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G = C4216Q8order 128 = 27

3rd semidirect product of C42 and Q8 acting via Q8/C4=C2

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

Aliases: C4216Q8, C43.10C2, C42.328D4, C4⋊Q827C4, C4.17C4≀C2, C4.16(C4×Q8), C4.54(C4⋊Q8), C42.C211C4, C42.271(C2×C4), C23.574(C2×D4), (C22×C4).766D4, C426C4.11C2, C4.94(C4.4D4), C4⋊M4(2).33C2, C22.28(C22⋊Q8), C42⋊C2.44C22, (C2×C42).1081C22, (C22×C4).1428C23, (C2×M4(2)).216C22, C23.37C23.20C2, C2.16(C23.67C23), C2.46(C2×C4≀C2), C4⋊C4.101(C2×C4), (C2×C4).215(C2×Q8), (C2×C4).1549(C2×D4), (C2×C4).611(C4○D4), (C2×C4).442(C22×C4), (C2×C4).208(C22⋊C4), C22.303(C2×C22⋊C4), SmallGroup(128,726)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4216Q8
C1C2C4C2×C4C22×C4C2×C42C43 — C4216Q8
C1C2C2×C4 — C4216Q8
C1C2×C4C2×C42 — C4216Q8
C1C2C2C22×C4 — C4216Q8

Generators and relations for C4216Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 244 in 142 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×6], C4 [×14], C22, C22 [×2], C22 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×20], Q8 [×4], C23, C42 [×2], C42 [×6], C42 [×12], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×Q8 [×2], C4⋊C8 [×2], C2×C42, C2×C42 [×2], C2×C42 [×2], C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C426C4 [×4], C43, C4⋊M4(2), C23.37C23, C4216Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C4≀C2 [×4], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C23.67C23, C2×C4≀C2 [×2], C4216Q8

Smallest permutation representation of C4216Q8
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 17 26 23)(2 18 27 24)(3 19 28 21)(4 20 25 22)(5 29 16 9)(6 30 13 10)(7 31 14 11)(8 32 15 12)
(1 27 3 25)(2 28 4 26)(5 10 14 32)(6 11 15 29)(7 12 16 30)(8 9 13 31)(17 24 19 22)(18 21 20 23)
(1 29 3 11)(2 8 4 13)(5 21 14 23)(6 27 15 25)(7 17 16 19)(9 28 31 26)(10 18 32 20)(12 22 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), (1,17,26,23)(2,18,27,24)(3,19,28,21)(4,20,25,22)(5,29,16,9)(6,30,13,10)(7,31,14,11)(8,32,15,12), (1,27,3,25)(2,28,4,26)(5,10,14,32)(6,11,15,29)(7,12,16,30)(8,9,13,31)(17,24,19,22)(18,21,20,23), (1,29,3,11)(2,8,4,13)(5,21,14,23)(6,27,15,25)(7,17,16,19)(9,28,31,26)(10,18,32,20)(12,22,30,24)>;

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4AD4AE4AF4AG4AH8A8B8C8D
order12222244444···444448888
size11112211112···288888888

44 irreducible representations

dim111111122222
type++++++-+
imageC1C2C2C2C2C4C4D4Q8D4C4○D4C4≀C2
kernelC4216Q8C426C4C43C4⋊M4(2)C23.37C23C42.C2C4⋊Q8C42C42C22×C4C2×C4C4
# reps1411144242416

Matrix representation of C4216Q8 in GL4(𝔽17) generated by

0100
16000
0040
00016
,
1000
0100
0040
0004
,
0100
16000
00130
0004
,
5500
51200
0004
0040
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,4,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[0,16,0,0,1,0,0,0,0,0,13,0,0,0,0,4],[5,5,0,0,5,12,0,0,0,0,0,4,0,0,4,0] >;

C4216Q8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{16}Q_8
% in TeX

G:=Group("C4^2:16Q8");
// GroupNames label

G:=SmallGroup(128,726);
// by ID

G=gap.SmallGroup(128,726);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,100,2019,248,124]);
// Polycyclic

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

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