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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:C8:7C22, (C2xQ8):22D4, Q8.7(C2xD4), C4:C4.337D4, C4:SD16:3C2, (C4xQ8):3C22, (C2xC8).18C23, C4.74(C22xD4), C4.35(C4:D4), C4:C4.384C23, (C2xC4).247C24, Q8.D4:14C2, C22:C4.138D4, (C2xQ16):17C22, (C2xD4).53C23, C23.444(C2xD4), D4:C4:76C22, C2.13(D4oSD16), Q8:C4:79C22, (C22xSD16):24C2, (C2xSD16):10C22, C4:1D4.58C22, C23.37D4:9C2, (C2xQ8).360C23, C23.24D4:24C2, C22.82(C4:D4), C42.6C22:5C2, (C22xC4).977C23, (C22xC8).341C22, C4.4D4.24C22, C22.507(C22xD4), C23.32C23:7C2, C22.29C24.10C2, (C22xD4).342C22, (C2xM4(2)).54C22, (C22xQ8).274C22, C42:C2.102C22, C4.157(C2xC4oD4), (C2xC4).467(C2xD4), C2.65(C2xC4:D4), (C2xC8.C22):16C2, (C2xC4).278(C4oD4), (C2xC4oD4).119C22, SmallGroup(128,1775)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — C42.16C23
C1C2C4C2xC4C22xC4C22xQ8C23.32C23 — C42.16C23
C1C2C2xC4 — C42.16C23
C1C22C42:C2 — C42.16C23
C1C2C2C2xC4 — 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, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C24, D4:C4, Q8:C4, C4:C8, C42:C2, C42:C2, C4xQ8, C4xQ8, C22wrC2, C4:D4, C4.4D4, C4:1D4, C22xC8, C2xM4(2), C2xSD16, C2xSD16, C2xQ16, C8.C22, C22xD4, C22xQ8, C2xC4oD4, C23.24D4, C23.37D4, C42.6C22, C4:SD16, Q8.D4, C23.32C23, C22.29C24, C22xSD16, C2xC8.C22, C42.16C23
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C4:D4, C22xD4, C2xC4oD4, C2xC4:D4, D4oSD16, 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+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4D4C4oD4D4oSD16
kernelC42.16C23C23.24D4C23.37D4C42.6C22C4:SD16Q8.D4C23.32C23C22.29C24C22xSD16C2xC8.C22C22:C4C4:C4C2xQ8C2xC4C2
# 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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