polycyclic

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

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