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G = C42.20C23order 128 = 27

20th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.20C23, (C2×D4)⋊9Q8, C4⋊Q87C22, D4.5(C2×Q8), C4⋊C810C22, C4⋊C4.341D4, D42Q83C2, D4.Q814C2, D4⋊Q820C2, C2.13(D4○D8), C4⋊C4.44C23, (C2×C8).28C23, C2.D850C22, C4.Q859C22, C4.32(C22×Q8), (C2×C4).279C24, C22⋊C4.142D4, (C4×D4).69C22, C23.448(C2×D4), C4.67(C22⋊Q8), C42.C21C22, C2.20(D4○SD16), (C2×D4).397C23, C23.25D45C2, M4(2)⋊C419C2, D4⋊C4.26C22, C42.6C229C2, (C22×C4).998C23, (C22×C8).181C22, C22.11C24.8C2, C23.37D4.3C2, C22.539(C22×D4), C22.10(C22⋊Q8), C23.41C233C2, (C22×D4).355C22, (C2×M4(2)).68C22, C42⋊C2.118C22, C4.89(C2×C4○D4), (C2×C4).481(C2×D4), (C2×C4).103(C2×Q8), C2.60(C2×C22⋊Q8), (C2×D4⋊C4).28C2, (C2×C4).481(C4○D4), (C2×C4⋊C4).605C22, SmallGroup(128,1813)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.20C23
C1C2C4C2×C4C22×C4C22×D4C22.11C24 — C42.20C23
C1C2C2×C4 — C42.20C23
C1C22C42⋊C2 — C42.20C23
C1C2C2C2×C4 — C42.20C23

Generators and relations for C42.20C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=e2=a2b2, ab=ba, cac-1=a-1b2, ad=da, eae-1=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2b2c, de=ed >

Subgroups: 404 in 199 conjugacy classes, 100 normal (38 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×14], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×11], D4 [×4], D4 [×6], Q8 [×2], C23, C23 [×8], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×5], C2×D4 [×6], C2×D4 [×3], C2×Q8 [×2], C24, D4⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×2], C4.Q8 [×2], C2.D8 [×2], C2.D8 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C42⋊C2 [×3], C4×D4 [×4], C4×D4 [×2], C22⋊Q8 [×2], C42.C2 [×2], C42.C2, C4⋊Q8 [×2], C4⋊Q8, C22×C8, C2×M4(2), C22×D4, C2×D4⋊C4, C23.37D4, C42.6C22, C23.25D4, M4(2)⋊C4, D4⋊Q8 [×2], D42Q8 [×2], D4.Q8 [×4], C22.11C24, C23.41C23, C42.20C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, D4○D8, D4○SD16, C42.20C23

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4L4M4N4O4P8A8B8C8D8E8F
order122222222244444···44444888888
size111122444422224···48888444488

32 irreducible representations

dim11111111111222244
type+++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2D4D4Q8C4○D4D4○D8D4○SD16
kernelC42.20C23C2×D4⋊C4C23.37D4C42.6C22C23.25D4M4(2)⋊C4D4⋊Q8D42Q8D4.Q8C22.11C24C23.41C23C22⋊C4C4⋊C4C2×D4C2×C4C2C2
# reps11111122411224422

Matrix representation of C42.20C23 in GL6(𝔽17)

1300000
1340000
000010
00111615
0016000
0011016
,
1600000
0160000
0001600
001000
00161612
00101616
,
2130000
14150000
0031400
00141400
00141466
00301411
,
1600000
1610000
0001600
0016000
00161612
0000016
,
1300000
1340000
000010
00111615
001000
0000016

G:=sub<GL(6,GF(17))| [13,13,0,0,0,0,0,4,0,0,0,0,0,0,0,1,16,1,0,0,0,1,0,1,0,0,1,16,0,0,0,0,0,15,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,16,1,0,0,16,0,16,0,0,0,0,0,1,16,0,0,0,0,2,16],[2,14,0,0,0,0,13,15,0,0,0,0,0,0,3,14,14,3,0,0,14,14,14,0,0,0,0,0,6,14,0,0,0,0,6,11],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,16,16,0,0,0,16,0,16,0,0,0,0,0,1,0,0,0,0,0,2,16],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1,16,0,0,0,0,0,15,0,16] >;

C42.20C23 in GAP, Magma, Sage, TeX

C_4^2._{20}C_2^3
% in TeX

G:=Group("C4^2.20C2^3");
// GroupNames label

G:=SmallGroup(128,1813);
// by ID

G=gap.SmallGroup(128,1813);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,1018,4037,1027,124]);
// Polycyclic

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

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