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G = C42.16C23order 128 = 27

16th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.16C23, C4⋊C87C22, (C2×Q8)⋊22D4, Q8.7(C2×D4), C4⋊C4.337D4, C4⋊SD163C2, (C4×Q8)⋊3C22, (C2×C8).18C23, C4.74(C22×D4), C4.35(C4⋊D4), C4⋊C4.384C23, (C2×C4).247C24, Q8.D414C2, C22⋊C4.138D4, (C2×Q16)⋊17C22, (C2×D4).53C23, C23.444(C2×D4), D4⋊C476C22, C2.13(D4○SD16), Q8⋊C479C22, (C22×SD16)⋊24C2, (C2×SD16)⋊10C22, C41D4.58C22, C23.37D49C2, (C2×Q8).360C23, C23.24D424C2, C22.82(C4⋊D4), C42.6C225C2, (C22×C4).977C23, (C22×C8).341C22, C4.4D4.24C22, C22.507(C22×D4), C23.32C237C2, C22.29C24.10C2, (C22×D4).342C22, (C2×M4(2)).54C22, (C22×Q8).274C22, C42⋊C2.102C22, C4.157(C2×C4○D4), (C2×C4).467(C2×D4), C2.65(C2×C4⋊D4), (C2×C8.C22)⋊16C2, (C2×C4).278(C4○D4), (C2×C4○D4).119C22, SmallGroup(128,1775)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.16C23
C1C2C4C2×C4C22×C4C22×Q8C23.32C23 — C42.16C23
C1C2C2×C4 — C42.16C23
C1C22C42⋊C2 — C42.16C23
C1C2C2C2×C4 — C42.16C23

Generators and relations for C42.16C23
 G = < a,b,c,d,e | a4=b4=c2=1, d2=b2, e2=a2b2, ab=ba, cac=a-1b2, ad=da, eae-1=ab2, cbc=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, de=ed >

Subgroups: 508 in 245 conjugacy classes, 100 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×15], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×11], D4 [×14], Q8 [×4], Q8 [×8], C23, C23 [×7], C42 [×2], C42 [×5], C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], SD16 [×12], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×10], C2×Q8, C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×4], C24, D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×4], C42⋊C2 [×2], C42⋊C2 [×2], C4×Q8 [×4], C4×Q8 [×2], C22≀C2 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4 [×2], C22×C8, C2×M4(2), C2×SD16 [×6], C2×SD16 [×4], C2×Q16 [×2], C8.C22 [×4], C22×D4, C22×Q8, C2×C4○D4, C23.24D4, C23.37D4, C42.6C22, C4⋊SD16 [×4], Q8.D4 [×4], C23.32C23, C22.29C24, C22×SD16, C2×C8.C22, C42.16C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D4○SD16 [×2], C42.16C23

Smallest permutation representation of C42.16C23
On 32 points
Generators in S32
(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)
(1 22 14 31)(2 23 15 32)(3 24 16 29)(4 21 13 30)(5 27 19 10)(6 28 20 11)(7 25 17 12)(8 26 18 9)
(2 13)(4 15)(5 10)(6 26)(7 12)(8 28)(9 20)(11 18)(17 25)(19 27)(21 23)(22 31)(24 29)(30 32)
(1 27 14 10)(2 28 15 11)(3 25 16 12)(4 26 13 9)(5 31 19 22)(6 32 20 23)(7 29 17 24)(8 30 18 21)
(1 4 16 15)(2 14 13 3)(5 8 17 20)(6 19 18 7)(9 25 28 10)(11 27 26 12)(21 29 32 22)(23 31 30 24)

G:=sub<Sym(32)| (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), (1,22,14,31)(2,23,15,32)(3,24,16,29)(4,21,13,30)(5,27,19,10)(6,28,20,11)(7,25,17,12)(8,26,18,9), (2,13)(4,15)(5,10)(6,26)(7,12)(8,28)(9,20)(11,18)(17,25)(19,27)(21,23)(22,31)(24,29)(30,32), (1,27,14,10)(2,28,15,11)(3,25,16,12)(4,26,13,9)(5,31,19,22)(6,32,20,23)(7,29,17,24)(8,30,18,21), (1,4,16,15)(2,14,13,3)(5,8,17,20)(6,19,18,7)(9,25,28,10)(11,27,26,12)(21,29,32,22)(23,31,30,24)>;

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), (1,22,14,31)(2,23,15,32)(3,24,16,29)(4,21,13,30)(5,27,19,10)(6,28,20,11)(7,25,17,12)(8,26,18,9), (2,13)(4,15)(5,10)(6,26)(7,12)(8,28)(9,20)(11,18)(17,25)(19,27)(21,23)(22,31)(24,29)(30,32), (1,27,14,10)(2,28,15,11)(3,25,16,12)(4,26,13,9)(5,31,19,22)(6,32,20,23)(7,29,17,24)(8,30,18,21), (1,4,16,15)(2,14,13,3)(5,8,17,20)(6,19,18,7)(9,25,28,10)(11,27,26,12)(21,29,32,22)(23,31,30,24) );

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)], [(1,22,14,31),(2,23,15,32),(3,24,16,29),(4,21,13,30),(5,27,19,10),(6,28,20,11),(7,25,17,12),(8,26,18,9)], [(2,13),(4,15),(5,10),(6,26),(7,12),(8,28),(9,20),(11,18),(17,25),(19,27),(21,23),(22,31),(24,29),(30,32)], [(1,27,14,10),(2,28,15,11),(3,25,16,12),(4,26,13,9),(5,31,19,22),(6,32,20,23),(7,29,17,24),(8,30,18,21)], [(1,4,16,15),(2,14,13,3),(5,8,17,20),(6,19,18,7),(9,25,28,10),(11,27,26,12),(21,29,32,22),(23,31,30,24)])

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E···4P4Q8A8B8C8D8E8F
order12222222244444···44888888
size11112288822224···48444488

32 irreducible representations

dim111111111122224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4D4C4○D4D4○SD16
kernelC42.16C23C23.24D4C23.37D4C42.6C22C4⋊SD16Q8.D4C23.32C23C22.29C24C22×SD16C2×C8.C22C22⋊C4C4⋊C4C2×Q8C2×C4C2
# reps111144111122444

Matrix representation of C42.16C23 in GL6(𝔽17)

2150000
11150000
00160150
000011
001010
001616160
,
100000
010000
00161500
001100
000101
001616160
,
100000
2160000
001000
00161600
000010
0010016
,
1600000
0160000
0001000
005000
000121212
00125125
,
2150000
11150000
00160150
000011
000010
0001160

G:=sub<GL(6,GF(17))| [2,11,0,0,0,0,15,15,0,0,0,0,0,0,16,0,1,16,0,0,0,0,0,16,0,0,15,1,1,16,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,16,0,0,15,1,1,16,0,0,0,0,0,16,0,0,0,0,1,0],[1,2,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,5,0,12,0,0,10,0,12,5,0,0,0,0,12,12,0,0,0,0,12,5],[2,11,0,0,0,0,15,15,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,1,1,16,0,0,0,1,0,0] >;

C42.16C23 in GAP, Magma, Sage, TeX

C_4^2._{16}C_2^3
% in TeX

G:=Group("C4^2.16C2^3");
// GroupNames label

G:=SmallGroup(128,1775);
// by ID

G=gap.SmallGroup(128,1775);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,352,1018,4037,1027,124]);
// Polycyclic

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

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