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G = C42.14C23order 128 = 27

14th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.14C23, (C2×D4)⋊28D4, C4⋊C85C22, D4.7(C2×D4), C4⋊D819C2, C4⋊C4.335D4, C2.8(D4○D8), (C4×D4)⋊3C22, (C22×D8)⋊9C2, (C2×D8)⋊17C22, C41D45C22, (C2×C8).16C23, C4.72(C22×D4), D4.2D414C2, C4.33(C4⋊D4), C4⋊C4.382C23, (C2×C4).245C24, C22⋊C4.136D4, (C2×SD16)⋊9C22, (C2×D4).52C23, C23.442(C2×D4), C4.4D45C22, (C2×Q8).39C23, D4⋊C464C22, C22.11C249C2, C22.29C248C2, Q8⋊C468C22, C23.24D48C2, C23.37D48C2, C22.80(C4⋊D4), (C22×C8).139C22, (C22×C4).975C23, C42.6C223C2, C22.505(C22×D4), (C22×D4).340C22, (C2×M4(2)).52C22, C42⋊C2.100C22, (C2×C8⋊C22)⋊16C2, C4.155(C2×C4○D4), (C2×C4).465(C2×D4), C2.63(C2×C4⋊D4), (C2×C4).276(C4○D4), (C2×C4○D4).117C22, SmallGroup(128,1773)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.14C23
C1C2C4C2×C4C22×C4C22×D4C22.11C24 — C42.14C23
C1C2C2×C4 — C42.14C23
C1C22C42⋊C2 — C42.14C23
C1C2C2C2×C4 — C42.14C23

Generators and relations for C42.14C23
 G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=a2b2, ab=ba, cac=a-1b2, ad=da, eae-1=ab2, cbc=dbd=b-1, be=eb, dcd=bc, ece-1=a2b2c, de=ed >

Subgroups: 620 in 263 conjugacy classes, 100 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×9], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×27], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×9], D4 [×4], D4 [×20], Q8 [×2], C23, C23 [×15], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×12], SD16 [×4], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×8], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×4], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×4], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4 [×2], C22×C8, C2×M4(2), C2×D8 [×6], C2×D8 [×4], C2×SD16 [×2], C8⋊C22 [×4], C22×D4 [×2], C2×C4○D4, C23.24D4, C23.37D4, C42.6C22, C4⋊D8 [×4], D4.2D4 [×4], C22.11C24, C22.29C24, C22×D8, C2×C8⋊C22, C42.14C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D4○D8 [×2], C42.14C23

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L4A4B4C4D4E···4L4M8A8B8C8D8E8F
order122222222222244444···44888888
size111122444488822224···48444488

32 irreducible representations

dim111111111122224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4D4C4○D4D4○D8
kernelC42.14C23C23.24D4C23.37D4C42.6C22C4⋊D8D4.2D4C22.11C24C22.29C24C22×D8C2×C8⋊C22C22⋊C4C4⋊C4C2×D4C2×C4C2
# reps111144111122444

Matrix representation of C42.14C23 in GL6(𝔽17)

0160000
100000
0040150
000011
0000130
0001640
,
100000
010000
001200
00161600
0004016
00131310
,
13110000
1140000
006600
00141100
00012314
00551414
,
1600000
0160000
001200
0001600
0004016
00013160
,
0160000
100000
0040150
000011
00160130
001140

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,15,1,13,4,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,13,0,0,2,16,4,13,0,0,0,0,0,1,0,0,0,0,16,0],[13,11,0,0,0,0,11,4,0,0,0,0,0,0,6,14,0,5,0,0,6,11,12,5,0,0,0,0,3,14,0,0,0,0,14,14],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,2,16,4,13,0,0,0,0,0,16,0,0,0,0,16,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,0,16,1,0,0,0,0,0,1,0,0,15,1,13,4,0,0,0,1,0,0] >;

C42.14C23 in GAP, Magma, Sage, TeX

C_4^2._{14}C_2^3
% in TeX

G:=Group("C4^2.14C2^3");
// GroupNames label

G:=SmallGroup(128,1773);
// by ID

G=gap.SmallGroup(128,1773);
# by ID

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

G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=1,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=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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