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G = C23.5C42order 128 = 27

5th non-split extension by C23 of C42 acting via C42/C4=C4

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

Aliases: C23.5C42, C8(C23⋊C4), (C22×C8)⋊8C4, C8(C4.D4), (C2×C8).383D4, C4.D46C4, C4.107(C4×D4), C23⋊C4.3C4, (C2×C4).5C42, C8(C4.10D4), C4.10D47C4, C22.25(C4×D4), C8.60(C22⋊C4), (C2×M4(2))⋊19C4, C23.2(C22×C4), C8(M4(2)⋊4C4), C82M4(2)⋊21C2, M4(2).16(C2×C4), C22.12(C2×C42), M4(2)⋊4C422C2, C4.46(C42⋊C2), C8(C23.C23), (C22×C4).652C23, (C22×C8).379C22, C42⋊C2.260C22, C8(M4(2).8C22), (C2×M4(2)).305C22, C23.C23.11C2, M4(2).8C22.12C2, (C2×C8).10(C2×C4), (C2×C8○D4).11C2, C2.15(C4×C22⋊C4), (C2×D4).158(C2×C4), (C2×C4).44(C4○D4), (C2×C4).1301(C2×D4), C22⋊C4.28(C2×C4), C4.109(C2×C22⋊C4), (C22×C4).71(C2×C4), (C2×Q8).140(C2×C4), (C2×C4).521(C22×C4), (C2×C4○D4).249C22, SmallGroup(128,489)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.5C42
C1C2C22C2×C4C22×C4C22×C8C82M4(2) — C23.5C42
C1C2C22 — C23.5C42
C1C8C22×C8 — C23.5C42
C1C2C2C22×C4 — C23.5C42

Generators and relations for C23.5C42
 G = < a,b,c,d,e | a2=b2=c2=d4=1, e4=c, ab=ba, ac=ca, dad-1=abc, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd >

Subgroups: 228 in 142 conjugacy classes, 74 normal (22 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×5], C8 [×2], C8 [×2], C8 [×6], C2×C4 [×2], C2×C4 [×6], C2×C4 [×8], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×6], C2×C8 [×6], M4(2) [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×2], C23⋊C4 [×4], C4.D4 [×2], C4.10D4 [×2], C42⋊C2 [×2], C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×4], C8○D4 [×4], C2×C4○D4, M4(2)⋊4C4 [×2], C82M4(2) [×2], C23.C23, M4(2).8C22, C2×C8○D4, C23.5C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C4×C22⋊C4, C23.5C42

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F···4O8A8B8C8D8E···8J8K···8V
order1222222444444···488888···88···8
size1122244112224···411112···24···4

44 irreducible representations

dim11111111111224
type+++++++
imageC1C2C2C2C2C2C4C4C4C4C4D4C4○D4C23.5C42
kernelC23.5C42M4(2)⋊4C4C82M4(2)C23.C23M4(2).8C22C2×C8○D4C23⋊C4C4.D4C4.10D4C22×C8C2×M4(2)C2×C8C2×C4C1
# reps12211184444444

Matrix representation of C23.5C42 in GL4(𝔽17) generated by

0400
13000
16149
160413
,
1000
0100
00160
413016
,
16000
01600
00160
00016
,
0010
413115
0100
0004
,
0200
15000
89213
80215
G:=sub<GL(4,GF(17))| [0,13,16,16,4,0,1,0,0,0,4,4,0,0,9,13],[1,0,0,4,0,1,0,13,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,4,0,0,0,13,1,0,1,1,0,0,0,15,0,4],[0,15,8,8,2,0,9,0,0,0,2,2,0,0,13,15] >;

C23.5C42 in GAP, Magma, Sage, TeX

C_2^3._5C_4^2
% in TeX

G:=Group("C2^3.5C4^2");
// GroupNames label

G:=SmallGroup(128,489);
// by ID

G=gap.SmallGroup(128,489);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,1018,172,4037]);
// Polycyclic

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

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