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G = C23.13D8order 128 = 27

6th non-split extension by C23 of D8 acting via D8/C4=C22

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C23.13D8, M5(2)⋊14C22, C4○D84C4, (C2×D8)⋊16C4, C4(D82C4), D82C44C2, (C2×Q16)⋊16C4, D8.11(C2×C4), C8.117(C2×D4), (C2×C4).141D8, (C2×C8).122D4, C8.7(C22×C4), C8.9(C22⋊C4), Q16.11(C2×C4), (C2×C4).52SD16, C4.64(C2×SD16), C22.18(C2×D8), C4.Q841C22, (C2×M5(2))⋊16C2, (C2×C8).225C23, C4○D8.17C22, (C22×C4).335D4, C4.57(D4⋊C4), C23.25D419C2, (C22×C8).234C22, C22.32(D4⋊C4), (C2×C8).84(C2×C4), (C2×C4○D8).12C2, (C2×C4).270(C2×D4), C4.57(C2×C22⋊C4), C2.35(C2×D4⋊C4), (C2×C4).152(C22⋊C4), SmallGroup(128,877)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C23.13D8
C1C2C4C2×C4C2×C8C22×C8C2×C4○D8 — C23.13D8
C1C2C4C8 — C23.13D8
C1C4C22×C4C22×C8 — C23.13D8
C1C2C2C2C2C4C4C2×C8 — C23.13D8

Generators and relations for C23.13D8
 G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=b, ab=ba, eae-1=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd7 >

Subgroups: 260 in 112 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×4], C22 [×3], C22 [×5], C8 [×4], C2×C4 [×6], C2×C4 [×7], D4 [×7], Q8 [×3], C23, C23, C16 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×6], D8 [×2], D8, SD16 [×4], Q16 [×2], Q16, C22×C4, C22×C4, C2×D4 [×2], C2×Q8, C4○D4 [×6], C4.Q8 [×2], C2.D8, C2×C16, M5(2) [×2], M5(2), C42⋊C2, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8 [×4], C4○D8 [×2], C2×C4○D4, D82C4 [×4], C23.25D4, C2×M5(2), C2×C4○D8, C23.13D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C2×D4⋊C4, C23.13D8

Smallest permutation representation of C23.13D8
On 32 points
Generators in S32
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 22)(2 31)(3 24)(4 17)(5 26)(6 19)(7 28)(8 21)(9 30)(10 23)(11 32)(12 25)(13 18)(14 27)(15 20)(16 29)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 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 29 22 16)(2 7 31 28)(3 19 24 6)(4 13 17 18)(5 25 26 12)(8 9 21 30)(10 15 23 20)(11 27 32 14)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F···4K8A8B8C8D8E8F16A···16H
order1222222444444···488888816···16
size1122288112228···82222444···4

32 irreducible representations

dim11111111222224
type+++++++++
imageC1C2C2C2C2C4C4C4D4D4D8SD16D8C23.13D8
kernelC23.13D8D82C4C23.25D4C2×M5(2)C2×C4○D8C2×D8C2×Q16C4○D8C2×C8C22×C4C2×C4C2×C4C23C1
# reps14111224312424

Matrix representation of C23.13D8 in GL4(𝔽17) generated by

0400
13000
00013
0040
,
0400
13000
0004
00130
,
16000
01600
00160
00016
,
00016
00160
51200
121200
,
0010
0001
0400
13000
G:=sub<GL(4,GF(17))| [0,13,0,0,4,0,0,0,0,0,0,4,0,0,13,0],[0,13,0,0,4,0,0,0,0,0,0,13,0,0,4,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,0,5,12,0,0,12,12,0,16,0,0,16,0,0,0],[0,0,0,13,0,0,4,0,1,0,0,0,0,1,0,0] >;

C23.13D8 in GAP, Magma, Sage, TeX

C_2^3._{13}D_8
% in TeX

G:=Group("C2^3.13D8");
// GroupNames label

G:=SmallGroup(128,877);
// by ID

G=gap.SmallGroup(128,877);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,352,1123,1466,136,1411,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^7>;
// generators/relations

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